Paper Title: Experimental continuation in nonlinear dynamics: recent advances and future challenges.



Experimental continuation in nonlinear dynamics: recent advances and future challenges]Experimental continuation in nonlinear dynamics: recent advances and future challenges













Ghislain Razeg.raze@uliege.be







*Aerospace and Mechanical Engineering Department, University of Liège, Allée de la Découverte, 9, Liège, 4000, Belgium








Experimental continuation encompasses a set of methods that combine control and continuation to obtain the full bifurcation diagram of a nonlinear system experimentally, including responses that would be unstable in the system without feedback control. Such control-based methods thus allow the experimenter to directly and exhaustively explore the dynamics of the system without the need for a good mathematical model. The objective of this paper is twofold, namely (i) to review and present the state-of-the-art methods in a unified manner and (ii) to introduce a novel experimental derivative-free arclength continuation procedure, termed arclength control-based continuation. These methods are also demonstrated and compared on an electronic Duffing oscillator and a clamped thin plate featuring geometrical nonlinearity. Finally, the current state of the art is reflected upon, and the challenges lying ahead for this growing field are discussed.





















[
    Gaëtan Kerschen
    August 5, 2024
===================





§ INTRODUCTION


The theory of nonlinear dynamical systems was developed based on Poincaré's seminal work. It witnessed extraordinary advances during the 20th century thanks to Arnold,  Holmes, Lorenz, Ruelle and Smale to name a few. A couple of decades ago, the theory spread across engineering thanks to a series of reference monographs thoroughly discussing the different phenomena and attractors nonlinear systems can exhibit. Nayfeh and Mook <cit.> applied perturbation methods to study nonlinear phenomena in single- and multi-degree-of-freedom systems. Guckenheimer and Holmes <cit.> adopted a different, geometrical viewpoint, appearing as an ideal companion to the perturbation approach. Kuznetsov <cit.> published a complete treatise on bifurcations.

Since the 1970s, impressive progress was also realized in computational nonlinear dynamics with the development of  nonlinear finite element methods <cit.> and of numerical continuation (NC) <cit.>. Numerical continuation amounts to calculating a bifurcation diagram, a key concept in nonlinear dynamics <cit.> representing the different branches of responses of a nonlinear system together with their stability and bifurcations as one or more parameters are varied. Today, the calculation of bifurcation diagrams of complex numerical models is within reach <cit.>. Noteworthy are also the advances in the area of nonlinear model reduction <cit.>. We also note that there now exists the possibility to perform direct, data-driven reduction onto invariant manifolds <cit.>.

Although not all challenges have been overcome yet, the theoretical understanding of nonlinear dynamical phenomena and their prediction using numerical models have reached a high level of maturity. Conversely, experimental nonlinear dynamics <cit.> received comparatively far less attention and is the main thrust of this paper. In this context, nonlinear system identification (NSI) which focuses on the development of mathematical models directly from experimental data is the most popular approach to date in the literature. There exists a wide variety of techniques see, e.g., <cit.>. Once a good model of the system has been obtained thanks to NSI, numerical continuation techniques can then be exploited to calculate a bifurcation diagram, as carried out, e.g., in <cit.>.  

Despite the recent advances in the field, NSI remains a particularly difficult exercise and imposes a number of challenges and constraints to the experimenter. An elegant, model-less alternative to NSI is experimental continuation
which leverages feedback control for the identification of a bifurcation diagram. This novel experimental strategy represents a true paradigm shift in nonlinear dynamics. There are two key differences with numerical continuation, namely (i) bifurcation diagrams have to be obtained in a model-less manner and (ii) feedback control must be applied to stabilize unstable periodic orbits which otherwise cannot be measured experimentally. The first experimental continuation method, control-based continuation (CBC), pioneered by Sieber et al <cit.>, can be seen as an experimental equivalent to pseudo-arclength continuation. The fact that CBC requires to calculate derivatives using finite differences from experimental data, a computationally-intensive and highly sensitive-to-noise process, motivated researchers to develop methods which do not rely on experimental derivatives. The three most effective derivative-free experimental continuation methods to date, namely the simplified CBC (SCBC) method <cit.>, phase-locked loops (PLLs) <cit.> and response-controlled testing (RCT) <cit.>, resort to natural parameter continuation, which simply increases the bifurcation parameter using the previous solution as an initial guess for the next solution. Thanks to these advances, the frequency responses and backbone curves of nonlinear systems around their primary resonances can now be identified in a robust and effective manner. 

The objective of this feature article is to offer the first extensive review of the developments in the EC area since its invention, to present the three derivative-free methods in an unified manner and to introduce a novel experimental continuation method, termed arclength control-based continuation (ACBC) <cit.>, the first derivative-free arclength continuation scheme.

The paper is organized as follows. Bifurcation diagrams and the means to compute them are introduced in Section <ref>. Section <ref> provides a brief introduction to basic concepts in feedback control. Section <ref> presents the core ideas of EC and details its first forms. Section <ref> then reviews the most recent methods and presents them under a unified framework. The novel ACBC approach is presented in Section <ref>. All the methods are compared experimentally with an electronic Duffing oscillator and a clamped plate featuring geometrical nonlinearity in Sections <ref> and <ref>, respectively. Finally, the conclusions of the present study and the challenges lying ahead of us are summarized in Section <ref>.



§ BIFURCATION DIAGRAMS: A CENTRAL TOOL IN NONLINEAR DYNAMICS



Nonlinear dynamical systems usually behave in a complex manner, requiring adequate tools to study their behavior. Bifurcation diagrams are ubiquitously used in nonlinear dynamics for that purpose. These diagrams typically present how one quantity characterizing the response of the system under study is influenced by a parameter, often called a bifurcation parameter. In this work, we focus on the periodic solutions of nonlinear systems and their stability (but fixed points can be seen as a special subcase).

In this section, the main features of analytical and numerical approaches to draw out bifurcation diagrams are summarized, and used to illustrate the challenges faced in experimental nonlinear dynamics.



 §.§ Periodic solutions


 We consider a generic, non-autonomous system governed by the set of first-order ordinary differential equations (ODEs)

    𝐱̇(t) = 𝐟(𝐱(t),t),

where 𝐱 is a vector gathering the states of the system under consideration,  t is the time, and 𝐟 is a state evolution law. We note that this framework encompasses the case of a set of second-order ODEs generally used in vibration problems

    𝐌𝐪̈(t) + 𝐂𝐪̇(t) + 𝐊𝐪(t) + 𝐟_nl(𝐪(t),𝐪̇(t)) = 𝐟_ext(t),

where 𝐌, 𝐂 and 𝐊 are linear structural mass, damping and stiffness matrices, respectively, 𝐟_nl is the vector of nonlinear forces and 𝐟_ext is the vector of external forces. Indeed, this set of second-order ODEs can be recast into a set of first-order ODEs (provided that the mass matrix is invertible) using

    [                                                   𝐱̇                                       = [ 𝐪̇; 𝐪̈ ] =           [      0      𝐈; -𝐌^-1𝐊 -𝐌^-1𝐂 ][  𝐪; 𝐪̇ ];                                                                                                           + [                      0; 𝐌^-1(𝐟_ext-𝐟_nl(𝐪,𝐪̇)) ];                                            = 𝐟(𝐱,t).                                                      ]


When the right-hand side of Equation (<ref>) is periodic in time, periodic solutions 𝐱 generally exist, satisfying

    𝐱(t+T) = 𝐱(t),     ∀ t ∈ℝ

where T is a period of the solution, being related to its angular frequency ω by ω T = 2π.



 §.§ Analytical solutions


    Equation (<ref>) with given initial conditions has a unique solution provided that 𝐟 satisfies mild smoothness conditions given by Picard's existence theorem. In some cases, analytical insight into these solutions can be gained.

    

  §.§.§ Exact solutions


        Due to the very nature of nonlinear systems, exact analytical solutions seldom exist. A few special or nongeneric cases can be treated, such as the unforced, undamped pendulum or Duffing oscillator with the use of Jacobi elliptic functions <cit.>.
        
        For illustration, we consider the free response q of an undamped Duffing oscillator, a typical academic example qualitatively representative of geometrically nonlinear structures, which is governed by the following ODE:
        
    mq̈(t) + kq(t) + k_3q^3(t) = 0,

        where m, k and k_3 are the mass, linear stiffness and cubic stiffness coefficients, respectively, of the oscillator and an overdot denotes a derivation with respect to time t. Dividing Equation (<ref>) by m,
        
    q̈(t) + ω_0^2 q(t) + α_3 q^3(t) = 0

        with ω_0^2 = k/m and α_3 = k_3/m. The exact solution of Equation (<ref>) is given by
        
    q(t) = A sn (Ω t + b |κ),

        where A is the amplitude of the free oscillations, sn is the elliptic sine function, b is a phase-like variable to be adjusted to the initial conditions (together with A), Ω is a frequency-like variable and κ is the elliptic parameter <cit.>. These two latter parameters are related to A by
        
    Ω = √(ω^2_0 + α_3 A^22),    κ = -α_3 A^22Ω^2.


        

    

  §.§.§ Perturbation methods


        Perturbation methods constitute a means to obtain approximate analytical solutions to nonlinear problems. They rely on a small parameter, usually denoted ϵ. In most cases, the external forcing, response amplitude, nonlinear coefficients and/or damping coefficients are assumed to scale with ϵ. When ϵ =0, a (generally known) solution forms the basis around which a power series expansion in terms of ϵ is performed.

        Perturbation methods are a powerful tool to study nonlinear systems analytically. The most famous ones are the method of multiple scales <cit.> and the averaging method <cit.>.
        These methods can provide insight into the system's behavior, but are inherently restricted by the limitations associated with power series expansions, and generally break down when one or several assumed small parameters become too large. 



 §.§ Numerical procedures


If analytical solutions are unavailable, intractable or inaccurate, periodic solutions for a fixed ω can be found by numerically integrating Equation (<ref>) until the transients die out. This approach is generally inefficient, and more sophisticated approaches can be used, such as the shooting <cit.>, harmonic balance <cit.> and orthogonal collocation <cit.> methods. These methods cast the ODEs into a set of algebraic equations that can be solved by well-known numerical routines such as the Newton-Raphson method. This usually yields accurate results. 

To gain a broader understanding of the system's behavior, a bifurcation diagram can be built by varying a bifurcation parameter. To find the family of periodic solutions that form a complete bifurcation diagram, numerical continuation can be used, as will be presented next.





 §.§ Numerical continuation



    Numerical continuation (also known as branch tracing or path-following method) denotes a class of numerical methods used to solve a system of algebraic equations with one (or more) free parameter. The solutions of these equations generally form a set whose dimension is equal to the number of bifurcation parameters. Since the most common case considers a single parameter, this set is generically a curve. We treat this case without loss of generality, because the case with multiple bifurcation parameters can be treated, e.g., by sampling the values of all but one parameter, and using a one-parameter continuation for all samples, thereby effectively "slicing" the solution set.
    
    Numerical continuation is an incremental procedure. The main idea behind it is to build a branch of solutions point by point (or piece by piece), using a constructive approach where the information at the previous point (or piece) of the curve is known to be well estimated and is used to find a suitable initial guess for the next one. This defines the prediction step. This initial guess is then refined to ensure that it lies on the sought curve within a prescribed tolerance, which defines the correction step. Figure <ref> schematically depicts the main ideas behind this predictor-corrector scheme, where x is one of the unknowns, and λ is a bifurcation parameter.

    

    Numerical continuation is a mature tool that is now treated in a variety of reference books and exploited in a large number of bifurcation analysis software, such as AUTO <cit.>, MATCONT <cit.>, COCO <cit.> and MANLAB <cit.>. In this work, we will focus on the two most popular variants of numerical continuation, namely natural parameter and arclength continuations. There also are numerous books discussing this topic, such as <cit.>.

    

  §.§.§ Natural parameter continuation

    

        Natural parameter continuation (also called sequential continuation) is the simplest form of continuation. It uses the bifurcation parameter of the problem to parametrize the solution curve. The implicit function theorem then gives conditions under which this choice is guaranteed to be sensible, i.e., when the solution curve is indeed (locally) parametrizable by the bifurcation parameter. When the solution is known for a given value of the parameter, the prediction step simply consists in increasing the parameter value, using the previous solution as initial guess for the next point. The correction phase generally consists in using an iterative solver such as the Newton-Raphson method. Figure <ref> represents a general schematics of a natural parameter continuation method.

        

        Natural parameter continuation fails at points where the implicit function theorem does not hold when one parametrizes a curve with the continuation parameter. Two notable cases are limit points and branch points, represented in Figures <ref> and <ref>, respectively. In the presence of the former, natural parameter continuation fails most of the time because there is no nearby solution at the imposed parameter value, as shown in Figure <ref>. With branch points, it is generally hard to predict and control which path the method will follow close to the bifurcation; this depends on the step length and on the problem at hand (cf. Figure  <ref>). To address these issues, more elaborate continuation approaches were developed.

         
    
    

  §.§.§ Arclength continuation


        The first ideas behind arclength continuation can be traced back to the seminal work of Haselgrove in the early sixties <cit.>, and were extensively developed and popularized by Keller <cit.>. It was (seemingly independently) popularized nearly twenty years later in the field of static mechanics with the famous works of Riks <cit.> and Crisfield <cit.>, who respectively proposed what is now known as the pseudo-arclength and arclength continuation procedures. Adaptations to dynamics came ten years later using the harmonic balance <cit.> and shooting <cit.> methods. The underlying idea of these methods is to parametrize the curve with an (approximation of its) arclength.
        
        These methods both use a more refined prediction step with either a tangent or secant predictor. The correction is then performed either on a hyperplane whose normal is the prediction vector (Figure <ref>), or on a hypersphere centered around the previous point and whose radius is equal to the prediction step length (Figure <ref>). Mathematically, λ is let free, and the correction locus adds another constraint to the problem to make it well-posed. Clearly, these methods offer better robustness than natural parameter continuation when the curve folds. In addition, the tangent predictor generally makes the method more likely to stay on the same branch when a branching behavior is encountered. These are some of the reasons for their current popularity and ubiquitous presence in numerical continuation packages.

         



 §.§ Stability



Equipped with the possibility to draw bifurcation diagrams, we now have to wonder whether the obtained family of solutions will be observable in the nonlinear system under study. By contrast with linear time-invariant (LTI) systems for which solutions are either all stable or all unstable, nonlinear systems can possess solutions of different stability. In addition, while unstable solutions of LTI systems grow unbounded with time, trajectories repelled by a nearby unstable nonlinear solution have the additional possibility to converge to other stable solutions.

In an experimental context, the unstable solutions of a system are impossible to observe in steady-state conditions without additional control. As for numerical aspects, unstable solutions can be computed, but an important question is to know whether this solution is stable or not. To answer this question, we look at the evolution of infinitesimally small perturbations around the equilibrium, which gives us results about the asymptotic stability of a solution. For more elaborate discussions on stability, we refer to, e.g., <cit.>.

We now assume that we know a periodic solution of Equation (<ref>) (obtained analytically or numerically) and denote it by 𝐱_*. A small perturbation 𝐬 is added to the equilibrium 𝐱_*, i.e.,

    𝐱(t) = 𝐱_*(t) + 𝐬(t),      |𝐬(t)| << |𝐱_*(t)|

Inserting this perturbed solution into Equation (<ref>) and expanding the result in Taylor series around 𝐬(t) = 0, we obtain

    [                𝐱̇_*(t) + 𝐬̇(t) = 𝐟(𝐱_*,t) + .∂∂𝐱𝐟(𝐱,t)|_𝐱=𝐱_*𝐬(t);                                                    + O(|𝐬(t)|^2). ]

Neglecting terms of orders higher than linear in |𝐬| and accounting for the fact that 𝐱_* is a solution of Equation (<ref>), one gets the first-order variational problem 

    𝐬̇(t) = .∂∂𝐱𝐟(𝐱,t)|_𝐱=𝐱_*(t)𝐬(t).

This is a linear, time-varying equation featuring the Jacobian matrix ∂𝐟/∂𝐱 and governing the evolution of the (small) perturbations. The solution of such an equation cannot be known in closed-form in general, and requires numerical integration. Nevertheless, the time-varying part of this equation is periodic, and this periodicity entails special consequences which are described by the Floquet theory <cit.>. The variational equation can be solved (using numerical time integration) with its principal fundamental matrix Φ(t) determined by

    Φ̇(t) = .∂∂𝐱𝐟(𝐱,t)|_𝐱=𝐱_*(t)Φ(t),   Φ(0) = 𝐈.

Due to the linearity and periodic nature of the first-order variational problem, it can be shown that the growth or decay of any small perturbation is related to the eigenvalues of this principal fundamental matrix evaluated at t=T (also known as the monodromy matrix) Φ(T), called the Floquet (or characteristic) multipliers. Specifically, if any Floquet multiplier is greater than one in modulus, the solution is asymptotically unstable, and it is stable otherwise.

The monodromy matrix is thus a determining factor for stability. Unfortunately, there generally exists no explicit expression of this matrix from ∂𝐟/∂𝐱, but the key point to remember is that this Jacobian matrix plays a pivotal role for the asymptotic stability of a solution. 



 §.§ Illustration with a Duffing oscillator


For illustration, we will consider the damped, forced Duffing oscillator. This example is well-known for the nonlinear dynamicist. However, it can exhibit rather complex dynamics associated with a wide variety of secondary resonances <cit.>.

The response q of the considered oscillator to an external excitation f is governed by a generalization of Equation (<ref>):

    mq̈(t) + cq̇(t) + kq(t) + k_2 q^2(t) + k_3q^3(t) = f(t),

where c is the damping coefficient of the oscillator, and we added a quadratic stiffness force with a coefficient k_2 that can be used to represent asymmetries in the system. When k_2≠ 0, the oscillator characterized by Equation (<ref>) is often called the Helmholtz-Duffing oscillator. Upon using the normalizations t̅ = ω_0 t and q̅=√(k_3/k)q, Equation (<ref>) can be put into the dimensionless form 

    q̅”(t̅) + 2ζ_0 q̅'(t̅) + q̅(t̅) + β_2 q̅^2(t) + q̅^3(t̅) = f̅(t̅),

with ζ_0=c/(2√(km)), β_2 = k_2/√(kk_3), f̅ = √(k_3/k^3)f, and a prime denotes derivation with respect to the dimensionless time t̅. For the examples shown hereafter, we chose ζ_0 = 0.05 and β_2=0 (hence we consider a Duffing oscillator), unless stated otherwise.



Figure <ref> features a bifurcation diagram obtained by coupling a harmonic balance procedure with numerical continuation <cit.>. This diagram represents the response amplitude of the Duffing oscillator under a harmonic forcing of amplitude f̅(t̅)=sin(ω̅t̅) (with a dimensionless frequency such that ω=ω_0ω̅), taking ω̅ as bifurcation parameter. This bifurcation diagram is also often called the nonlinear frequency response (NFR). In addition to the well-known primary (1:1) resonance that bends rightward due to the hardening nonlinearity of the oscillator, a series of peaks appear throughout the bifurcation diagram and are associated to non-primary resonances <cit.>. Specifically, odd superharmonic (m:1, m∈ℕ_0) resonances appear on the main branch of the NFR at frequencies lower than that of the primary resonance, whereas the even superharmonic resonances bifurcate from it through symmetry-breaking bifurcations.



Figure <ref> represents the NFR of the Duffing oscillator under a higher-amplitude harmonic forcing, namely f̅(t̅)=3sin(ω̅t̅). In addition to the primary and superharmonic resonances, subharmonic (1:l, l∈ℕ_0) and ultrasubharmonic (m:l, m,l∈ℕ_0, (m,l)=1) resonances appear, and they are isolated from the main NFR. We note that these isolated resonances were not obtained directly from numerical continuation of the main branch; stochastic procedures were used to find a set of initial conditions resulting in a subharmonic or ultrasubharmonic response. Once a suitable initial point was found, numerical continuation was used to trace out the complete isola. Thus, even in this simple oscillator, complex nonlinear features can appear.





 §.§ Consequences for open-loop testing





We now illustrate how the complexity featured by nonlinear systems as deceptively simple as the Duffing oscillator impacts the experimental results obtained while testing a system.

During a test, a drive signal is generated by the experimenter and fed to the system. In vibration testing, this drive signal is fed to an actuator (generally through an amplifier) whose purpose is to excite the structure under test, either via a force or via base excitation. In both cases, the drive is seldom proportional to the forcing signal because the amplifier and the actuator have their dynamics <cit.>, hence we distinguish the drive from the actual forcing. For a harmonic excitation, the two parameters that the user can control are the drive amplitude and frequency. The response of a nonlinear system is generally characterized for multiple values of these parameters. For instance, the drive amplitude may be fixed and the frequency varied. This variation can be continuous or discrete, representing swept-sine (SWS) or stepped-sine (STS) excitations, respectively. The set of points obtained from different frequencies can be interpreted as the NFR. Alternatively, one can fix the frequency and vary the drive amplitude and obtain a so-called S-curve <cit.>.



We apply a SWS with logarithmically varying frequency to the Duffing oscillator, as shown in Figure <ref>. As the frequency increases, a smooth change in amplitude is observed, except in the vicinity of resonances. The most noticeable feature is the occurrence of 
a sudden jump-down at a frequency ω̅≈ 7.2. Conversely, by progressively decreasing the frequency, a jump-up occurs around ω̅≈ 2.1. The jump-up and jump-down phenomena occur where the NFR folds, as seen in Figure <ref>. Indeed, SWS can be seen as an experimental equivalent to natural parameter continuation. 

Closing up on the low-frequency region, we can observe most odd superharmonic resonances, whereas the even ones are sometimes partially missed due to the transient nature of the excitation. Sub- and ultrasubharmonic resonances are completely missed by the (simulated) testing procedure because of their isolated nature. 



























 §.§ Challenges for experimental nonlinear dynamics


To summarize this section, we have seen that a combination of theoretical and numerical techniques can be used to obtain bifurcation diagrams that provide a great deal of information about the response of nonlinear systems. Thanks to the maturity of these approaches, and in particular the numerical continuation procedure, stable and unstable parts of a bifurcation diagram can be obtained.

As for experiments, the classical open-loop testing procedures cannot offer the same picture. There are in particular specific challenges associated with nonlinear dynamics:

    
  * Unstable solutions are by nature impossible to follow with open-loop approaches.
    
  * Jumps are thus susceptible to occur when a stability change is encountered through bifurcations.
    
  * Several solutions are missed during a test, resulting in an incomplete bifurcation diagram. 
    
  * Isolated solutions are challenging to reach with classical continuation approaches. In an open-loop setting, they can be obtained in some cases by using a stochastic approach <cit.>, or leveraging jumps <cit.>. 


A novel class of testing methods, herein called control-based methods, has emerged slightly before the beginning of the last decade <cit.>. The salient capabilities of these methods are that, through suitable automatic adjustments of the drive signal, they (i) allow the experimenter to prescribe given parameters of the system to extract complete bifurcation diagrams and (ii) can stabilize the equilibria of the system under test. These methods are thus paradigm-shifting for nonlinear vibration testing and allow the experimenter to perform experimental continuation.




§ FEEDBACK CONTROL: A BRIEF INTRODUCTION



    Before delving into control-based methods, we review a few essential aspects about feedback control, which is a key enabler for experimental continuation.

    

 §.§ Linear feedback control


   Humans have always wanted to control their environment, i.e., to stir the systems that surround them toward desired states of operation. A natural way to perform this is to use feedback control, wherein the output of a system to be controlled (usually called plant) is sensed and used to adapt its input, eventually (and hopefully) reaching a desired output through the action of that input.  When feedback is devoid of human interaction, one speaks of automatic control. We shall refer a system with and without feedback to as closed- and open-loop systems, respectively.

   Feedback is naturally used by most animals, but it can be surprising to realize that this concept was purposefully exploited rather late and formalized even later in human history <cit.>. Starting from the twentieth century, the study of feedback control has known a great evolution, mostly motivated by technological needs and advances, leading to the ubiquitous presence of controlled systems around us today. For a historical perspective on this subject, we refer, e.g., to <cit.>.

    Figure <ref> presents the typical architecture of a linear control system. The plant, excited by the drive (or control signal) u and unknown disturbances d, is monitored through its response y (typically an element of 𝐱 or a linear combination of them). This response is compared to a desired reference y_* (in the sequel, a subscript with a star indicates a target quantity) and fed to a controller, that automatically adapts the drive u. The controller can be designed to force y to follow the reference y_* as closely as possible, to reject the effect of the external disturbance d, or to do both these things at the same time.
    
    

    In this linear time-invariant setting, the plant response can be expressed as the superposition of the effect of the drive u and the disturbance d. We use a Laplace transform formalism to express this as
    
    Y(p) = H_yu(p) U(p) + H_yd(p) D(p),

    where Y, U and D are the Laplace transforms of y, u and d, respectively, H_yu and H_yd are the transfer functions of the plant, and p is Laplace's variable. The drive is the controller output given by
    
    U(p) = C(p)  (Y_*(p) - Y(p)),

    where C is the controller's transfer function, and Y_* is the Laplace transform of y_*. Injecting Equation (<ref>) into Equation (<ref>) yields the closed-loop relation
    
    
    [                (1+H_yu(p)C(p)) Y(p); = H_yu(p)C(p) Y_*(p) + H_yd(p) D(p) ]

    In general, the controller transfer function C is chosen to be large compared to the other transfer functions. Hence,
    
    [     lim_|C(p)|→∞ H_yu(p)C(p) Y(p); = lim_|C(p)|→∞ H_yu(p)C(p) Y_*(p) ]

    Hence, Y(p)≈ Y_*(p), and y(t) ≈ y_*(t) if the controller has a large enough gain. It is thus theoretically possible to achieve perfect tracking between y_* and y. In practice, finite gains are chosen for several reasons. First, closed-loop stability may require finite gains (and in general, unknown, unmodeled dynamics will impose a finite gain). Second, sensor noise can pollute the sensed signal y and transmit fully to (and even be amplified by) the closed-loop system. Third, a large controller gain can require a large drive u, exceeding the actuator capabilities. There may also be other practical reasons depending on the application at hand.

    

 §.§ Nonlinear control

    

        Nonlinear control encompasses feedback control of systems where either the plant or the controller is nonlinear, or both. Naturally, its development was closely related to linear feedback control and nonlinear dynamics theories. The inherent difficulties associated with nonlinear dynamical systems unsurprisingly arise (and are arguably even more problematic) in nonlinear control systems, making them a contemporary active research field. For historical perspectives, we refer to <cit.>. There exists a variety of reference books on nonlinear control <cit.>, with one specifically aimed toward nonlinear vibration control <cit.>.

        One of the challenges associated with nonlinear (and linear) control (except for adaptive and optimal nonlinear control, to a certain extent) stems from the need for a representative model of the plant for the controller design. Although substantial progress has been made in NSI, there still remains inherent challenges associated with this endeavor <cit.>, and the complexity associated with modeling a nonlinear system generally reverberates into nonlinear control. This motivated the use of model-less control approaches, which emerged in the context of chaos control.

        In the early nineties, a seminal article written by Ott, Grebogi and Yorke proposed a new model-less method to control the behavior of a chaotic system <cit.>. This OGY algorithm sparked a lot of interest in the late twentieth and early twenty-first centuries; reviews on chaos control methods can be found in <cit.>. A notable contribution by Pyragas later proposed a simplified continuous delayed feedback law to achieve the same goal <cit.>. By feeding back the difference between an observable and its time-delayed self, this control law was also shown to be able to stabilize the targeted periodic orbit. We note that choosing this delay to be equal to the period of the target makes the control non-invasive on that orbit, an important notion that we shall come back to and define later. Chaos control can be considered as the ancestor of the first control-based methods discussed in this work.





§ EXPERIMENTAL CONTINUATION: THE PIONEERS AND FIRST EFFORTS




    Control-based methods combine control and continuation to trace out a complete bifurcation diagram of a nonlinear system. This idea was first proposed for chaotic systems (or chaotic at least in some region of the investigated parameter range) by combining the OGY algorithm (or its variants) to a natural parameter continuation and was coined under the term "tracking" <cit.>. Methods combining continuous feedback and continuation to trace out bifurcation diagrams made up by the collection of periodic orbits of nonchaotic systems were introduced following the seminal works of Sieber and Krauskopf <cit.> and are the focal point of the present work.




 §.§ Control-based approaches: generalities


    

  §.§.§ Why are closed-loop tests necessary?


        Besides the academic thrill of tracing out complete experimental bifurcation diagrams, one may legitimately wonder what interest there is in attempting to follow unstable solutions in a general setting, given that they will be practically impossible to observe in the bare system. There are multiple reasons for this, and some of them are given below, the most important arguably being the first one.
        
        
            
  * Continuation: unstable branches are connecting various stable branches, which might not (or not fully) be reached with open-loop system testing methods. For instance, in a Helmholtz-Duffing oscillator, SWS tests may systematically miss the highest amplitude of the NFR, as illustrated in Figure <ref>. There can also be stable responses that can only be accessed through unstable branches (so-called islands of stability <cit.>).
            
  * Global dynamics: unstable equilibria of saddle type carry important information about the global dynamics. They lie on the separatrices of the system and provide valuable information about the basins of attraction of the system, and especially their boundaries. Loosely speaking, if a stable and an unstable equilibria are "close", one would expect the basin of attraction of the former to be small. On a related note, due to the hyperbolic nature of the dynamics around such sets, saddles also play an important role in transient dynamics.
            
  * Qualitative changes through bifurcations: branching on a stable path through bifurcations might lead the system to a completely different type of behavior, whereas following an unstable path may yield a system response that is similar to its pre-bifurcation behavior, easing the analysis and interpretation of the results.
            
  * Data diversity: in an NSI or model updating framework, measuring unstable solutions offers more data points for building an accurate model of the system under test.
            
  * System interconnections: finally, when one considers nonlinear substructuring or nonlinear control, due to the feedback action enacted upon the system by its surroundings, unstable equilibria might become stable, or play a prominent role in the appearance of new stable equilibria, and thus become obviously practically relevant.
        

        

    



    

        
    
        
    
        

    

  §.§.§ Common architecture and non-invasiveness

    

        Figure <ref> presents the general architecture of a control-based testing method (note the similarity with Figure <ref>). As in open-loop testing, the plant (e.g., in the vibration testing case, a structure under test, and a set of actuators and sensors) is excited by a drive signal u. The response of the plant is then analyzed through any of its observables (e.g., in the vibration testing case, a displacement q or a force f) to obtain a representative signal y which is fed back to a controller. This controller compares this representative signal to a reference y_*, and adapts the drive signal to enforce as much as possible an equality between the reference and actual signals.
        
        The target (or a set of targets) z_* is predetermined by the experimenter depending on the goals of the tests to be performed, and the selected experimental continuation method. In some methods, the continuation procedure will adapt y_* through a continuation procedure. In other methods (discussed in Section <ref>), one simply sets y_*=z_*, and the continuation block is trivial.
        
        

        

        A feedback controller can and generally will completely modify the dynamics of the system. Since the goal of experimental continuation is to test the plant, such a modification should be avoided. A requirement put on the control (in addition to its two main goals) is thus that it should be non-invasive, i.e., the equilibria of the plant in the closed-loop system should be identical to those of the open-loop system. A sufficient condition to enforce this is to subject the closed-loop system to the same excitation as in open-loop conditions. Hence, the total input to the plant in closed-loop can be seen as the superposition of an open-loop excitation signal used to investigate the dynamics of the plant, and a stabilizing signal which vanishes once the system reaches the targeted open-loop equilibrium. 
        

        





  §.§.§ Stabilizing effect



As outlined in Section <ref> and particularly in Figure <ref>, the main difference between an open-loop test and a test with a control-based method is the presence of a feedback signal. We thus reconsider the results of Section <ref> with this difference, staying at an abstract level for generality. A way to account for this feedback is to modify Equation (<ref>) as

    𝐱̇(t) = 𝐟̃(𝐱(t),𝐮(t;𝐱(t)),t),

where the right-hand side 𝐟̃ now depends on 𝐮, the control signal. Among other possibilities, 𝐮 may either be determined from a prescribed continuous-time evolution

    𝐮̇(t;𝐱(t)) = 𝐠_1(𝐱(t),𝐮(t;𝐱(t)),t),

be updated by a discrete-time law, or be a direct function of the states and time

    𝐮(t;𝐱(t)) = 𝐠_2(𝐱(t),t).


In our setting, the control is non-invasive if, at an equilibrium of the open-loop system 𝐱_*(t),

    𝐮(t;𝐱_*(t)) = 0

and

    𝐟̃(𝐱(t),0,t) = 𝐟(𝐱(t),t),     ∀𝐱, t.

If these two conditions hold, we thus have

    𝐟̃(𝐱_*(t),𝐮(t;𝐱_*(t)),t) =
        𝐟(𝐱_*(t),t) = 𝐱̇_*(t),

i.e., 𝐱_* is an equilibrium of both the open- and closed-loop systems. We note that in experiments, being exactly on the equilibrium 𝐱_* is not possible due to, e.g., noise. Equation (<ref>) can therefore be seen as purely theoretical and can be replaced by the condition that the control signal should be smaller and smaller as the system approaches its equilibrium state. Instead of Equation (<ref>), non-invasiveness can therefore be satisfied if the control is required to satisfy the Lipschitz continuity condition 

    || 𝐮(t,𝐱(t)) || ≤ L || 𝐱(t)-𝐱_*(t)||

around 𝐱_*(t), were L is a finite positive constant and ||·|| is the Euclidean norm.

Following the same developments as in Section <ref>, we may study the stability of this equilibrium by introducing small perturbations, and linearizing the dynamics to reveal the Jacobian of the controlled system 

    .d𝐟̃d𝐱|_𝐱=𝐱_*(t) =. (∂𝐟̃∂𝐱+ ∂𝐟̃∂𝐮d𝐮d𝐱)|_𝐱=𝐱_*(t)

(where the total derivative is taken with fixed t), and, using Equations (<ref>) and (<ref>), we find

    .d𝐟̃d𝐱|_𝐱=𝐱_*(t) = . (∂𝐟∂𝐱+ ∂𝐟̃∂𝐮d𝐮d𝐱)|_𝐱=𝐱_*(t).

We thus observe that the Jacobian of the original system is modified by the effect of the control signal 𝐮. Consequently, according to the discussion in Section <ref>, the stability of 𝐱_* may be altered by the action of the controller. Unfortunately, going beyond this discussion while staying in a general framework may prove complicated, first because of the non-trivial relation between the Jacobian and the Floquet multipliers, but also because of the different nature of the control signals used by the different approaches.

We may nevertheless illustrate this section with a specific example. Figure <ref> presents the counterpart of Figure <ref> when the Duffing oscillator is stabilized with a differential controller with a fixed reference signal q̅_* correctly ensuring the non-invasiveness condition, i.e., f̅ = k̅_d (q̅'_* - q̅'), where f̅ is sinusoidal and q̅' corresponds to an open-loop (stable or unstable) equilibrium solution to that forcing (as done in the control-based continuation method, CBC, introduced in the next section). With the differential gain k̅_d, the open-loop unstable branches can be partially (Figure <ref>) or totally (Figure <ref>) stabilized. When the gain is not large enough, the remaining unstable solution co-exists with other stable attractors for which the control is invasive (keeping q̅_* fixed to make the control non-invasive on the unstable solution). Interpretations of the stabilizing effects of the CBC in terms of folding and closed-loop NFR can be found in <cit.> and <cit.>, respectively.

 


The stabilizing and non-invasive properties of the control signal may intuitively seem like conflicting requirements to the newcomer: how can this signal stabilize anything if it is supposed to vanish? The key element of the answer lies in understanding that while it does indeed vanish when the system is exactly at the equilibrium, it does not when the system is near this equilibrium. To illustrate this, a simple and evocative example is that of an inverted pendulum on a cart, where the cart motion is controlled to ensure that the pendulum stays in its upright position. This control therefore stabilizes what would normally be an unstable equilibrium and acts against any perturbation. It is also non-invasive, because the cart stays still when the pendulum is upright in the (theoretical) absence of perturbations. Another illustrative example is given in <cit.>.


    

 §.§ Control-based continuation

    

        

        We now introduce the first experimental continuation method, namely CBC. Inspired by previous works <cit.>, it was proposed by Sieber and Krauskopf <cit.> in a very general setting by seeking an experimental equivalent to pseudo-arclength continuation. The two key innovations of this work were (i) to use a PID-type controller[We note that other types of controller can be used, as long as they satisfy the non-invasiveness requirement.] to stabilize the unstable responses of the system under test and (ii) to formulate a zero problem with measurable functions that indicate a non-invasive control. Figure <ref> schematizes this original version of the CBC method.

        This method is organized around the reference signal x_*, which is set so as to make the control non-invasive by prescribing the force applied on the structure f to a preset profile f_* (alternatively, there can be versions where the drive u is specified instead of f), i.e., 
        
    f(t;𝐰_*) - f_*(t) ≈ 0.

        This non-invasiveness requirement is dealt with by using truncated Fourier series of the signals of interest,
        
    x_*(t) ≈𝐡(t)𝐰_*,

        where the column vector 𝐰_* gathers the Fourier coefficients of x_*, and 𝐡(t) is the associated vector of 2h+1 harmonic functions
        
    𝐡^T(t) = [         1;  sin(ω t);         ⋮; sin(hω t);  cos(ω t);         ⋮; cos(hω t) ],

        where superscript T denotes the transposition operator. By adjusting the Fourier coefficients 𝐰_* of the reference signal fed to the controller, the difference between the excitation f and a desired excitation profile f_* is sought to be as close to zero as possible. A well-posed problem can be obtained if the force profile is also discretized with Fourier series, i.e.,
        
    f(t;𝐰_*) ≈𝐡(t)𝐛(𝐰_*,ω),

        and Equation (<ref>) becomes
        
    𝐛(𝐰_*,ω) - 𝐛_* = 0,

        with 𝐛_* the desired force profile in a Fourier basis. For instance, if f_*(t) = sin(ω t), 𝐛_* = 𝐞_2, where 𝐞_i is the i^th canonical basis vector of ℝ^2h+1. Equation (<ref>) defines a zero problem with a nonlinear function 𝐛(𝐰_*,ω), where the number of unknowns (𝐰_* and ω) exceeds the number of equations by one. To solve this problem, a numerical pseudo-arclength continuation algorithm can be used. Since this algorithm relies on Newton-type methods to find the zeros of a residual function, it requires derivatives, which are not known in an experimental setup, but can be approximated using finite differences. Hence, in this work, this original version of the CBC is called CBC-FD. From the output of this continuation method, x_*(t) can be reconstructed from 𝐰_* using Equations (<ref>) and (<ref>).

        In a vibration testing context, as the CBC-FD method prescribes the forcing profile, so is its amplitude, and in general this method provides the NFR of a system, although variants can be conceptualized to provide other types of response curves. Figure <ref> presents the main resonance of an energy harvester from <cit.>, showing that this method is indeed able to obtain a full branch of an NFR.

        
        

        

  §.§.§ CBC-FD applications

            
            The first experimental validation of CBC-FD was with a vertically forced pendulum <cit.>. An alternative way to compute the residual function using a time delay (similarly to <cit.>) to handle non-fundamental harmonics was also proposed in <cit.>, and later used on energy harvesting systems <cit.>. This extended time delay approach is simpler than the CBC-FD algorithm presented herein, but has less stabilization capabilities. The CBC-FD method was validated on a system with impacts <cit.> and on a geometrically nonlinear beam <cit.>. Interestingly, similar ideas have recently been introduced in quasi-static tests to avoid snapthrough and snapback phenomena <cit.>, in real-time hybrid testing <cit.>, and in testing procedures for microwave circuits <cit.>.

            

            Strategies to determine the stability of the found periodic orbit relying on temporarily switching the control off were investigated in <cit.>, and the effect of noise was thoroughly studied in <cit.>. To the authors knowledge, Bureau et al <cit.> were also the first to measure a complete isola experimentally, as shown in Figure <ref>. The possibility to use B-splines (instead of Fourier decomposition) was recently explored by <cit.> for slow-fast systems. An in-depth dissertation on CBC-FD can be found in <cit.>, and a software package, CONTINEX, is available in COCO <cit.>.

        

  §.§.§ Limitations


            The CBC-FD method leverages pseudo-arclength continuation. Folding is almost guaranteed to be absent, which makes this method inherently more robust (continuation-wise) than traditional SWS tests. Unfortunately, this robustness comes at the expense of substantial complexity and longer tests due to the operations required by the method, such as the computation of a Jacobian through finite differences. This motivates the derivative-free methods that are treated in the next sections.




§ DERIVATIVE-FREE EXPERIMENTAL NATURAL PARAMETER CONTINUATION



We now treat the set of state-of-the-art methods that can be classified as "derivative-free", in the sense that they do not rely (explicitly) on a Jacobian of the residual function, and thus do not require finite differences and the likes (e.g. Broyden updates). We first present them in a unified framework, showing that they can be seen as derivative-free experimental natural parameter continuation methods (cf. Section <ref>), before reviewing them.



 §.§ Continuation



Similarly to its numerical counterpart presented in Section <ref>, experimental continuation can be performed in two main variants. The first one, arclength-type continuation, is used in the CBC-FD method as presented in Section <ref>. The second one is natural parameter continuation, for which a series of relevant parameters can be chosen, generally based on how representative they are of the excitation or response of the system.

We now specialize the discussion to nonlinear vibrations, given that most of the methods treated herein were developed in this area. The discussion can however be translated to other disciplines by considering relevant variables. To represent the state of a vibrating structure, four distinct variables are generally used:

    
  * f, an amplitude of the external forcing applied to the system,
    
  * ω, the frequency of this forcing,
    
  * a, the amplitude of some harmonic of the response at some point of the structure (typically, the same harmonic as the forcing of the collocated degree of freedom) and
    
  * θ, the phase of this harmonic.

There are of course more variables than this, but these four are of particular interest in control-based methods. In the most basic set-ups described in Section <ref>, the operator controls f and ω, and a and θ are naturally determined by the evolution of the system (or, in a mathematical setting, they are the unknowns of the problem, whereas f and ω are given parameters). For linear systems, this evolution is often represented with a fixed f and a varying ω in Bode diagrams. Due to the invariance of the linear properties with respect to f, the results are usually normalized with the latter. For nonlinear systems however, variations of f retain their full importance, resulting in more than a mere scaling of the response.

Control-based methods change the paradigm of Section <ref>, in the sense that f and/or ω can be automatically adjusted to prescribe a and/or θ as parameters set by the experimenter. But in all cases, there are always two parameters among the four variables that can be set by the operator, while the two remaining variables adapt according to the dynamics of the system and can be used to represent its response.



In addition, one of the two chosen parameters is usually kept constant during a test, while the other is varied. The fixed parameter determines the type of response, whereas the changing parameter is most of the time used as the abscissa when plotting the response of the system. For illustration, Figure <ref> represents all possible responses of a Duffing oscillator under tonal excitation as a two-dimensional manifold in the three-dimensional (ω, f, a) and (ω, f, θ) spaces, together with response curves obtained when one of these parameters is fixed (which can be seen as "slices" of that manifold). We note that the terminology "backbone curve" refers herein to a special case of a constant-θ curve where θ=-π/2, which is often close to the amplitude maximum of the NFR <cit.>.



With this view in mind, we can classify most of the experimental continuation methods (that shall be presented hereafter in this section) following Table <ref>. In this table, we can distinguish methods that require no, one or two control procedures. By control procedures, we mean, in a broad sense, any procedure that will adjust the input to the systems, using continuous feedback control (for SCBC and PLL) or discrete iterative methods (for SCBC and RCT). Methods simultaneously imposing f and ω require no control procedure, and are thus always open-loop. Most of the existing control-based methods require just one control procedure, with the exception of the SCBC method applied to the search of specific phase resonance <cit.>, SCBC combined with PLL (CBPLL) <cit.>, or PLL with amplitude control (PLL-A) <cit.>.

Fixing a parameter while varying another one to parametrize the response makes the methods listed in Table <ref> inherently equivalent to natural parameter continuation. As such, they are all susceptible to folding issues. This is well-known for the open-loop procedures, but should also be kept in mind when using the control-based methods listed in Table <ref>. 

We can now present in more detail the main derivative-free experimental natural parameter continuation methods, namely, SCBC, PLL and RCT.

        

 §.§ Simplified control-based continuation



        
        
        A substantial simplification of the original CBC algorithm was proposed by Barton et al <cit.> and is schematized in Figure <ref>. They noted that, since the PID controller is linear and time-invariant, a single-harmonic input x_*-x to the PID yields a single-harmonic drive u. They thus proposed to set the fundamental amplitude of the reference signal to a target value a_*,
        
    x_*(t) = a_* sin(ω t) + 𝐡_nf(t)𝐰_*,nf

        and to set its other harmonic coefficients, 𝐰_*,nf, equal to those of x, thus satisfying the non-invasiveness requirement. Thus, if 𝐰 are the Fourier coefficients of x (and 𝐰_nf its non-fundamental Fourier coefficients), Equation (<ref>) is changed into
        
    𝐰_*,nf - 𝐰_nf(𝐰_*,ω) = 0.

        They also simplified the continuation algorithm by noting that Equation (<ref>) can be solved with a simple Picard iteration scheme, thereby removing the need for experimental finite differences. We refer to this simplified implementation as simplified CBC (SCBC).

        The main reason for using a_* as a bifurcation parameter instead of ω is that it removes the aforementioned folding issue in most cases of interest, provided the controller's gain are properly chosen. Indeed, it can be observed in Figure <ref> that S-curves can be parametrized with the amplitude a. It should be noted that the actual amplitude a will not be exactly equal to the target amplitude a_*, but will tend to it as the controller gains are increased. We finally note that, thanks to its broad theoretical framework, the SCBC can be performed with another bifurcation parameter than a_*, but the latter is preferred in the vast majority of cases.
        
        A SCBC test is performed at a constant frequency ω, by progressively varying the value of the target amplitude a_*, obtaining so-called S-curves. Collecting these curves for different target amplitudes gives a surface (in the (ω,f,a_*) space) that can be sliced at a constant force level to obtain an NFR. This procedure essentially amounts to fitting experimental data and performing interpolation on the fit. Figure <ref>, reproduced from <cit.>, illustrates such a process. 

        


        

  §.§.§ SCBC applications

            
            The SCBC method was applied to systems with friction <cit.> and beams <cit.>, and advanced versions were developed by Renson et al, enabling the tracking of backbone curves <cit.>, fold bifurcations <cit.> and internal resonances <cit.>. A simplified implementation using adaptive filters and circumventing the use of Picard iterations was proposed in <cit.>, thereby speeding up the process. Interestingly, the SCBC approach can also be used in a purely numerical setting to solve challenging problems <cit.>.

            Noise can be handled using local Gaussian process regression <cit.>. From a NSI perspective, noise effects were shown to be smaller with CBC than with traditional testing methods <cit.>, and successes using machine learning and/or Gaussian regression processes using SCBC results were demonstrated in <cit.>. Procedures to determine the open-loop stability were developed based on data-driven linearization <cit.>. A systematic method to choose the gains of the PD controller to guarantee the absence of folding was also developed for a Duffing oscillator <cit.>. Finally, we note that the SCBC found applications beyond structural mechanics, such as aeroelastic systems <cit.>, biochemical processes <cit.>, micro-electromechanical systems <cit.> and pedestrian flow control <cit.>. An in-depth dissertation on the SCBC method can be found in <cit.>.
        
    

 §.§ Phase-locked loops


        Resonance conditions are oftentimes complicated to reach in nonlinear systems due to their closeness to unstable regions. This is what motivated Babitsky <cit.> to introduce velocity feedback, creating auto-resonant systems that automatically maintain themselves at resonance. This idea was later extended to continuation by Sokolov and Babitsky <cit.> who proposed the concept of phase-controlled systems: the phase shift between the excitation and the response θ can be used as a parameter for the NFR. An  experimental implementation of this concept was realized with a delayed acceleration feedback by Mojrzisch, Wallaschek and Bremer <cit.>. The unfolding of the NFR also comes with feedback stabilization, offering the possibility to obtain its unstable parts. 
        
        Delayed displacement, velocity or acceleration feedback is very simple to realize in practice. It can be used, e.g., for (linear) experimental modal analysis <cit.>. However, it is generally invasive and stabilization can be tricky to achieve. The latter point has recently been discussed in <cit.> and treated by measuring the structural response at multiple locations. An alternative to this type of approach is to use phase-locked loops (PLLs).

        Invented nearly one hundred years ago, PLLs are extensively used in electronics for signal synchronization purposes and are actually thought to be one of the largest engineering application of feedback control in the world <cit.>. Given a reference signal, PLLs are able to adjust the frequency and phase of another signal so as to lock the phase between these two signals to a prescribed value. Coupling PLLs with (electro-) mechanical systems was achieved even prior to the works of Babitsky for structural control as a solution to modal spillover <cit.> and to track resonance in fatigue testing <cit.>. Phase parametrization of frequency response functions (FRFs) using PLLs was proposed in <cit.>, and the first time PLLs were used in an experimental continuation framework for nonlinear structural systems was the relatively recent work of Peter and Leine <cit.>. The main advantage that PLLs have over delayed feedback is their natural non-invasiveness.

        
        

        A PLL test set-up consists of a structure under test, a phase detector, a PID controller, and the equivalent of a voltage-controlled oscillator, as depicted in Figure <ref>. The difference between a reference θ_* and θ, the phase shift between the response x and either the drive u or the force f, is fed to the PID controller, which automatically adapts the frequency of excitation ω to reach the desired phase. Indeed, if θ_* is constant, since ω is the output of the PID,
        
    [                                ω (t) =                ω (0) + k_p(θ(t) - θ_*);                                        + k_i∫_0^t (θ(τ) - θ_*) dτ + k_dθ̇(t), ]

        where k_p, k_i and k_d are the proportional, integral and derivative gains of the controller, respectively. Differentiating Equation (<ref>) with respect to time,
        
    ω̇ (t) = k_pθ̇(t) + k_i (θ(t) - θ_*) + k_dθ̈(t),

        and assuming steady-state conditions (ω̇ = θ̇ = 0) shows that, at steady state, θ(t) = θ_* as long as k_i≠ 0. Once ω reaches a steady value, the control is naturally non-invasive and can stabilize unstable responses. By fixing the forcing amplitude f_* and varying the phase θ_*, one can obtain an NFR. Alternatively, by fixing the phase θ_* (typically to obtain phase resonance) and varying the force f_*, one can obtain a constant-phase curve (typically, a phase-resonance backbone curve).

        

  §.§.§ PLL applications

            
            The main thrust for PLL application was (and still is) nonlinear modal analysis, given how easily it can be performed with this control-based method. Early works provided a theoretical study based on normal forms that was developed and validated in <cit.>, and an NFR synthesis method based on the measured phase resonance curve was proposed in <cit.>. Since then, the PLL has been applied to a large variety of structures, including beams with friction <cit.>, impacts <cit.>, repelling magnets <cit.> or featuring extremely large vibration amplitude <cit.>, a T-structure <cit.>, plates with internal resonances <cit.> and a turbine blade with friction <cit.>. This latter example is an indication of technological maturity, and some of the associated results are plotted in Figure <ref>. In addition, an adaptation for multi-input-multi-output test case was sketched in <cit.> based on a centralized modal-based control scheme, and a framework for phase resonance testing of base-excited systems was proposed in <cit.>.

            

            Finally, we note that PLL was compared to velocity feedback in <cit.> and to CBC in <cit.>. All methods were shown to be capable of providing relevant results to the experimenter. In-depth dissertations on PLL can be found in <cit.>.
        
    

 §.§ Response-controlled stepped-sine testing


        The most recent control-based method is the RCT approach, whose basic principles were first drafted in <cit.> and then elaborated in <cit.> together with a theoretical justification. Similarly to the SCBC and PLL, its underlying motivation came from the observation that the frequency-force curves are generally unfolded when the experiment is performed with a constant amplitude response a_* and varied excitation frequency ω. It can thus be seen as a dual of the SCBC method, as illustrated by Table <ref>. The general schematics of the RCT method are given in Figure <ref>.

        
        
        A key strength of RCT is that, due to numerous cases of modal tests where the structure is base-excited <cit.>, conventional linear vibration testing software generally implement a possibility to perform the test at a constant response amplitude. The control principle used in Siemens Testlab (the commercial software used for all RCT demonstrations) is conceptually explained in <cit.>, although a detailed formulation of the control scheme does not seem to be available. Simply put, it is an automatic trial-and-error procedure that adapts the drive amplitude at a given frequency to match the desired response amplitude. A test at a given target amplitude is started at a given frequency with a trial drive amplitude maintained constant over several periods of excitation, and the harmonic response of the structure is estimated, e.g., with a digital tracking filter. A frequency response function between the response and drive is estimated (or updated) based on the ratio of these amplitudes, and the next drive amplitude is calculated based on this update. The process is repeated until the response amplitude reaches its target (within tolerance bounds) or until a maximum number of corrections have been carried out. The test then carries on to the next frequency. With this method, the control is naturally non-invasive, because the drive is always tonal.

        
        For a given target amplitude, the test yields a harmonic force curve (HFC). Collecting them for multiple target amplitudes, one obtains a harmonic force surface (HFS), and, similarly to SCBC, NFRs can then be extracted using interpolation of this HFS.
        
        Another interesting feature of the RCT is the individual linear-like look of the frequency-force curves, making it possible to identify them by standard linear system identification tools providing modal characteristics. This quasi-linearity is supported by the single-nonlinear mode theory <cit.>, which states that if a structure approximately vibrates in a tonal fashion according to one of its mode and if the amplitude of one degree of freedom is maintained constant, the amplitude at the other degrees of freedom will remain constant. This effectively "freezes" the effect of geometrical nonlinearities throughout the structure, giving an intuitive understanding for the observed quasi-linear behavior. This approach also offers an alternative to harmonic force surface interpolation for obtaining NFRs by synthesizing them using identified modal properties.
        
        The RCT method thus provides significant advantages, such as a simplified post-processing procedure leveraging quasi-linearity, and the possibility to use the method directly with commercial vibration measure and control apparatus. When the assumptions for quasi-linearity do not hold, the post-processing might become more complicated but the method still offers a valid control-based framework.

        

  §.§.§ RCT applications

            
            In spite of its relative youth, the RCT has successfully been applied to a variety of examples, including beam setups <cit.>, a bolted missile <cit.>, a stack piezoelectric actuator <cit.> and a highly-damped control fin actuation mechanism <cit.>. We refer to <cit.> for an in-depth dissertation on RCT. The RCT has not been experimentally compared to any other control-based method yet, and such a comparison is carried out in this work.






    
    
    
    
    
    
    


        






§ A DERIVATIVE-FREE ARCLENGTH CONTROL-BASED CONTINUATION PROCEDURE



    According to the discussion in Section <ref>, current control-based methods require a choice from the experimenters. They can use simple, derivative-free methods such as those presented in Table <ref>, but the test is then exposed to potential folding issues and the associated possible occurrence of a jump. Some examples will be given later in Section <ref>. Alternatively, they can opt for the more general CBC-FD method, which is however more complex, sensitive to noise and makes the test longer. The main reason for these two downsides is the need for experimental derivatives evaluated through finite differences or Broyden updates.
    
    The purpose of this section is to propose a new method which is almost as generic as the CBC-FD method in the sense that it does not assume any parametrization a priori, but which also circumvents the need for experimental derivatives. This procedure is called arclength CBC (ACBC) herein, was originally presented in <cit.>, is schematically depicted in Figure <ref> and explained hereafter.

    
        

    

 §.§ Non-invasive control


        As in the CBC, a PID-type controller with a reference signal is used to stabilize the structure under test. This reference signal is determined based on two criteria. First, its non-fundamental harmonics are set by the non-invasiveness requirements similarly to the SCBC method. Second, its fundamental harmonic is used together with the excitation frequency to perform continuation.

        Since the method works along the principles of CBC, non-invasiveness is guaranteed as long as the reference and actual signals have identical non-fundamental harmonics. As in <cit.>, the non-invasive character of the control signal is ensured by the use of adaptive filters. These filters are able to automatically perform a truncated Fourier decomposition of a signal x with h harmonics, i.e.,
        
    x(t) ≈𝐡(t)𝐰,

        where 𝐰 approximate the 2h+1 Fourier coefficients of x.
        
        The  Widrow-Hoff least mean squares (LMS) algorithm can be used to update the Fourier coefficients 𝐰 of a signal x through the simple discrete-time law
        
    [                        𝐰((n+1)t_s) - 𝐰(nt_s); = μ t_s (x(nt_s) - 𝐡(nt_s)𝐰(nt_s))𝐡^T(nt_s), ]

        where μ is the filter gain and t_s is the sampling time.

        Assuming the Fourier coefficients converge, the control can thus be made non-invasive by defining the target signal as
        
    [                                   x_*(t)                                        =              a_* sin(ω t) + b_* cos(ω t);                                                                                       + 𝐡(t)(𝐈 - 𝐞_2𝐞_2^T - 𝐞_h+2𝐞_h+2^T)𝐰;                                                                                 = a_* sin(ω t) + b_* cos(ω t) + 𝐡(t)𝐏_nf𝐰. ]

        𝐏_nf can be interpreted as a matrix whose action retains only the non-fundamental harmonics so that, if Equations (<ref>) and (<ref>) hold, the signal x_* - x is mono-harmonic.
        
    

 §.§ Simplified continuation problem


        Having secured the non-invasiveness of the method, three parameters remain to be set by the method, namely a_*, b_* and ω. Without loss of generality, one of the Fourier coefficients can be set to zero, say b_* = 0. The two remaining unknowns are set by enforcing the fundamental forcing amplitude f to be equal to the target one f_*. 
        
        The method can probably be best understood by looking at the force imposed on the structure as a function of the frequency and target amplitude. On this representation of the HFS depicted in Figure <ref>, the NFR can be seen as a level set of this surface, where the forcing amplitude f corresponds to the target f_*. In this example, we consider for simplicity infinitely large control gains, such that a = a_*. However, we note that the foregoing discussion also holds for finite gains provided they are large enough.

        The new approach for experimental continuation leverages this, and is inspired from arclength continuation <cit.>. Because the NFR is one-dimensional in the (ω,a_*) space, a sufficiently small ellipse with semi-axes Δω and Δ a_* and centered on a point of the branch intersects it twice, as depicted in Figure <ref>. The intersections are characterized by f-f_* = 0, whereas every other point of the ellipse does not (in general) satisfy this equality. The continuation procedure can thus be formulated as follows: given a current point (ω_n,a_n) and an ellipse centered around it in the (ω,a_*) space, the next point of the NFR (ω_n+1,a_n+1) is found as the intersection of the ellipse and the level set where f = f_*, as seen in Figure <ref>. To do so, the procedure sweeps along the ellipse by automatically varying the eccentric anomaly α of a point on the ellipse. The procedure can then be repeated using ellipses centered around each next point, thereby obtaining a set of points on the NFR in the same fashion as Crisfield's arclength continuation <cit.>.

         
        
        A key advantage of this approach is that the harmonic forcing amplitude can easily be measured online, and there are several strategies to find the eccentric anomaly that solves the zero problem f-f_* = 0 without need for experimental derivatives: 
        
            
  * A first simple possibility consists in sweeping the angle α (t) at a constant rate r_α until f=f_*. 
            
  * A continuous integral control law α̇ = -k_α (f-f_*) can be used. An advantage of this law is that it automatically makes the algorithm converge toward a point where f=f_*, and orients the curve so that the method does not turn back. Indeed, this law makes the point where f=f_* and ∂ f/∂α >0 a stable equilibrium, whereas the other solution is unstable. As a consequence, this law converges to intersections which pierce the NFR outward as α is increased. However, since the sweep rate depends on the error f-f_*, the sweeping rate of the ellipse depends on the amplitude of the profile f-f_*, and is typically low and high away and close to resonances, respectively.
            
  * A discontinuous integral control law α̇(t) = -r_αsign (f-f_*) combines the advantages of the sweeping and integral control methods. However, the discontinuity of this control law can make the convergence toward a solution where f-f_*=0 difficult.
        
        The sweeping law can be chosen freely by the user depending on their preference, and is denoted by α̇ = s(f-f_*) in the sequel. The sweeping rate should be high for a fast continuation, but is nevertheless limited by the system's transients, as well as the adaptive filters speed to guarantee a non-invasive control.
        
    

 §.§ The algorithm


        Algorithm <ref> summarizes the proposed derivative-free ACBC procedure. In this implementation, the prediction is made with the previous converged value of α, which amounts to implementing a secant predictor. ρ represents a relative tolerance, and σ < 1 is a factor that makes the algorithm stop sweeping in more strict conditions than the convergence check in order to anticipate transient effects. If this preemptive procedure is not enough to meet the convergence conditions, one can adapt the sweeping law by making it slower.

        One can also decide to measure the NFR in one direction or another (with globally increasing or globally decreasing forcing frequencies) by selecting a positive or negative value for k_α or r_α.

        

        The proposed algorithm is equivalent to the CBC-FD in terms of generality of the continuation approach, as it does not make any assumption on the unfolding property of a parameter. A key advantage of this approach is its derivative-free nature, brought by the combined use of the adaptive filters and the arclength continuation procedure in a two-dimensional space.

    

 §.§ Illustration with a Helmholtz-Duffing oscillator


        

        The Helmholtz-Duffing example in Figure <ref> is revisited with a numerical simulation of the ACBC algorithm in Simulink with k̅_d=1, k_α=1, h=15, μ=0.0025, Δa̅_* = Δω̅ = 0.05 and t̅_steady=50. Clearly, the ACBC algorithm is able to capture the full NFR in this example, including its open-loop unstable parts. 

    
    

    
        
    
    
    
    
    
        




§ ILLUSTRATION WITH AN ELECTRONIC DUFFING OSCILLATOR



    Having reviewed the different control-based approaches for experimental continuation, we illustrate them with two experimental examples.

    A first demonstration is made with an analog electronic circuit that replicates the behavior of a Duffing oscillator <cit.>. This circuit is described in more details in the supplementary materials of this article and in <cit.>. The main advantages of using such an experimental demonstrator are its near perfect repeatability and actuation (i.e., the drive and the force are almost exactly equal), which allows us to test the inherent features of control-based methods. Yet, this set-up is subject to practical experimental challenges, such as noise, unknown non-ideal behavior, and real-time constraints.

    

 §.§ Experimental setup


    A picture of the experimental apparatus, comprising the electronic Duffing oscillator and an acquisition and control system, is given in Figure <ref> and schematized in Figure <ref>. The oscillator takes a forcing signal V_in as input, and outputs a displacement signal V_x and a velocity-proportional signal V_v, with V_v≈ -10^-3V̇_x. Compared to Figure <ref>, the acquisition and control system encompasses the continuation, controller and signal analyzer blocks. For all methods except the RCT, this system was a MicroLabBox from dSPACE (as in Figure <ref>), whereas the RCT was performed with a Scadas Mobile from Siemens.

    


    The parameters c, k and k_3 (cf. Equation (<ref>)) of the electronic Duffing oscillator can be tuned with knobs (potentiometers), and their theoretical values can be computed based on the electrical parameters of the board (see supplementary materials). They are reported in Table <ref>. They were compared to an identification with the nonlinear frequency subspace identification (FNSI) method <cit.> using data from a swept-up sine test. A close agreement between the theoretical and identified values can be observed, with a discrepancy of up to 8% for the damping coefficient. These parameters correspond to a natural frequency ω_0=131.5 rad/s (20.9 Hz) and a damping ratio ζ_0=0.005. This damping ratio can be considered rather small although it is still representative for a number of metallic structures. It was initially chosen to result in large amplitudes for moderate forcing, and strong nonlinearity, but it also revealed some limitations of various methods that seemed interesting to report herein.

    


    
    

 §.§ Implementation details

    
    All methods (except for the RCT) were programmed in a fully online way with dSPACE RTI for Simulink. Almost all methods require a Fourier decomposition. This decomposition was performed online by adaptive filters, which were shown to be quick, effective and accurate <cit.>. For a tonal signal at frequency ω, an expression for an optimal gain (minimizing the settling time of the Fourier decomposition) was derived in <cit.>[Note that the symbol μ used herein corresponds to μ t_s in <cit.>.]. Here, based on a linearization of this expression with respect to the (small) sampling time, a frequency-dependent gain was used
    
    μ = ωμ̅,

    where the optimal dimensionless gain based on this linearization is μ̅=2. For a signal with multiple harmonics, this value was observed to lead to a slightly too aggressive filter, and a smaller value was usually selected. Another procedure to tune μ was recently proposed in <cit.>.
        
    Some iterative methods propose specific procedures to detect if the system is at steady-state. In this work, we simply waited a certain time t_steady before subjecting the system to another set of parameters. 

    All CBC methods were implemented with the same differential controller. It should be noted that the velocity signal was used instead of the displacement one, thereby avoiding the use of an approximate derivation with a high-pass filter (as can commonly be done in practical PID implementations).

    Stepped methods (CBC-FD, SCBC, ACBC and RCT) require the definition of steps Δω and Δ a_*, which are based on the desired resolution of the NFR. For CBC methods, since the feedback is made with a velocity-type signal, the steps were defined in terms of this signal, Δ |V_v|_1 for simplicity, where |·|_k denotes the k^th harmonic amplitude of a signal. For the RCT, a (nearly) equivalent step in terms of displacement Δ |V_x|_1 was defined instead. The two are related by Δ |V_v|_1 = 10^-3ωΔ |V_x|_1.

    Most of the parameters were tuned on a trial and error basis. A fine-tuning toward optimal performance was not sought given the very large number of parameters and their variety. The results given therein are thus not to be interpreted as the best each method can yield, but rather as an order of idea of how they compare.

    In the sequel, most of the results will be presented in terms of the total amplitude of the response over a period, being represented by the operator |·| (without subscript), i.e.,
    
    |x| = max_t∈ [0,2π/ω]abs(x(t)),

    where abs is the absolute value operator.


    

 §.§ Primary resonance


        The primary resonance of the oscillator was first tested. The time for the two methods where parameters are swept, namely SWS and PLL, were chosen based on the fastest stepped method (in this case, ACBC). All parameters are given in the supplementary materials. We focus on measuring an NFR at a specific excitation f=0.03 V between 50 and 200 rad/s.
        
        

  §.§.§ Comparison of the NFR obtained with different measurement approaches

        

            
             To start this analysis off, Figure <ref> shows the velocity signal NFR directly obtained from methods that ran without issues, namely, the SWS, PLL and ACBC approaches. The agreement between the two latter is excellent, even at the resonance peak. Some discrepancies are observable with the SWS data: as usual, transient effects are the cause for a slightly lower peak amplitude. We also note that the SWS data includes higher harmonics as well. The PLL test experiences large noise at low frequency, which is due to the 2:1 superharmonic resonance of the oscillator. While it is too small to be noticeable in the amplitude of the NFR, Figure <ref> shows that the phase of the response evolves in a non-monotonous way at low frequencies, due to this superharmonic resonance. Once the phase becomes unique again (in particular near the primary resonance), this is no longer an issue.
            
            
            
            

        

  §.§.§ Challenges for the pseudo-arclength CBC method

        

            The results obtained with the CBC-FD method are compared to those of the ACBC method in Figure <ref>. Convergence issues were encountered with the former method in the vicinity of the resonance peak, requiring to restart the method from the high-frequency limit (200 rad/s) to approach the peak from both sides.
            
            
            
            The difficulties encountered with CBC-FD are most likely due to two main causes. First, owing to the small damping of the oscillator, the NFR curve experiences a sharp turn near the fold bifurcation. As a consequence, the pseudo-arclength method can struggle to converge to the curve during a correction phase. This issue could be solved by using an arclength continuation scheme, or taking smaller steps. The second issue is linked to the use of finite differences, and again, is mostly problematic in the vicinity of the resonance peak. Indeed, close to this peak, the residual function varies extremely rapidly and has a large curvature (as will also be illustrated by Figure <ref>). Similar issues were encountered in <cit.>. Smaller finite difference steps could have been taken, but this would shift the issues to zones away from resonance, where the residual function is much less sensitive to variations in the input, and where large enough finite difference steps need to be taken to overcome noise. This issue could be solved using an adaptive finite difference step by, e.g., prescribing an ideal change in the residual function (suitably larger than the noise level).

            
        

  §.§.§ Resolution requirements for the methods using interpolation


            

            As explained in the respective works where the methods were proposed, both SCBC and RCT obtain a collection of response curves that can be combined into a response surface. NFR curves can be extracted from this surface by slicing it with a plane characterized by |f|=f_*, which essentially amounts to performing interpolation on experimental data. An off-the-shelf solution used in this work was to use the  function in Matlab to fit the measured force and amplitude as 2D spline functions of the frequency and target amplitude, as f̂(ω,a_*) and â(ω,a_*), respectively (where a hat denotes an interpolated function). The curves satisfying f̂(ω,a_*)-f_* = 0 were then found using the  function. Figure <ref> shows such curves obtained for the SCBC. Points of these curves were finally fed to the fitted function â(ω,a_*) to obtain the NFR curve.

             

            The results of the SCBC and RCT obtained with this method are compared to those of the ACBC in Figure <ref>. In this case, we analyze the fundamental harmonic amplitude rather than the total (multi-harmonic) one, because the latter is more complicated to extract with the results from TestLab with the RCT. These two amplitudes do not differ much anyway. While the interpolated NFRs are close to the one from ACBC away from the resonance peak, there is a clear discrepancy between the two types of results in the vicinity of the peak. This issue is mainly due to the interpolation, and to an insufficient resolution of the (ω, a_*) test grid: near resonance, the function f̂(ω,a_*) varies rapidly and requires a very fine resolution for the interpolation to be correct. Similar issues were encountered in <cit.> for the detection of fold bifurcations from the response surface. This can be solved using either a finer grid (but this requires a longer test time), or using an adaptive sampling strategy instead of a regular grid. Figure <ref> supports this hypothesis by showing the results obtained by narrowing the test frequencies from 120 to 180 rad/s with steps of 2 rad/s, and amplitude steps of 0.05 V, effectively doubling the resolution in both directions. Clearly, the agreement between both methods is improved, at the expense of a longer test time (increasing by a factor 4). The interpolation with the SCBC still inadequately represents a range of about 10 rad/s near the resonance peak, though.
            
            

            In addition to this issue, we noted that the SCBC method could encounter convergence struggles, as will be discussed in Section <ref>. As for the RCT, another issue related to stability was encountered, which is why the curve in Figure <ref> appears incomplete. This issue is discussed in Section <ref>.

        

  §.§.§ Invasiveness of the SCBC in the vicinity of superharmonic resonances

        
        
            The SCBC method was implemented with both the original scheme based on Picard iterations <cit.>, as well as a scheme that exploits the output of an adaptive filter <cit.> to make the control non-invasive. The former was observed to be much slower than the latter at some measurement points, forcing a raise of the tolerance to 5% for the Picard scheme (instead of 1% for the adaptive scheme) to allow for convergence. As shall be shown hereafter, this occurs in the vicinity of strong superharmonic resonances. Away from them, both methods performed similarly. The difference between the two methods in terms of results presented in Figure <ref> were negligible, which is why only the results of the method based on adaptive filters were shown.

            Figure <ref> compares the results of the two methods in terms of the response surface. Differences are observable at low frequencies, while the agreement is almost perfect for the rest of the surface (not shown here). Figure <ref> shows the relative error, given by the invasiveness (the norm of the left-hand side of Equation (<ref>) divided by a_*) made by the Picard scheme, featuring the largest error in the zones of largest discrepancies. The distribution of these zones clearly follows a trend, making superharmonic resonances a potential suspect for this effect. The invasiveness of the SCBC with adaptive filters was also checked a posteriori, and it was also observed that the control shows the largest degree of invasiveness in the same regions, in spite of the expected action of the adaptive filter. This was not detected during the test, because the check for non-invasiveness was performed based on the error made by the signal synthesized by the adaptive filter, x(t)-𝐡(t)𝐰(t).

             
            
            To ascertain that superharmonic resonances can be an issue, the 2:1 and 3:1 resonances were tracked with the PLL, by locking the phase of the second and third harmonic to -π and -π/2, respectively[The resonant phase lag for the 2:1 resonance differs from that observed in a perfect Duffing oscillator due to the inevitable presence of asymmetries, making it more similar to a Helmholtz-Duffing oscillator in this regard.], and varying the excitation amplitude (implementation details are given in the supplementary materials). Overlaying the obtained results[The forcing amplitude was converted to an equivalent imposed amplitude for the reference signal via a_* = f/k_d + a.] with those of the SCBC test with Picard iterations in Figure <ref> clearly highlights that the issue indeed occurs close to the superharmonic resonances. The non-fundamental harmonic coefficients are set with a Picard iteration scheme, and a failure of this scheme in the vicinity of a resonance is coherent with numerical experience <cit.>.
            
        

  §.§.§ Instabilities with the RCT approach

        
        
            We now turn to the incomplete aspect of the NFR curve obtained with the RCT, which is due to stability issues during the test. Figure <ref> plots the actual amplitude of the displacement as a function of the target amplitude and forcing frequency. Zones with large errors are clearly apparent. The points with a relative error larger than 20% were rejected when performing the interpolation, explaining the incomplete aspect of the NFR curve obtained with the RCT in Figure <ref>. These zones also appear to organize into a particular pattern reminiscent of the zone of instability of the open-loop oscillator, delineated by the loci of fold bifurcations.

             

            To verify this hypothesis, the experimental fold bifurcation tracking procedure proposed by <cit.> was implemented (implementation details are given in the supplementary materials), and the resulting loci of fold bifurcations are overlaid in Figure <ref>. The near complete depletion of points between the two fold bifurcation curves confirms that the issue is very likely to come from the instability of the oscillator. It thus appears that the RCT was not able to stabilize unstable solutions in this case, and the oscillatory look of the amplitude observed in the large-error region in Figure <ref> corresponds to alternating convergence of the response to the upper and lower stable branches. The presence of many unstable points makes the controller struggle and systematically reach its maximum number of iterations, explaining why the RCT took about an order of magnitude longer than the other methods (cf. Table <ref>).

             Some points lay inside the unstable zone delineated in Figure <ref>, and an analysis of the time series revealed that they are associated with transient phenomena. The resulting amplitude thus cannot be considered as a steady-state quantity. For illustration, Figure <ref> depicts the time series associated with the point (ω,a_*)=(170,0.9) for which the amplitude indicated by the TestLab software results is |V_x|_1 = 0.9135, hence seemingly correct enough. The time series of the amplitude estimated by TestLab was unfortunately unavailable and was retrieved through time-frequency analysis (using the short-time Fourier transform from the  function in Matlab). Its estimation is overlaid together with the target amplitude. It appears that the amplitude of the considered point, indicated by the middle step in Figure <ref>, is calculated during transients (right before the end of that step) and is thus not representative of the steady-state response of the oscillator. Assessing this in a systematic way is unfortunately difficult given the little amount of information that can be extracted from TestLab results. 

            
            
            The accumulation of points at the left of the leftmost branch, and the lack of points at the right of the rightmost branch in Figure <ref>, which are both stable regions, was first thought to be due to the stepped-up frequency during the test. However, a stepped-down frequency surprisingly showed the same trend. The reason for this puzzling trend was not investigated further.
            
            Finally, we note that there are points in the upper left corner of Figure <ref> which also feature a large error. Similarly to the case exposed in Figure <ref>, this is most likely due to superharmonic resonances, where folding is also expected to occur at high amplitudes.
            
            Because the RCT works with a black-box commercial software, there is no potential implementation issue. Different parameters were tried out without success, as a zone of instability always seemed to appear in the same range (more details are given in the supplementary materials). The authors believe that the main difference between this work and the original <cit.> is the presence of an electrodynamic shaker. This is known to stabilize unstable responses in some cases <cit.>, which might be the case therein. This issue would deserve further investigation but was deemed beyond the scope of this work. Nevertheless, if this explanation is correct, the extraction of the folded part of the NFR of the structure alone is more related to continuation aspects than stabilization ones, since the RCT itself performs continuation on curves that do not fold but does not appear to inherently change the stability of the system under test (composed of the structure and the shaker). 

        

  §.§.§ Testing times


        We conclude this first series of tests by providing the times needed to perform each experiment in Table <ref>. In terms of testing times, the fact that we focused on one NFR puts direct NFR measurement techniques, namely SWS, PLL, CBC-FD and ACBC, at an advantage compared to the other methods. It should thus be kept in mind when looking at the total time taken by each methods that SCBC and RCT allow for the extraction of multiple NFRs in one measurement run, and a more meaningful measure of the time efficiency of a method can be given by the average time taken per measurement point.

        

    

 §.§ Superharmonic resonance

    

        We now turn to the 3:1 superharmonic resonance of the Duffing oscillator. This type of resonance is seldom studied in control-based methods, but actually challenges all of them.

        

  §.§.§ Numerical illustration

        

        Before moving forward with the experiments, a numerical study has been made around the 3:1 superharmonic resonance of the Duffing oscillator. Variants of a harmonic balance method coupled with numerical continuation <cit.> have been used to compute constant forcing amplitude, forcing frequency and response amplitude curves with the theoretical parameters in Table <ref>. They are displayed in Figure <ref>. 

        

        Clearly, all methods based on natural parameter continuation, namely SWS, SCBC, PLL and RCT will fail, because folding is present in all cases. Similar observations were made in <cit.>. As for the ACBC approach, success is a priori uncertain given that the first harmonic amplitude curve features a loop that does not form a contour level.
    
        A solution nevertheless exists for the PLL, as exploited in <cit.>, which consists in replacing the phase lag of the first harmonic by that of the third harmonic. As Figure <ref> evidences, such a parametrization does not exhibit any fold in the vicinity of the 3:1 resonance, making it a viable approach. 

                
        
        Finally, we noted experimental difficulties with the CBC-FD method as well. In the vicinity of the 3:1 resonance, non-invasiveness was found to be crucial for the accuracy of the results and for the success of the method itself. To investigate the impact of invasiveness, a series of numerical simulations were performed by imposing a non-tonal forcing to the oscillator. To the nominal fundamental harmonic forcing was added a forcing on the third harmonic of 1% relative amplitude and fixed phase. Performing the continuation for multiple values of the phase, we obtained an idea of locus of responses if a 1% relative tolerance is allowed on the residual, as depicted in Figure <ref>. In spite of the small tolerance, the loop associated with the resonance can vary by up to about 50% in size, stressing the importance of a non-invasive control. 

        

        Numerical simulations of the Duffing oscillator tested with the CBC-FD uncovered a bottleneck linked to this issue. If too large a relative tolerance was chosen (such as 1%), the continuation procedure was progressing in an erratic fashion inside a locus similar to that depicted in Figure <ref>. The resulting irregular curve posed challenges for the continuation by making inappropriate predictions which could not be corrected. Alternatively, choosing a small tolerance made the curve less erratic. However, inaccuracies on the residual (coming from the Fourier decomposition) were observed to increase the vicinity of the resonance, eventually exceeding the tolerance threshold, rendering the convergence of the method impossible.

        

  §.§.§ Experimental testing


            In view of the issues outlined in Sections <ref>, <ref> and <ref>, the SCBC and RCT methods were not deemed suitable for the 3:1 superharmonic resonance. A SWS was performed in spite of its deficiencies to obtain a reference method, together with the CBC-FD, ACBC and PLL (with the phase of the third harmonic), with a forcing amplitude of 1 V. All parameters are given in the supplementary materials.
            
                

            Figure <ref> shows the results of the most successful method, namely the PLL, for the 3:1 resonance, together with swept-sine results for reference. The PLL is able to robustly go through the unstable branches and trace the complete NFR, and it was the only method able to do so thanks to the use of the phase of the third harmonic.

            The NFR follows a loop, which is easier to see in Figure <ref> where only the first harmonic amplitude is depicted. The results of the CBC-FD and ACBC are also shown therein. For the former, convergence of the Newton-Raphson procedure did not occur at some point. This lack of convergence could be due to the increased noise level observed around the 3:1 resonance, making the finite-difference estimation of the Jacobian inaccurate. As for the ACBC, the control was seen to be invasive, signalling that the adaptive filters were failing to fulfill their purpose, as in Section <ref>. The estimated NFR consequently significantly diverges from what is estimated to be its true shape, similarly to the issue raised with Figure <ref>. In addition, the fact that the NFR loops around the 3:1 resonance (Figure <ref>) indicates that the HFS cannot be parametrized solely by ω and a_* in this case, making the proposed ellipse sweeping procedure uncertain.

            The success of the PLL based on the phase lag of the third harmonic could lead to think that the SCBC and RCT would potentially benefit from a similar unfolding provided they work with the amplitude of the third harmonic. Unfortunately, as Figure <ref> reveals, such an unfolding does not necessarily occur for this harmonic amplitude, making it uncertain whether these methods would be able to handle superharmonic resonances.

            This series of tests on a superharmonic resonance showed that the assumptions of all methods based on natural parameter continuation can fail even with a single-degree-of-freedom system. Such resonances are nevertheless usually less important and/or relevant than primary resonances, that remain the main interest in nonlinear vibration testing. 



§ ILLUSTRATION WITH A CLAMPED THIN PLATE



    We now move to a real structure with challenging nonlinear dynamics, namely multiple nonlinear modes with a distributed geometrical nonlinearity. This set-up does not feature the same exceptional repeatability as the electronic Duffing oscillator and therefore gives a more realistic idea of what can be expected from a test with a control-based method.

    

 §.§ Experimental setup


        A thin rectangular steel plate of dimensions 500 mm × 300 mm × 0.5 mm and clamped on its shortest edges was considered as the next experimental setup. The plate was excited with an electrodynamic shaker (TIRA TV 51075) in current mode, and the response of the plate was monitored with an impedance head (DYTRAN 5860B) as well as a laser vibrometer (Polytec NLV-2500-5). In addition to these measurements from the structure, the voltage across the electrodes of the shaker was also monitored.
        
         

        Figure <ref> shows pictures of the experimental clamped plate. The clamping device was realized with two cantilever arms fixed on an optical table. The plate was then fixed to the arms with bolts, nuts and wide washers to distribute the load as evenly as possible on the clamped side. The cantilever arms were observed to be rigid enough to provide an approximate clamping condition for the bending dynamics of the plate. However, they were not rigid enough with regards to the plate axial stiffness, and a bar was added on top of the plate to stiffen the assembly, allowing for the observation of clear nonlinear midplane stretching effects. Details about the dimensions of the setup are given in the supplementary materials.

        Compared with the electronic Duffing oscillator, several new challenges arise. The most problematic one is the occurrence of shaker-structure interactions. These interactions have two main consequences when a harmonic drive signal is fed to the amplifier of the shaker: (i) the fundamental forcing harmonic amplitude is frequency- and amplitude-dependent and (ii) harmonics appear in the force. These issues were overlooked in this work by considering the signal fed to the amplifier, V_in (proportional to the current in the shaker), as the input to the system (as opposed to the force measured by the impedance head). The price to pay for this convenience is that shaker-structure interactions can then substantially dampen the structural modes if V_in is considered as the input. The shaker was thus placed following a trade-off between conflicting requirements: on the one hand, it should be placed on a responsive location of the targeted modes to excite them well, but on the other hand it should be placed close to the clamping to minimize shaker-structure interactions.

    

 §.§ Linear modal analysis


        A low-level pseudorandom excitation signal was applied to the plate to compute the FRFs when the system behaves linearly. The PolyMAX modal identification method <cit.> was then used to extract the linear modal characteristics of the first nine modes (below 100 Hz). The characteristics of the first five modes, which will be tested in nonlinear regimes of motion, are gathered in Table <ref>.

                

        

        
        

        Figures <ref> and <ref> show the two FRFs related to the NFRs of interest in the sequel, namely the shaker voltage and laser velocity.

        
        

        Figure <ref> shows the FRF of the force applied to the structure, as measured by the impedance head. Clear shaker-structure interactions are visible, as the FRF is far from being constant. This is due to the high flexibility of the plate combined with the relatively high mass of the impedance head <cit.>. Nevertheless, the shaker does not appear to dampen the modes of interest excessively. Controlling the force would prove at best extremely challenging anyway, because this FRF has a nonminimum phase zero (a pair of complex conjugate zeros with positive real part) around 137 Hz, which is very likely to destabilize any attempt of linear feedback control <cit.>.

        The dynamics of the plate in higher-frequency regions feature a high modal density. While this might not be a limitation for several methods, preliminary SWS tests showed that, in a nonlinear regime of motion, extremely complex phenomena took place therein, with the appearance of internal and combination resonances. Both these phenomena are generally associated with the appearance of quasiperiodic oscillations, that none of the state-of-the-art control-based method can handle. This high frequency range was thus not investigated further in this work.

    

 §.§ Implementation details


        The CBC requires feedback control similar to direct velocity feedback. As is well known in the field of linear control of flexible structures, collocation of the applied forcing and the feedback signal is almost mandatory to hope for a stable closed-loop system <cit.>. Loosely speaking, collocation allows for the feedback controller to work with power conjugate quantities, thereby controlling the power flow out of the structure. In particular, the control law can be designed to avoid power input to the structure which could lead to closed-loop instabilities. In most works about the CBC, one-degree-of-freedom-like structures were considered, thereby dismissing this question. For the plate, flexibility effects bear a strong importance. To allow for collocated control, one possibility is to measure the motion of the moving part of the shaker <cit.>. Alternatively, the strategy used in this work consists in sensing the voltage across the electrodes of the shaker (which is measured by the shaker amplifier). Since the current and voltage are power conjugates, a truly collocated control is achieved. More specifically, this is confirmed in Figure <ref> , where the phase bounded between ± 90^∘ ensures an interlacing pole-zero pattern, typical of a collocated sensor-actuator pair <cit.>. A first-order, unit-gain low-pass filter with a cut-off frequency at 500 Hz was nevertheless used on the voltage signal to avoid the excitation of high-frequency dynamics and the undesirable effects associated with digital control.

        For the PLL and RCT, such a collocation is not necessary. Since the phase evolution around the first considered mode shows better monotonicity for the velocity signal (cf. Figure <ref>), the latter was chosen as the feedback signal for PLL.

        Finally, the CBC-FD and RCT methods were unsuccessful. Concerning the former, this is probably due to the stronger noise in this setup. This lead the method to diverge quite far away from the NFR at the start of the test at low amplitudes. Since this behavior was worrisome near resonance, the CBC-FD was not tested in this case. Better results can be hoped for if noise is handled properly <cit.>, but this is beyond the scope of the present work. The RCT method also had stability issues similar to those encountered for the electronic Duffing. However, since these systematically lead to an overload, the RCT could not be performed fully to guarantee the structural integrity of the plate.

        The tested methods, namely SCBC, PLL and ACBC, were compared once again with a SWS for reference. All parameters are given in the supplementary materials.

        

 §.§ Broadband NFR


        We start off the nonlinear analysis with the aim to obtain an NFR encompassing multiple resonant modes by investigating the frequency range between 10 and 60 Hz. In this case, several of the control-based methods are impractical or even inapplicable. The PLL is inherently hampered by the fact that the phase is non-monotonous (cf. Figures <ref> and <ref>). In addition, the phase sensitivity away from resonance is rather small, indicating that phase parametrization is inadequate, and phase noise may cause large frequency noise. As for SCBC and RCT, their main drawback is that they require a certain response amplitude at all frequencies. Near antiresonances, reaching such an amplitude would require unreasonably large forcing levels. Furthermore, since the frequency band is quite large, the structure would have to be excited at maximum response amplitude for a long time, which is obviously undesirable. Hence, while these three methods are suitable to measure the response of a structure around its resonance, they are unsuited for broadband NFR testing. This could nevertheless be different for the two latter if an adapted sampling procedure was chosen instead of a regular grid in the (ω, a_*) space.

        

        

        Using the ACBC for different levels of excitation, one can obtain a global picture of the plate dynamics. Figures <ref> and <ref> feature the voltage and velocity NFRs, respectively, for five different drive amplitudes. Although the test started at 10 Hz, the frequency range shown therein starts at 20 Hz because the first mode is barely visible and barely excited by the shaker (as seen in Figure <ref>), thus remaining linear and being rather uninteresting. Each of these NFRs were obtained under 8 minutes of testing.
 

        

        The ACBC method was tested at the highest level of excitation (V_in = 0.05 V) against SWS for reference. Figure <ref> presents the voltage NFR. The second and third modes are hardening, while the fifth mode is initially softening, but seems to start its hardening phase. The softening trend may come from a small but inevitable initial curvature of the plate. The results agree almost perfectly with SWS, except for the inevitable hysteresis, small transients near the jump phenomena, and for the fourth mode. The ACBC sometimes had trouble converging in the vicinity of the fourth mode. The issue is thought to come from its poor observability and controllability (this mode being almost absent from the considered driving-point NFR at a linear level, cf. Figure <ref>). Superharmonic resonances are appearing throughout the test. They are strong enough to be noticed as peaks in the NFR, but not enough to form loops; hence the ACBC method was able to characterize them.

        

        Figure <ref> shows the velocity NFR at the laser measurement point. Mode 5 is more visible therein, as well as the superharmonic resonances.

        All modes feature an interesting and strongly nonlinear behavior at the highest considered forcing amplitude. To close this analysis, we investigate the dynamics of the second mode more in depth. It should be noted that since the frequency of mode 5 is about twice that of mode 2, the structure is in near-internal resonance conditions, resulting in an unconventional shape of the resonance peak.

    

 §.§ Second mode


          

        We now focus on a single resonance, the second flexible mode of the assembly. Tests that were performed for the Duffing oscillator are repeated for the considered methods, namely SWS, SCBC, ACBC and PLL. All tests were performed on the same day within a time span of 3 h 30 min, and Table <ref> provides the time taken by each method (NB: as in the Duffing case, the times for SWS and PLL were chosen based on the fastest method). Figure <ref> presents the results for the voltage NFR. All methods agree relatively well away from resonance, and in particular along the unstable branch (except of course for SWS), but small discrepancies are visible near resonance. These discrepancies are more marked for the velocity NFR, as illustrated by Figure <ref>, where two peaks are generally visible.

        

        

        Adaptive filters provide means to analyze the reason for the presence of a double peak in the NFR. As Figure <ref> reveals, the second harmonic becomes more prominent around 26.5 Hz, which is slightly more than half the linear resonance frequency of mode 5. A 2:1 superharmonic resonance of mode 5 thus occurs right before the primary resonance of mode 2, explaining the appearance of the second peak.

        
        
        In addition, Figure <ref> shows the evolution of the phase of the first harmonic for the voltage and velocity NFRs obtained with the ACBC. Similarly to the FRFs (Figures <ref> and <ref>), the voltage NFR does not feature a monotonous phase decrease around the resonance, whereas the velocity NFR shows this trend overall. Moreover, the 2:1 resonance manifests itself through a small phase trough, which is responsible for a small jump of the PLL method. With attention to detail, this jump can be observed in Figure <ref>, approximately between 26.5 and 27 Hz, wherein the PLL curve has less resolution.

          

        To assess the source of discrepancies, ten repeatability tests were performed for the ACBC and PLL. Statistics on the point cloud formed by the measurement were then obtained by fitting it with a spline with eleven control points and Gaussian process regressions <cit.>, as explained in the supplementary materials. Figure <ref> presents the results for the voltage NFR. Except for an outlier for the ACBC, both types of test yield relatively repeatable results, although the differences between test partially explain the discrepancies observed in Figure <ref>. The remaining reasons may come from the repeated tests at high amplitude (it can be deduced from Figure <ref> that the plate undergoes a displacement amplitude of more than twice its thickness at the laser measurement point at resonance) over a relatively long time span. Indeed, there was at least 30 min of heavy testing in between each curve in Figure <ref>. It is also interesting to note that the averaging process makes the small peak associated with the superharmonic resonance appear more clearly in both cases.

        
        



§ CONCLUSION



With the growing need to exhaustively test nonlinear systems, experimental continuation approaches are emerging as robust and efficient means to obtain bifurcation diagrams experimentally, encompassing the open-loop unstable responses of the system under test. By reviewing their essential ingredients, namely continuation and control, this article presented the main variants in experimental continuation in a unified manner. In particular, a common framework for the derivative-free experimental continuation approaches which are usually presented in an individual way, namely SCBC, PLL and RCT, was given. It also objectively compared these different methods with an electronic Duffing oscillator and a thin clamped plate. The main conclusions coming from this work and the literature are outlined below, summarizing the strengths and weaknesses of the different methods in their current form.

SWS is typically the first type of test that one tries on a system. In spite of its inherent deficiencies, the fact that this method is open-loop makes it vastly simpler than the other methods presented in this work. It is an excellent approach to obtain preliminary results and to have an idea of the orders of magnitude involved with the system under test.

CBC offers the most mature and comprehensive suite for experimental continuation. With its numerous variants, it is possible to trade between simplicity and advanced nonlinear testing capabilities. The controller being  separated from the continuation algorithm, it is possible to have an intuitive understanding of its effects. Yet, CBC can be the most complex of the approaches outlined herein.  

PLL offers a quick and easy testing solution with a simple architecture. It is particularly effective and fast for testing around a specific resonance, obtaining the NFR close to this resonance, or tracking its backbone. Yet, tuning its parameters can sometimes prove difficult for the unstable parts of the NFR, and the fact that the phase folds as soon as there are multiple resonances (including secondary ones) inherently limits this approach for broadband testing. 

The RCT is an attractive method for the possibility to implement it with commercial software, and the simple postprocessing associated with tests made at constant vibration amplitude. There remains yet to be formal proofs of its stabilization properties (or the conditions for them), as examples herein showed that this is not guaranteed.

Finally, we look back at the proposed ACBC procedure. The fact that it is based on arclength continuation and derivative-free makes it a powerful and general tool, allowing the experimenter to obtain complex bifurcation diagrams, as featured in the plate example. Nevertheless, limitations were also highlighted, e.g., when the bifurcation diagram features a loop, or when adaptive filters fail to secure non-invasiveness. Future works will address these issues.

Since the pioneering work of Sieber and Krauskopf <cit.>, substantial progress has been made over the last twenty years in experimental continuation, but great challenges also lie ahead. Some of them are listed below:

    
  * Gain selection and adaptivity: In almost all works, the control parameters are selected manually, often on a trial and error basis. A few theoretical studies have been made on a single-degree-of-freedom Duffing oscillator <cit.>, and a general but time-consuming parameter space exploration procedure was proposed in <cit.>. More systematic procedures should be developed to enable the use of control-based methods for non-expert users. We note that recent works proposed to use adaptive procedures to control the system with solid theoretical properties <cit.>. These methods have stringent requirements for an experimental implementation, such as full-state feedback, but are very promising and represent a significant step forward. Alternatively, practical ways of setting the parameters of a PLL with limited information from the plant were proposed in <cit.>. Finally, the data-driven approach based on parameter sweeps recently proposed in <cit.> could form another interesting solution.
    
  * Stability and bifurcations: When a test is successfully carried out, the closed-loop stability of every equilibria is ascertained. However, since the system of interest is the open-loop one, it is interesting to know whether the measured equilibria are open-loop stable or not. Procedures based on temporarily shutting the control down <cit.> or on real-time linearization <cit.> were proposed for the CBC, and similar procedures could be developed for the other control-based approaches. Going beyond that, different bifurcations could also be detected <cit.> and even tracked in an experimental setting, as in the pioneering works of Renson et al <cit.> and Shen et al <cit.> (in the field of statics).  
    
  * Isolated solutions: From a numerical standpoint, isolated solutions can be challenging to find with traditional continuation tools, but may be reachable in experiments. As a few works demonstrated <cit.>, control-based methods could offer robust tools to trace out a complete isola, but reaching or even detecting them in a systematic way still represents a challenge.
    
  * Aperiodic responses: None of the control-based method is currently able to deal with aperiodic (quasiperiodic or chaotic) responses, even though they may generally appear in nonlinear systems. As stated in Section <ref>, it is possible to stabilize unstable periodic orbits embedded in a chaotic attractor <cit.>, but, to the authors' knowledge, little work has been done to stabilize an aperiodic attractor itself.
    
  * Actuator-system interaction: In most cases, a structure will not be perfectly actuated. This problem pertains to the breadth of nonlinear vibration testing problems <cit.> and not just to control-based methods. Although some attempts have been made <cit.>, no ideal solution currently exists in the literature.
    
  * Large-scale systems: Except a few notable exceptions, control-based methods were almost always tested on systems of relatively low dimensionality, being either single-degree-of-freedom-like, or focusing on a single resonance widely spaced from the others. To achieve their full potential and provide convincing results for a future industrial use, control-based methods should turn toward more challenging, large-scale systems. The use of multiple sensors and/or actuators and the study of underactuated problems also remain largely uncovered.






















































































































































































































§ STATEMENTS AND DECLARATIONS






  * Funding: Gaëtan Abeloos and Ghislain Raze are a FRIA Grantee and a Postdoctoral Researcher, respectively, of the Fonds de la Recherche
Scientifique - FNRS, which is gratefully acknowledged.

  * Competing interests: The authors have no relevant financial or non-financial interests to disclose.




  * Author Contributions: 
Ghislain Raze: literature search, formal analysis, data analysis, investigation and writing - original draft preparation. Gaëtan Abeloos: conceptualization, methodology and writing - review and editing. Gaëtan Kerschen: literature search, funding acquisition, supervision and writing - review and editing.


  * Data Availability:  The data collected and used
in this research is available in <cit.>.




 Paper Title: Enumerating Finite Braid Group Orbits on $SL_2(\C)$-Character Varieties.







 









cd


  




















definition
*definition*Definition
definitionDefinition
*theorem*Theorem

theorem[definition]
Theorem
lemma[definition]
Lemma
corollary[definition]
Corollary
proposition[definition]
Proposition
*proposition*Proposition

example[definition]Example

definitionsubsection

remark
remark[definition]Remark
*remark*Remark

claim
claim[definition]Claim
*claim*Claim

 


solution


	
	
	Enumerating Finite Braid Group Orbits on SL_2()-Character Varieties
    Amal Vayalinkal
    August 5, 2024
===================================================================

§ ABSTRACT 
		We analyze finite orbits of the natural braid group action on the character variety of the n times punctured sphere. Building on recent results relating middle convolution and finite complex reflection groups, our work implements Katz's middle convolution to explicitly classify finite orbits in the SL_2()-character variety of the punctured sphere. We provide theoretical results on the existence of finite orbits arising from the imprimitive finite complex reflection groups and formulas for constructing such examples when they exist. In the primitive finite complex reflection groups, we perform an exhaustive search and provide computational results. Our contributions also include Magma computer code for middle convolution and for computing the orbit under this action when it is known to be finite. 
	

	

§ INTRODUCTION

	Dynamics of the natural braid group action on character varieties has been studied for over a century. The growing interest in classifying the finite orbits of this action has led to a wealth of new techniques and examples. Inspired by Boalch <cit.> who explored finite complex reflection groups as a source of finite orbits, Lam-Landesman-Litt <cit.> showed, up to mild conditions, all finite orbits come from finite complex reflection groups via Katz's middle convolution operation <cit.>. With the goal of completing their analysis, we search the finite complex reflection groups to enumerate all possibilities and study them. We classify all finite orbits arising via middle convolution from the finite complex reflection groups. 
	
	Let Σ_0, T+1 denote the T+1-punctured sphere. Given T and n, a rank n representation of the fundamental group of Σ_0, T+1 can be identified with a tuple of matrices (A_1, … A_T+1) ∈ GL_n()^T+1 satisfying 
    ∏_i = 1^T+1 A_i = Id

	
	The mapping class group of Σ_0, T+1, denoted  _0, T+1 has a well-defined action on such representations. It is well known that _0, T+1 is isomorphic to the spherical braid group on T+1 strands, denoted B_T+1.  This group has T generators σ_1,  …, σ_T and acts on T+1-tuples in the following way: 
	
    σ_i(A_1, A_2, …, A_T, A_T+1) = (A_1, …, A_i-1, A_iA_i+1A_i^-1, A_i, A_i+2, …, A_T+1)
	
	If we further consider such representations up to simultaneous conjugation, this action descends to an action on the character variety i.e. on the isomorphism classes of representations.
	
		Conjugacy classes of tuples [A_1, ⋯, A_T+1] with finite orbit under this action are called MCG-finite and we say a tuple is irreducible if the associated representation is irreducible.
	

		Recently, Lam-Landesman-Litt <cit.> classified finite orbits of rank two tuples with at least one matrix of infinite order.  They proved that such Zariski dense representations come from pullbacks of curves or arise via middle convolution from finite complex reflection groups.
	
	The first kind, “pullback" representations were classified by Diarra <cit.>. 
	
	Finite irreducible complex reflection groups, which can be viewed as generalizations of Coxeter groups, were classified by Shephard and Todd in 1954 as belonging to an infinite family G(m, p, n) for each rank n or being one of 34 exceptional cases denoted G_3 to G_37 [9]. In view of this classification, Lam-Landesman-Litt <cit.> prove that for T+1 ≥ 7, MCG-finite representations do not come from groups of rank five or greater. Their paper leaves open for analysis the infinite family of rank three and four and most of the exceptional groups of rank three and four, as well as the case when T + 1 < 7, which we investigate.
	
	Katz <cit.> originally introduced middle convolution in the language of perverse sheaves and one of our contributions is implementing computer code to explicitly compute the middle convolution given a tuple of matrices, following the algebraic reformulation of Dettweiler-Reiter <cit.>. We also provide code for computing the braid group orbit for MCG-finite 4-tuples and 5-tuples in the  SL_2()-character variety of the punctured sphere. This computer code is publicly accessible online at the following link:  <amalvayalin.github.io/research.html> and we hope it encourages further applications of this operation.
	
	
		A 𝐧𝐢𝐜𝐞 T-tuple is a tuple (up to simultaneous conjugation) from a finite complex reflection group such that its middle convolution produces a rank 2 MCG-finite T+1-tuple. 
	
	For the precise definition of when a tuple is nice, see Definition <ref>. An explanation for how we get MCG-finite T+1-tuples from a T-tuple is provided in Remark <ref>. Since middle convolution depends on a parameter λ (see Section <ref>), the same nice T-tuple can give rise to distinct conjugacy classes of rank 2 MCG-finite T+1-tuples, as discussed in Remark <ref>.
	
	With this definition in mind, our problem of studying rank 2 MCG-finite tuples changes to studying nice tuples in the finite complex reflection groups. 
	Using properties of complex reflection groups, as well as prior work on this topic, it remains to study nice n and n+1-tuples in the rank n = 3, 4 primitive groups, nice n+1-tuples in the imprimitive groups of rank 2 <n ≤ 5 and  nice n-tuples in imprimitive groups of rank n ≥ 3. More details on why these are the only remaining cases is provided in Remark <ref>, Remark <ref> and <cit.>.
	
	
	In the imprimitive groups, we either prove no such nice tuples exist, or provide a system of equations for an infinite family of solutions. We found nice tuples in all the primitive reflection groups of rank three. Among the rank four primitive reflection groups, all but G_29 and G_31 contain nice tuples. For each nice tuple in the primitive groups, we provide representatives (up to simultaneous conjugation and braid group orbit equivalence) before and after middle convolution. We also list the size of the subgroup generated by the rank 2 matrices, and in most cases we provide an approximate `size' for the braid group orbit. 
	 
		There are no nice n-tuples in G(m, p, n) for n ≥ 5. 
	
	
		In the rank 3 imprimitive groups, an infinite family of nice 3-tuples exist in G(m, 1, 3) and G(m, m, 3) and in each case we provide conditions on the eigenvalues of the product of the matrices in the tuple. Nice 4-tuples exist in G(m, p, 3) for m >1 if and only if one of the following holds: 
		
			
  * p = m and m ≠ 3. 
			
  * p = 1 and (m, 3) = 1.
			
  * p ≠ 1, m and p = 3. 
		
		In all three cases the parameter for middle convolution is a root of unity of order m.
	
	 
		In the rank 4 imprimitive groups, nice 4-tuples and nice 5-tuples exist in G(m, p, 4) if and only if p = m and m =2 or m = 4. 
		
	The proof of these theorems, as well as further conditions on the eigenvalues of the product of the matrices and equations for how to construct such examples are provided in Section <ref>.
	
		In our computations for the primitive groups, nice tuples within each group are partitioned into different 𝐭𝐲𝐩𝐞𝐬 based on eigenvalues (see Definition <ref> and Remark <ref>). 
	 
		For each primitive reflection group of rank three, we summarize the distinct rank 2 MCG-finite tuples (up to simultaneous conjugation and braid group equivalence) produced by taking middle convolution of the different nice tuples (up to type equivalence). We also note how many of the rank 2 tuples generate an infinite subgroup. 
		
			
  * In G_23, there are three nice 3-tuples, resulting in six distinct MCG-finite 4-tuples after middle convolution. Three of the six generate infinite subgroups. This group also has one nice 4-tuple, which after middle convolution produces one MCG-finite 5-tuple that generates an infinite subgroup. 
			
  * In G_24,  there is only one nice 3-tuple, which produced three distinct MCG-finite 4-tuples after middle convolution, all generating infinite subgroups. There does not exist any nice 4-tuples in this group. 
			
  * In G_25, there are four nice 3-tuples, resulting in eight distinct MCG-finite 4-tuples after middle convolution. Five of these generate infinite subgroups. This group also has three nice 4-tuples, which produced two distinct MCG-finite 4-tuples after middle convolution. Only one of these 4-tuples generates an infinite subgroup. 
			
  * In G_26, there are three nice 3-tuples, resulting in seven distinct MCG-finite 4-tuples after middle convolution. Five of these generate an infinite subgroup. In this group there are no nice 4-tuples. 
			
  * In G_27, there are three nice 3-tuples, resulting in nine distinct MCG-finite 4-tuples after middle convolution. Six of these generate an infinite subgroup.
			This group also has one nice 4-tuple, which produced one MCG-finite 5-tuple after middle convolution. This 5-tuple does not generate an infinite subgroup.
		
	
	
	 
		For each primitive reflection group of rank four, we summarize the distinct rank 2 MCG-finite tuples (up to simultaneous conjugation and braid group equivalence) produced by taking middle convolution of the different nice tuples (up to type equivalence). Only nice tuples in G_32 produces an MCG-finite tuple that generates an infinite subgroup after middle convolution. 
		
			
  * In G_28, there is only one nice 4-tuple, which produces one MCG-finite 5-tuple after middle convolution. 
			
  * In G_29 there are no nice 4-tuples nor any nice 5-tuples. 
			
  * In G_30, there are three nice 4-tuples, each producing one distinct MCG-finite 5-tuple after middle convolution. For each of these 5-tuples, after tensoring with a character so the matrices are in SL_2( ) we end up with one matrix being identity, which means these tuples reduce to MCG-finite 4-tuples instead. There are no nice 5-tuples in this group. 
			
  * In G_31 there are no nice 4-tuples nor any nice 5-tuples.
			
  * In G_32 there are three nice 4-tuples, resulting in four distinct MCG-finite 5-tuples after middle convolution. One of these 5-tuples generates an infinite subgroup. After tensoring with a character so the matrices are in SL_2(), for two of the remaining three 5-tuples, we get the identity matrix, so these reduce to MCG-finite 4-tuples instead. This group also contains at least two nice 5-tuples, producing at least one example of an MCG-finite 6-tuple.  The 6-tuple did not generate an infinite subgroup. The search for nice 5-tuples in this group was too large computationally, so this list may be incomplete. 
					
		
	

 §.§ Prior Work

	MCG-finite representations correspond to algebraic solutions to nonlinear differential equatons, such as the Painlevé VI equation and more generally the  Schlesinger equations <cit.>.
	Regarded as a nonlinear analogue of the hypergeometric equation, the Painlevé VI (PVI) equation posed over 100 years ago, has inspired much work on the study of finite orbits. Algebraic solutions to the PVI equation correspond to irreducible MCG-finite representations when n = 2 and T+1 = 4 and motivates progress towards a nonlinear Schwartz list <cit.>. Among the many advances made in this area, notable contributions on finite orbits come from Hitchin <cit.>, Dubrovin-Mazzocco <cit.>, Boalch <cit.>, Kitaev <cit.> and an eventual computer-aided classification by Lisovy-Tykhyy <cit.>. 
	
	The work of Dubrovin-Mazzocco originally studied the PVI equation in connection with reflection groups <cit.>. Expanding on this idea,  solutions constructed from finite reflection groups through special cases of middle convolution, like the “Okamoto transform,"  were explored by others in <cit.>, <cit.>, <cit.>. More generally, a computer-aided classification was provided by Tykhyy in <cit.>, with explicit orbits listed when T+1 = 4, 5 and conjectural descriptions of the different possibilities for T+1 ≥ 6.  We found exact matches for orbits given in this list, except in two cases where the closest match differs in just one choice of parameter. As both these examples almost match with the same orbit they provided, we suspect this difference might be due to a typo. Details on these two tuples are provided in Section <ref> and Section <ref> and the differing coordinates are marked with an asterisk in the tables.
	
	The connection between finite complex reflection groups and middle convolutions was further explored more recently in the work of Lam-Landesman-Litt <cit.> who proved that all MCG-finite tuples with at least one matrix of infinite order are either pullback representations (classified in <cit.>) or come from finite complex reflection groups via middle convolution. 
	
	
	Our work investigates the finite complex reflection groups for MCG-finite tuples and examines their middle convolution in relation to the braid group orbit. In the applicable cases, we also explore how the examples we find relate to the classifications provided by Lisovy-Tykhyy <cit.> and Tykhyy <cit.>. Note that the analysis in these two papers exclude the tuples where all the matrices share a common eigenvector; these correspond to non-irreducible cases and were classified by Cousin-Moussard <cit.>.
	
	

§ BACKGROUND

	

 §.§ Middle Convolution
 
	Defined by Katz <cit.>, middle convolution  is an invertible functor in the category of perverse sheaves depending on some parameter λ∈^× and is denoted MC_λ. Middle convolution can also be viewed as a functor in the category of local systems on the punctured sphere to itself, and with this interpretation, Katz proved that all irreducible rigid local systems on the punctured sphere can be constructed from rank 1 local systems via scaling or middle convolution.  
	
	Letting k be a field and F_T denote the free group on T generators, middle convolution can equivalently be understood as a functor on the category of finite dimensional k[F_T] modules to itself as described by Dettweiler-Reiter <cit.>. The relation to local systems on the T+1-punctured sphere is by taking k = and viewing a tuple of T matrices as defining the action of [F_T] on the module i.e. a representation of [F_T]. In this way irreducible tuples correspond to irreducible representations.
	While middle convolution preserves irreducibility <cit.>, it typically changes the dimension of the module (rank of the local system on T+1-punctured sphere) and does not preserve basic properties e.g. finiteness of the subgroup generated by the matrices (i.e. monodromy group). Important to the study of finite orbits, one desirable property of middle convolution is that it is equivariant with the action of the braid group B_T <cit.>. One way to construct nontrivial finite orbits is to take irreducible tuples generating a finite subgroup, hence finite braid group orbit, and taking the middle convolution to (hopefully) get a tuple generating an infinite group which will necessarily be irreducible and have finite braid group orbit. 
	
	The purely algebraic analogue by Dettweiler-Reiter <cit.> gives an explicit description of middle convolution in terms of tuples of matrices and relates some properties of middle convolution to the matrices in the tuple.
	
		Given a tuple of T matrices A = [A_1, A_2, …, A_T] with A_i ∈ GL_n(), the 𝐩𝐫𝐨𝐝𝐮𝐜𝐭 of such a tuple is the product of the matrices in the tuple, in the ordering given by the tuple. So the product of the tuple 𝐀 is A_1A_2⋯ A_T. The 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 of the tuple A is the tuple [A_T^-1, …, A_1^-1], denoted 𝐀^-1. The 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐩𝐫𝐨𝐝𝐮𝐜𝐭 of a tuple is the matrix product of the inverse tuple.
	
	
		Given a tuple of T matrices A = [A_1, A_2, …, A_T] with A_i ∈ GL_n(), we denote the middle convolution of 𝐀 with parameter λ∈^× as MC_λ(A) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃]. Note that MC_λ(A) will again be a tuple of T matrices, well-defined up to simultaneous conjugation, with Ã_̃ĩ∈ GL_n'() for some n' not necessarily equal to n. 
	
	
	
		Middle convolution does not preserve the product of the matrices as it is only well-defined up to simultaneous conjugation. 
	
	
	Letting V denote the underlying vector space associated to the [F_T] module defined by a tuple 𝐀. Then the relation between n and n' is as follows. 	
	
		Let V denote the [F_T]-module defined by the tuple A = [A_1, A_2, …, A_T] and let λ∈^×. 
		
			
  * If λ≠ 1 then 
    (MC_λ(V)) = ∑_i = 1^T(A_i -I) - ((V) - (λ· A_1A_2 ⋯ A_T -I))

		
	 
	
		Proofs of this statement, as well as further details and properties of the middle convolution can be found in <cit.> and <cit.>. 
	
	To get an MCG-finite T+1-tuple of rank 2, we want to find a T-tuple of reflections (hence (A_i -I) = 1) such that their product has a nontrivial eigenvalue of multiplicity T-2. This is why we search for nice tuples, Definition <ref> in Section <ref>.

	Another property of middle convolution that will greatly simplify our computations is the following. 
	 (Dettweiler and Reiter, 2000)
		If 𝐀 is an irreduble tuple of rank greater than 1, then the modules MC_λ^-1(𝐀^-1) and MC_λ(𝐀)^-1 are isomorphic. 
	
	
		Proved in <cit.>.
	
	
	Our code for middle convolution uses the formulation given by Dettweiler-Reiter in <cit.> as opposed to their formulation in <cit.> which only differs by changing the order of blocks in the matrices. The code requires a matrix group W containing the matrices in the tuple 𝐀 and the base ring of W should contain the parameter λ used for middle convolution. Note that the final function Compute_MC will automatically check if our intermediate constructions act invariantly on the subspaces K and L, as defined in the notation of <cit.>. The full computer code can be found online at  <amalvayalin.github.io/research.html>.
	
	

 §.§ Finite Complex Reflection Groups
 
	A real reflection can be identified as a matrix with one eigenvalue -1 and all other eigenvalues are 1, and pseudo-reflections refers to matrices with one eigenvalue a root of unity and all other eigenvalues 1.  A finite complex reflection group (FCRG) is a finite group generated by pseudo-reflections. 
	As we are only working with complex groups, the distinction between real or complex eigenvalues is not relevant and henceforth reflections can refer to real reflections or pseudo-reflections.
	
	All irreducible finite complex reflection groups  were classified by Shephard-Todd <cit.>, as belonging to an infinite family of imprimitive reflection groups or being one of 34 exceptional primitive cases. 
	 
		The rank of a reflection group is the smallest dimension of the vector space on which the group acts irreducibly.
		Every finite complex reflection group of rank n has a minimal generating set consisting of either n or n+1 reflections. The group is said to be well-generated if it can be generated by n reflections. 
	
	As we want irreducible MCG-finite T+1-tuples, we know that our nice T-tuple (see Defintion <ref>) must at least be of length n to generate an irreducible FCRG, so in each group we only need to search for T-tuples with T > n. Among the rank three and four primitive irreducible groups, only G_31 is not well-generated (see Cartan matrices in <cit.>. An imprimitive group G(m, p, n) is well-generated if p = m or p = 1. 
	
	Associated to each FCRG of rank n is a set of n invariants called the degrees of the reflection group <cit.>, <cit.>. An important property of these invariants relevant for our purposes is their relation to the eigenvalues of the group elements. Note that we already know the eigenvalues for any element of the group must be roots of unity since the group is finite. 
	
		Let W be a FCRG of rank n and let {d_i}_i = 1, …, n denote the degrees of W. Suppose λ is some primitive d-th root of unity that is an eigenvalue for some element of W. Then, d | d_i for some degree d_i. Moreover, the maximum dimension of the eigenspace corresponding to λ for any element of the group is the number of degrees d_i that is divisible by d, and for some element in W the eigenspace corresponding to λ achieves this maximal dimension. 
	
	
		Proved in <cit.>.
	
	The theorem above will be helpful in our computations for the primitive groups as we know how large our field needs to be to ensure the code finds all eigenvalues. 
	
	Further properties specific to imprimitive reflection groups are described in Section <ref>.
	
	

 §.§ Representations and Character Varieties

	
	Given a finitely generated group Γ, a representations of Γ is a homomorphism ρ: Γ⟶ G where G < GL_n() is some algebraic group. Considering the vector space V the matrices ρ(γ) act on, ρ is said to be irreducible if there is no nonzero subspace U ⊊ V that is left invariant by the matrix ρ(γ) for all γ∈Γ. Two representations ρ and ρ' are isomorphic if there exists some g ∈ G such that ρ(γ) = g^-1ρ'(γ) g for all γ∈Γ. 	The character of a representation  ρ is the map χ_ρ: Γ⟶ given by χ_ρ(γ) = ρ(γ). Isomorphic representations have the same character since trace is fixed under conjugation. The character variety 𝔛(Γ, G) parametrizes the isomorphism classes of representations ρ: Γ⟶ G in terms of suitable traces.  
	
	In the case of the punctured sphere Σ_0, T+1, the fundamental group is π_1(Σ_0, T+1) = γ_1, …, γ_T+1|γ_1*γ_2*⋯ *γ_T+1 = 1. Any representation of Γ = π_1(Σ_0, T+1) is identified by ρ(γ_i) = M_i ∈ G such that M_1 · M_2 ⋯ M_T+1 = I is the identity matrix. Two such representations  ρ and ρ' are equivalent if the tuple [ρ(γ_1), …, ρ(γ_T+1)] is related to [ρ'(γ_1), …, ρ'(γ_T+1)] by simultaneous conjugation.
	
	Taking G = SL_2(), we can consider the SL_2()-character variety of π_1(Σ_0, T+1). 
	
	Given a tuple of of matrices [M_1, M_2, …, M_T+1], let 
    t_i = (M_i)      t_ij = (M_iM_j)      t_ijk = (M_iM_jM_k) *for 1≤ i < j< k ≤ T+1

	Since π_1(Σ_0, T+1) = F_T we have a natural inclusion 𝔛(π_1(Σ_0, T+1), SL_2()) ⊆𝔛(F_T, SL_2()). This means for our purposes, M_T+1 will always be (M_1M_2⋯ M_T)^-1 so t_T+1 = (M_1M_2⋯ M_T) as all the matrices M_i will be in SL_2() where matrices have the same trace as their inverse.
	
	
		Let [M_1, M_2, …, M_T+1] ∈ SL_2()^T+1 such that Π M_i = I. Then up to simultaneous conjugation, any such (T+1)-tuple can be identified by the following trace coordinates: 
		
			
  * For T = 3: (t_1, t_2, t_3, t_12, t_13, t_23, t_4) 
			
  * For T = 4: (t_1, t_2, t_3, t_4, t_12, t_13, t_14, t_23, t_24, t_34, t_123, t_124, t_134, t_234, t_5)
		
	
	
		For proof and further discussion of character varieties of the punctured sphere see <cit.> and <cit.>. 
	
	Such trace coordinates are also sometimes referred to as the 𝐬𝐢𝐠𝐧𝐚𝐭𝐮𝐫𝐞 of the tuple. 
	
	

§ METHODS
 
	For the rest of this written work, fix ζ_d = e^2π i/d as a primitive d-th root of unity. 
	
	
	Since the product of [A_1, ⋯, A_T+1] must be the identity, we know any MCG-finite tuple is determined by the first T matrices, up to simultaneous conjugation. As middle convolution does not preserve products (see Remark <ref>), our idea is to take middle convolution of the first T matrices to get 2× 2 matrices and then induce a representation of π_1(Σ_0, T+1) by taking the T+1-th matrix to be the inverse product. The condition for middle convolution being rank 2, (stated in Theorem <ref>) depends on the dimension of the subspace fixed by, or equivalently the multiplicity of the eigenvalue 1 of each matrix in the tuple and the eigenvalues of their product. For much better control on the former, we consider reflections since they only have one nontrivial eigenvalue, of multiplicity 1. 
	
	Thus, for each irreducible finite complex reflection group W of rank n, we want to find all irreducible T-tuples of reflections A = [A_1, A_2, …, A_T] in W such that for a suitable choice of parameter λ, MC_λ(A) has rank 2. Given T, the formula given in Theorem <ref> tells us that the product Π_i = 1^T A_i of reflections in A should have a nontrivial eigenvalue λ^-1 of multiplicity T-2 so that MC_λ(A) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] is a T-tuple of rank 2. We give a name to tuples satisfying these conditions.
	
		A T-tuple A = [A_1, A_2, …, A_T] of matrices from a finite complex reflection group W is called 𝐧𝐢𝐜𝐞 if A satisfies the following: 
		
			
  * All A_i are reflections in W
			
  * W = A_1, A_2, …, A_T
			
  * (A_1A_2 ⋯ A_T)^-1 has an eigenvalue λ≠ 1 of multiplicity T-2. 
		
	
	
	The condition of generating the whole group is to ensure that we are not “over-counting" as some reflection groups contain other reflection groups as subgroups, and we only want to study irreducible MCG-finite tuples. 
	
	
		Given a nice T-tuple 𝐀 = [A_1, A_2, …, A_T], we know there exists some λ≠ 1 such that MC_λ(𝐀) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] is a T-tuple of rank 2. taking the T+1-th matrix to be 
    A_T+1 = (Π_i = 1^T Ã_̃ĩ)^-1
 we get a T+1-tuple [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃, A_T+1] whose product is the identity. As 𝐀 comes from a finite group we know it has finite braid group orbit, and since middle convolution is equivariant with the braid group action (see Section <ref>), we know that [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃, A_T+1] must be MCG-finite. Moreover, if {A_1, A_2, …, A_T} generates an irreducible group, then as middle convolution preserves irreducibility, we know our T+1-tuple is irreducible as well. Thus, we have used a nice T-tuple to produce an irreducible MCG-finite T+1-tuple. 
	
	 
		For some nice T-tuples, it is possible that multiple choices of λ have multiplicity T-2 and hence the different choices of parameter λ gives rise to possibly different rank 2 MCG-finite T+1-tuples. 
	
	
	 
		The eigenvalue condition of multiplicity T-2 tells us that in any complex reflection group of rank n, for T > n+2 the middle convolution will not be rank 2 so there are no nice T-tuples.  By Definition <ref>, since we need at least n reflections to generate the whole group, we know there are no nice tuples when T < n. For T = n+2, the product of the T matrices would be a scalar matrix (as there must be an eigenvalue of multiplicity n) which we can take to be the identity, reducing to the T= n+1 case. 	So for each n ∈{3, 4}, we only search the primitive reflection groups for nice tuples of length T = n or T = n+1. 
	
	
		

 §.§ Computing The Braid Group Orbit

	The braid group can be presented as B_T+1 = σ_1, …, σ_T |σ_iσ_i+1σ_i = σ_i+1σ_iσ_i+1, σ_iσ_j = σ_jσ_i where the first relation holds for 1 ≤ i≤ T-3 and the second for i-j ≥ 2. A generator σ_i acts on T+1-tuples in the following way: 
	
    σ_i(M_1, M_2, …, M_T, M_T+1) = (M_1, …, M_i-1, M_iM_i+1M_i^-1, M_i, M_i+2, …, M_T+1)
	
	
	Let F_+ denote the set of all words in positive powers of these generators. Further, let Orb(B_T+1, x) denote the orbit of some tuple x under the action of B_T+1 and Orb(F_+, x) denote the set of images of x under the action of words in F_+. 
	
	
		For a tuple x of matrices, Orb(B_T+1, x) is finite if and only if Orb(F_+, x) is finite. 
	
	
		Since F_+ ⊂ B_T+1, one direction is obvious. For the other direction first note that for any fixed g ∈ B_T+1, if the set {g^i x : i ≥ 0} is finite, then we can rewrite g^-1x = g^n x for some n ≥ 0, so we get that {g^i x: i ∈} must be finite. Further, since the elements of B_T+1 act one at a time, it is enough to study how words in positive powers of the generators act on x. Thus, if Orb(F_+, x) is finite, then the braid group orbit must be finite as well. 
	
	
	To help with computing the braid group orbit, we need a way to capture the equivalence relation of tuples being related by simultaneous conjugation. Using character varieties, we can write down trace coordinates for our tuple, which we know how to do from Theorem <ref> when the matrices are in SL_2().
	
	
		Given a nice tuple 𝐀 = [A_1, A_2, …, A_T], let  MC_λ(𝐀) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] denote the middle convolution. If (a_1, a_2, …, a_T+1) ∈ (^×)^T+1 is such that a_i Ã_̃ĩ∈ SL_2() for i = 1, … T and a_T+1 ((Ã_̃1̃Ã_̃2̃⋯Ã_̃T̃)^-1) ∈ SL_2() with Π^T+1 a_i = 1, the tuple 
		
    Â = [a_1 Ã_̃1̃, a_2 Ã_̃2̃, …, a_T+1(Ã_̃1̃Ã_̃2̃⋯Ã_̃T̃)^-1]

		is said to be 𝐢𝐧𝐝𝐮𝐜𝐞𝐝 by MC_λ(𝐀) or by  𝐀 if the parameter λ is clear. Â is a tuple in (SL_2())^T+1 which multiplies to the identity.
	
	When we say trace coordinate of a tuple MC_λ(𝐀) we will always mean the trace coordinates of the induced tuple Â unless otherwise specified.
	
	We want to compute the orbit of induced tuples Â up to simultaneous conjugation. This is equivalent to computing the orbit of the point associated to Â in the character variety 𝔛(π_1(Σ_0, T+1), SL_2()). Note that since all of our tuples come from FCRG and middle convolution is equivariant with the action of the braid group we already know all the orbits are finite. 
	
	The code for the algorithm described below can be found online at <https://amalvayalin.github.io/research.html>.
	
	Given a tuple 𝐌 = [M_1, …, M_T+1] of SL_2()^T+1, such that their product is the identity, the braid group B_T+1 orbit is computed as follows. 
	
	We first compute the signature of 𝐌, and then apply each of the T generators to the tuple 𝐌. If σ_i(𝐌) has a new signature, then we store the tuple in To_Test and the signature in Signatures and add 𝐌 to Tested.  We then apply σ_i again to the (atmost T) new tuples in To_Test we got from the action on 𝐌, once again storing only tuples which have new signatures. We repeat this process until the sets To_Test and Tested are the same. The function returns the set Tested, which should contain only one representative per distinct signature, and can optionally return the set Signatures as well. 
	Note that here our definition of signature is the trace coordinates, which  differs from the definition used in Tykhyy's classification in terms of residues <cit.>.
	
	

 §.§ Finding Nice Tuples
 
	
	
	Nice T-tuples correspond to  MCG-finite T+1-tuples in the following way: given a nice tuple [A_1, …, A_T], we get a corresponding representation  [A_1, …, A_T, A_T+1] of the T+1-punctured sphere if we take A_T+1 = (A_1A_2 ⋯ A_T)^-1. 
	Thus, it is enough to search each complex reflection group for nice tuples.
	
	By Theorem <ref>, an eigenvalue of multiplicity T-2 for an element in W must be a d-th root of unity for d dividing at least T-2 degrees of W. 
	Any reflection group can be constructed in Magma by providing the generating reflections; the irreducible finite complex reflection groups can be easily constructed in Magma by providing the Shephard-Todd number or the corresponding parameters for G(m, p, r). In most cases Magma generates the group W via the reflection representation over the trace field of W <cit.>. 
	
	
	One issue is that not all characteristic polynomials will split over the trace field, as Magma computes eigenvalues for matrices over the field of entries. Since our search depends on the eigenvalues of the product of matrices in the tuple(s), we must change the base field of W to the smallest cyclotomic field containing all the possible eigenvalues. The largest this extension will be (ζ_d), where d is the least common multiple of all the degrees of W, but in certain cases smaller fields might suffice. For large d computation time is typically slower, but we have expanded the field to the LCM of the degrees unless otherwise stated. For example, in the case of W = G_32, the degrees are (W) = {12,18,24,30 } and so ((W)) = 360 which slows down computation time considerably. For this group, we prove in Section <ref> that it is enough to search for 4-tuples over (ζ_12) by showing any eigenvalue of multiplicity 2 must be a 12-th root of unity. 
	
	Before outlining the steps of our search, we define some terms. 
	
	 
		Given a tuple of matrices 𝐀 = [A_1, A_2, …, A_T], the 𝐭𝐲𝐩𝐞 of 𝐀 is determined by the eigenvalues of (A_1A_2 ⋯ A_T)^-1 counted with multiplicity. Tuples whose inverse product has the same eigenvalues with multiplicity are said to be of the same type.  An 𝐞𝐱𝐞𝐦𝐩𝐥𝐚𝐫 of a type is a choice of nice tuple of that type. Given a tuple of some type A, the type of the inverse tuple is called the 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐭𝐲𝐩𝐞 of A and the two types together make an 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐩𝐚𝐢𝐫. 
	

		The use of types allows us to partition a set of nice tuples based on the eigenvalues of the inverse product. Since there are many nice tuples of the same type, we choose exemplars for each type to perform the in-depth computations on. Note that the choice of exemplar for any type is arbitrary and it may be possible that different exemplars result in different orbits. To ensure this does not happen, we go back through all the tuples for each type and make sure that it belongs to the braid group orbit we compute for the middle convolution of the exemplar. More details on the braid group computations can be found in Section <ref>.  
	
	In some groups we get types that are inverse pairs. In these cases we only provide the computations for one exemplar from each pair. To ensure the inverse type would not produce a new orbit, we check that the inverse tuple of the middle convolution lies in the orbit of the middle convolution of the exemplar; this is enough by the relationship between inverse tuples and inverse parameters given in Theorem <ref>. 
	The code for how we checked each type is available at <https://amalvayalin.github.io/research.html>.
	 
		Aside from a few  exceptional cases that were computationally too large, we verified that for each choice of the parameter λ, the rank 2 MCG-finite tuples coming from nice tuples of the same type are in the same braid group orbit, as are exemplars from the inverse type. In this sense, types can be viewed as `almost' equivalence classes for nice tuples up to inverse pairs.
	
	
	After coercing the matrix group into a larger field if necessary, the general process for searching for T-tuples is as follows.
	
		
  * Fix a representative Q_i for each conjugacy class of reflections in W and fix an ordering [Q_1, ⋯, Q_n].
		
  * Given T, construct the list M of distinct sets of T reflections in W, using the ordering of Q to eliminate some tuples equivalent up to simultaneous conjugacy. 
		
  * Search M for all sets such the matrices generate W (means the tuple is irreducible) and the product of the matrices, for some order of multiplying, has an eigenvalue of multiplicity T-2. Each such set is then organized into a tuple based on the order of multiplication and stored in a list X of irreducible tuples. 
		
  * For each nice tuple x ∈ X, compute the eigenvalues of the inverse product of x and partition X into different types corresponding to the different sets of eigenvalues of the inverse product. 
		
  * Choose exemplars 𝐀, 𝐁... for each type and compute the middle convolution MC_λ_1(𝐀), MC_λ_2(𝐀), MC_λ(𝐁)... for suitable choice(s) of λ.
		
  * Compute the subgroup of GL_2() generated by MC_λ(𝐀), MC_λ(𝐁), …
		
  * For each middle convolution of the exemplars, find a suitable character [a_1, …, a_T+1] and construct the induced tuple Â
		
  * Compute the braid group orbit of Â and identify the SL_2() subgroup generated by the matrices of Â
		
  * Check the other exemplars and inverse tuple do not produce new orbits
		
  * Use the tuples in the braid group orbit to compute the parameter and residue signatures of the full equivalence class as defined in the previous classifications to try and match our examples to theirs. 
	
	We recorded all the details for our exemplars and the middle convolution computations for each of the primitive reflection groups of rank 3 and 4. The middle convolution tables are listed in the Appendix. The types and exemplars are listed in the section for each group, as is the comparison tables with the other classifications.  
	More details on what we use to compare with the existing classifications of <cit.> and <cit.> are given in Sections <ref> and <ref> .
	
		

§ IMPRIMITIVE REFLECTION GROUPS

	Imprimitive reflection groups G(m, p, n) are parameterized by m, p , n ≥ 1 with p | m. These groups can be viewed as n × n generalized permutation matrices, which each row and column containing only one non-zero entry which is either 1 or an m/p-th root of unity.  For m > 1 and (m, p, n) ≠ (2, 2, 2) these groups are always irreducible, meaning they act irreducibly on ^n (see <cit.>). 
	
	An element A  of G(m, p, n) can be written as A = [a_1, a_2, …, a_n |σ] for σ∈ S_n and p | a_1 + a_2 … + a_n. This element can be identified with the matrix where the (i, σ(i))-th entry of A is ζ_m^a_i and all the other entries are 0 <cit.>. Note that all these a_i should be read modulo m since we are working with powers of an m-th root of unity. 
	Additionally, there is a natural surjective homomorphism : G(m, p, n) ⟶ S_n given by sending A ↦σ. 
	
	One observation that follows from the definition is that for any p,  G(m, m, n) ≤ G(m, p, n) and G(m, p, n) ≤ G(m, 1, n) as subgroups.  
	
	Just as in the primitive case, the degrees of imprimitive groups tell us about the possible eigenvalues.  Writing q = m/p, we have: 
	
    (G(m, p, n))    = {m, 2m, …, (n-1)m, qn}
    
    		|ZG(m, p, n)|    = q (p, n)

	where ZG(m, p, n) is the center of G(m, p, n) and a cyclic subgroup.  
	
	
	Reflections in the group G(m, p, n) will be of one of the following two types:
	
		
  * Type 1: These are elements such that σ = (ij) is a transposition with a_k ≡ 0  m for k≠ i, j and a_i + a_j ≡ 0  m. All reflections of Type 1 are of order 2 and lie in the subgroup G(m, m, n) of G(m, p, n). Using the notation from <cit.> we write a Type 1 reflection as s(i, j; a_i) with the understanding that this is the same as s(j, i; -a_i).
		
  * Type 2: These are elements such that σ = e and for some i, a_i ≢0   m and a_k ≡ 0  m for all k ≠ i. These matrices are diagonal matrices of order m/(m, a_i) and will be denoted s(i; a_i). Type 2 matrices can only exist when p < m. 
	

	
		Let S be a set of reflections with minimal possible cardinality such that S generates G(m,p,n). Then S satisfies the following conditions: 
		
			
  * When p = 1, S contains n-1 reflections of Type 1 and one reflection of Type 2 of order m
			
  * When p = m, S contains n reflections of Type 1
			
  * When p ≠ 1, m, S contains n reflections of Type 1 and one reflection of Type 2 of order m/p
		
	
	
		Stated in <cit.> with proofs in <cit.> and <cit.>.
	
	
	
		For any reflection r ∈ G(m, p, n), the homomorphism : G(m, p, n) ⟶ S_n, one of the following must be true: 
		
			
  * (r) is a transposition and r has order 2
			
  * (r) is the identity.
		 
	
	
		This is true by the definition of reflections and the definition of the homomorphism. For a more thorough examination of related properties, we refer to <cit.>.
	
	
	Given any set of reflections generating an irreducible imprimitive group the permutations associated to the  reflections in the set must generate a transitive subgroup of S_n, or else the group cannot be irreducible. Note that in the case where the associated permutations are transpositions (i.e. the reflections are Type 1), the only transitive subgroup of S_n generated by transpositions is S_n itself <cit.>. 
	
	A subgroup of a reflection group generated by reflections is called a reflection subgroup. We modify some notation from <cit.> to state relevant theorems for identifying reflection subgroups in the imprimitive groups. Let X = {s(a_h, a_h+1; b_k)} be a set of reflections in G(m, m, n). We can associate to X the graph Γ_X with nodes {a_i} and edges (a_h, a_h+1) for s(a_h, a_h+1; b_k) in X. Letting X denote the reflection group generated by the reflections in X, we see that this group is irreducible if and only if the graph Γ_X is connected. 
	Suppose Γ_X is such that the graph contains exactly one cycle, given by r edges corresponding to {s(a_h, a_h+1; b_h)} for 1 ≤ h ≤ r for some 2 ≤ r ≤ n. Then we define the parameter δ(X) to be the absolute value of ∑_1^r b_h. 
	If p ≠ m, we say X is a nice reflection set of G(m, p, n) if X contains exactly one reflection of Type 2.
	
	
		Note that s(a_h, a_h+1; b_h) = s(a_h+1, a_h; -b_h), so if X is such that the graph contains only one cycle of length r, then the cycle corresponds to some permutation (a_1a_2 ⋯ a_r) (in cycle notation) and the signs of b_h must be chosen accordingly when computing δ(X). In particular, if the cycle is of length two, i.e. given as a double edge between two nodes corresponding to reflections s(a_1, a_2; b_1) and s(a_1, a_2; b_2) then δ(X) = |b_1 -b_2|.
	

	
		With the notation as defined above, we have the following relations for the reflection subgroup generated by X.  
		
			
  * Let X be reflection set of G(m, m, n) such that the graph Γ_X is connected and contains exactly one cycle. Then X = G(m, m, n) if and only if the integer δ(X) is coprime to m.
			
  * Let X be a nice reflection set of G(m, p, n) with s(a_1; b) the one reflection of Type 2, such that Γ_X is connected with n_1 nodes and exactly one cycle. Then X = G(m/m_1, (b, m)/m_1, n_1) where m_1 = (b, m, δ(X)).
			
  * Let X be a nice reflection set of G(m, p, n) with s(a_1; b) the unique reflection of Type 2 such that the graph Γ_X is a connected tree with n_1 nodes. Then X = G(m/(b, m), 1, n_1)
		
	
	
		The first is proven in <cit.> and the second is proven in <cit.>. Note that in both these papers they refer to cycles as circles. The third statement is proven in <cit.>, where they define rooted trees to capture the presence of Type 2 reflections. We refer to these paper for more details on the graph theoretic interpretation of generating sets and their connection to reflection subgroups of the imprimitive reflection groups.
	

		
		Among the imprimitive groups, we have to check for n-tuples and n+1-tuples in G(m, p, n). By Theorem <ref>, the only possible groups that can contain nice n-tuples are the well-generated groups G(m, 1, n) and G(m, m, n). We prove there are no nice n-tuples in either of these groups when n ≥ 5 in Section <ref>. By results in <cit.>, we know that n+1-tuples can only exist in G(m, p, n) for n < 5. This work will fully analyze the existence of nice n+1-tuples for 2< n <5.
	
	
		

 §.§ Nice 3-Tuples in G(m, p, 3)

	In this case we are seeking nice tuples [r_1, r_2, r_3] from G(m, p, 3) such that their product has a nontrivial eigenvalue of multiplicity 1. Note that since r_1, r_2, r_3 = G(m, p, 3) we know that p = 1 or p = m are the only possibilities as these are the only well-generated imprimitive groups (see Definition <ref>)
	
		In G = G(m, 1, 3)  for a nice tuple [r_1, r_2, r_3] with r_1, r_2 of Type 1 and r_3 of Type 2 with nontrivial eigenvalue ζ_m' of order m, letting A = r_1r_2r_3, the eigenvalues of A must be the distinct cube roots of ζ_m'. 
	
	
		Say we have three reflections 
    r_1r_2r_3 = A
 for some A ∈ G with Eigv(A) = {λ_1, λ_2, λ_3} with some λ_i ≠ 1. Note that since the r_i generate G, exactly two of them are Type 1 corresponding to distinct transpositions and the third is Type 2 with order m, say with nontrivial eigenvalue ζ_m'. Using the homomorphism : G ⟶ S_3, we know that (A) is even and since it cannot be the identity it must be a 3-cycle.
		Since A is a generalized permutation matrix, we know (A) = 0 and we get the following equalities must be true: 
		
    ζ_m'    =  (r_1r_2r_3) = (A) = λ_1λ_2λ_3 
    
    			0    = (A) = λ_1 + λ_2 + λ_3

		Since λ_i are all roots of unity, we know their sum being zero means the argument differs by 2 π/3.
		Further writing A = [a_1, a_2, a_3 | (123)], without loss of generality, gives us the matrix: 
		
    A = [       0 ζ_m^a_1       0;       0       0 ζ_m^a_2; ζ_m^a_3       0       0 ]

		Thus we can compute the characteristic polynomial as p_A(x) = -(x^3 - ζ_m^a_1 + a_2 + a_3) and since the constant term of the characteristic polynomial is the determinant of A, we get that ζ_m' = ζ_m^a_1 + a_2 + a_3. Thus λ_i must be the distinct cube roots of ζ_m'.
	
	
		There are nice 3-tuples in G(m, 1, 3) for all m >1. 
	
	
		The reflections in the tuple [s(2, 3; 1), s(1, 2; 1), s(3; 1)] will generate G(m, 1, 3) by Theorem <ref>. By the preceding lemma, the product will have eigenvalues corresponding to the three distinct cube roots of ζ_m, all of which cannot be 1.  
	

	
		In G = G(m, m , 3), if we have r_1r_2r_3 = A for reflections r_i generating G, then the eigenvalues of A are {λ^-2, λ, -λ}
	
	
		Since all the r_i are of Type 1, using : G ⟶ S_3 we see that (A) must be a transposition, say (ij) ∈ S_3. Writing A = [a_1, a_2, a_3 | (ij)] we see that (A) = ζ_m^a_k where k ≠ i, j and (A) = -1. Letting {λ_1, λ_2, λ_3} denote the eigenvalues of A, we get that 
		
    ζ_m^a_k   = λ_1 + λ_2 + λ_3 
    
    			-1    = λ_1 λ_2λ_3

		The characteristic polynomial of A is p_A(x) = -(x^2 -ζ_m^a_i + a_j)(x- ζ_m^a_k). Let λ_k = ζ_m^a_k so then we get that λ_i = -λ_j since these are the roots of x^2 - ζ_m^a_i + a_j. Using the determinant equation we get 1 = λ_i^2 λ_k hence λ_i^2 = λ_k^-1 = ζ_m^-a_k = ζ_m^a_i + a_j so both λ_i and λ_j square to the inverse of λ_k.
	
	
	
		Nice 3-tuples exist in G(m, m, 3) for all m >1. 
	
	
		In this case we can consider reflections [s(2, 3; b_1), s(1, 2; b_2), s(1, 2; b_3)] with b_2 and b_3 chosen so that |b_2 -b_3| is coprime to m. Taking b_1 = 1, b_2 = 1 and b_3 = 0, we get δ(X) = |b_2 -b_3| is coprime to m and so by Theorem <ref>, we know these reflections will generate G(m, m, 3). The characteristic polynomial of the product r_1r_2r_3 = A will in this case be p_A(x) = -(x-ζ_m)(x^2 -ζ_m^-1). Since the roots of the quadratic will be distinct, we know we will have at least two distinct eigenvalues, so the only worry is if the eigenvalues might be {ζ_m, ζ_m, 1} as then there is no nontrivial eigenvalue of multiplicity 1. If 1 is a root of the quadratic, then we know ζ_m^-1 = 1 and ζ_m = 1 which is not possible. Hence, such nice tuples always exist. 
	
	
	

 §.§ Nice 4-Tuples in G(m, p, 3)

	We are seeking nice tuples of four reflections from G(m, p, 3) such that their product has an nontrivial eigenvalue λ of multiplicity 2. This can be written as the identity 
	
    r_1r_2r_3r_4 = λ r_5^-1

	where the r_i generate G(m, p, 3) and r_5 is some reflection, not necessarily in G(m, p, 3). 
	
	
		In the notation of equation <ref> above, for any such nice tuple, it must be that r_5^-1 is in G(m, 1, 3) and λ is an m-th root of unity. Moreover, in any G(m, p,3 ), (including p = 1 and p= m) it must be that exactly four of these reflections are Type 1 and only one is of Type 2. 
	
	
		Note that since λ is an eigenvalue of multiplicity 2 for some matrix in G(m, 1, 3) it must be that the order of λ divides at least 2 degrees in G(m, 1, 3) = {m, 2m , 3m}. Thus the order of λ divides m and so λ I is in the center of G(m, 1, 3). This tells us that r_5 is always in G(m, 1, 3). To generate any of G(m, p, 3) we know the set {r_i}_i ≤ 5 must contain at least 2 reflections of Type 1 by Theorem <ref>.  For any p, rearranging the equation and using the homomorphism : G(m, 1, 3) ⟶ S_3 we get that (r_1r_2r_3r_4r_5) = (λ I) = e. Since e is even we know there must be exactly 2 or 4 reflections of Type 1. The product of two transpositions can only be the identity if they are the same transposition and we need at least two distinct transpositions to generate a transitive subgroup of S_3, hence there are exactly 4 reflections of Type 1 and one reflection of Type 2 for any G(m, p, 3).  
	
	Note that this does not prove that λ is primitive m-th root, however in the next three theorems we will conclude this is indeed always the case. 
	Up to rearranging the equation <ref>, we may assume r_5^-1 is the unique reflection of Type 2 and label its nontrivial eigenvalue ζ which we know must be some m-th root of unity, not necessarily primitive. 
	
	 
		To study the tuples in the cases where they might exist, we first study the number of ways to factorize the identity in S_3 into a transitive product of 4 transpositions. In S_3 any two distinct transpositions will generate a transitive subgroup which must be all of S_3 so fixing the first transposition as (12) we only need to find, up to simultaneous conjugacy, tuples of 3 transpositions that multiply to (12). Using computer search to find such conjugacy classes of 3-tuples, then adjoining with (12) on the left yields four options: 
		
			
  * [(12), (12), (23), (23)]
			
  * [(12), (23), (12), (13)]
			
  * [(12), (23), (23), (12)] 
			
  * [(12), (13), (23), (13)]
		
		Note that these tuples are multiplied starting at the leftmost transposition, although that does not matter in this case. 
		Converting these tuples to Type 1 reflections we can compute the product for each option. 
		
			
  * s(1, 2; b_1)s(1, 2; b_2)s(2, 3; b_3)s(2, 3; b_4) = [           ζ_m^b_1 -b_2                      0                      0;                      0 ζ_m^-b_1 +b_2 +b_3-b_4                      0;                      0                      0         ζ_m^-b_3 + b_4 ]
			
  * s(1, 2; b_1)s(2, 3; b_2)s(1, 2; b_3)s(1, 3; b_4) = [  ζ_m^b_1 +b_2 -b_4                  0                  0;                  0      ζ_m^-b_1 +b_3                  0;                  0                  0 ζ_m^-b_2 -b_3+ b_4 ]
			
  * s(1, 2; b_1)s(2, 3; b_2)s(2, 3; b_3)s(1, 2; b_4) = [ ζ_m^b_1 +b_2 -b_3-b_4                     0                     0;                     0         ζ_m^-b_1 +b_4                     0;                     0                     0        ζ_m^-b_2 + b_3 ]
			
  * s(1, 2; b_1)s(1, 3; b_2)s(2, 3; b_3)s(1, 3; b_4) = [  ζ_m^b_1 +b_3 -b_4                  0                  0;                  0 ζ_m^-b_1 +b_2 -b_3                  0;                  0                  0     ζ_m^-b_2 + b_4 ]
		
		Equations for {b_i} can be found by setting each of these matrices equal to λ r_5^-1:
		
    [  λ  0  0;  0  λ  0;  0  0 λζ ]

		for some root ζ with order dividing m. The choice of putting ζ in the third row is arbitrary and ζ could have been placed in any diagonal entry.
	
	 
		For any nice 4-tuple in G(m, p, 3), p≠ 1 it must be that either λ or λζ in the notation of Remark <ref> has the same order as ζ_m^δ(X), where X is any generating reflection set given by the nice 4-tuple.
	
	
		By Lemma <ref>, given any nice 4-tuple in G(m, p, 3) satisfying equation <ref>, we can rearrange all the Type 1 reflections to be one of the factorizations given in Remark <ref> up to simultaneous conjugation. As any reflection set X generating G(m, p, 3) p ≠ 1 will only need at least three Type 1 reflections, we see that for each of the four factorizations above, there are multiple ways to choose X so that the associated graph only contains one cycle. For example, the first factorization gives us the possibilities: X = {s(1, 2; b_1), s(1, 2; b_2), s(2, 3; b_i)} for i = 3, 4 and X = {s(1, 2; b_i), s(2, 3; b_3), s(2, 3; b_4)} for i = 1, 2. So we know δ(X) must be of the form
		
    δ(X) = 
    			|b_1 -b_2| 
    
    			|b_3 -b_4|

		By the equations each factorization must satisfy, we know ζ_m^b_1 -b_2 = λ and ζ_m^-b_3 +b_4 = λζ so one of these must have the same order as ζ_m^δ(X). The argument is similar for the third factorization since the formula for δ(X) only differs in the indices. The possible δ(X) values from the second and fourth factorizations only differ in the indices, so we only justify the result for the second factorization. In this case the options for δ(X) are 
		
    δ(X) = 
    			|b_1 +b_2 -b_4| 
    
    			|b_1 -b_3| 
    
    			|b_2 +b_3 -b_4|

		Up to taking absolute value, the first two options must equal λ and the third must be λζ so one of these has the same order as ζ_m^δ(X).
	

	
		Nice 4-tuples exist in G(m, m, 3) if and only if m ≠ 3. For m ≠ 3, we have: 
		
			
  * If (m, 3) = 1 then order(λ) = order(ζ) = m
			
  * If (m ,3) = 3 then order(λ) = m = 3 order(ζ) with order(ζ) ≠ 1. 
		 
	
	
		To generate G(m, m, 3), we know that our nice 4-tuple must contain a generating reflection set X such that δ(X) is coprime to m, which means either λ or λζ has order m, by Lemma <ref>. If λ does not have order m, then λζ has order m, but taking the determinant of equation <ref>, we know that 1 = λ^3 ζ so λ^2 has order m which means λ has order m. Thus, λ is always a primitive m-th root. Moreover, we know m ≠ 3 since the determinant tells us λ^3 = ζ^-1 and ζ≠ 1 as it is the nontrivial eigenvalue of the Type 2 reflection r_5^-1.
		
		If (m, 3) = 1 then we know λ^3 = ζ^-1 is also a primitive m-th root and hence so is ζ. Using factorization 1, we can take the tuple [s(1, 2; 1), s(1, 2; 0), s(2, 3; 2), s(2, 3; 0)], so λ = ζ_m and ζ = ζ_m^-3. This tuple generates G(m, m, 3)  by Theorem <ref> since taking X = {s(1, 2; 1), s(1, 2; 0), s(2, 3; 2)} gives δ(X) = 1 which is coprime to m.  
		
		If (m, 3) = 3 then λ^3 has order m/3 and thus so too will ζ. Again taking the tuple as above will work, since δ(X) = 1 will still be coprime to m and the product will give λ = ζ_m and ζ = ζ_m^-3 which now is not a primitive m-th root.  
	
	
		
		Nice 4-tuples exist in G(m, 1, 3) if and only if (m, 3) = 1. In this case the order of λ is m.
	
	
		Let [r_2, r_3, r_4, r_5] be a nice 4-tuple in G(m, 1, 3). Let r_5 be the unique reflection of Type 2 with nontrivial eigenvalue ζ^-1 of order m, so we can write ζ^-1 = ζ_m^-b with (-b, m) = 1. Let d = order(λ). We know d| m since λ I is in the center ZG(m, 1, 3). Taking determinants we get the equation
		
    λ^3 = ζ^-1

		so m | d since (ζ^-1)^d = 1, hence d = m. Since ζ^-1 is a primitive m-th root, so too is λ^3 and therefore we know (m, 3) = 1.
		
		Let m be such that (m, 3) = 1, so we know m-3 is coprime to m. Taking b = m-3, we can consider the tuple [s(1, 2; 0), s(2, 3; 1), s(2, 3; -1), s(3; m-3)]. Taking X = {s(1, 2; b_2), s(2, 3; b_3), s(3; -b)} we see by Theorem <ref> that X = G(m, 1, 3) since (m-3, m) = 1.
		Taking the product, we get r_2r_3r_4r_5 = λ r_1^-1 for some Type 1 reflection r_1 = s(1, 2; -1) with λ = ζ_m.
	

	
		Suppose p ≠ 1 and p≠ m. Then there exists a nice 4-tuple in G(m, p, 3) if and only if p = 3. In this case λ must be of order m.
	
	
		Let [r_2, r_3, r_4, r_5] be a nice 4-tuple in G(m, p, 3). Letting r_5 denote the unique reflection of Type 2, we know it must have order q = p^-1m so we can denote its nontrivial eigenvalue as ζ = ζ_q^b = ζ_m^pb with (b, m) = 1. 
		Further, we have the product r_2r_3r_4r_5 = λ r_1^-1 with r_1 ∈ G(m, m, 3) ≤ G(m, p, 3), hence λ I is in the center ZG(m,p , 3). Letting d = order(λ) we see that since |ZG(m, p, 3)| = q (p, 3), we must have that d | q (p, 3). Taking determinants, we get λ^3 = ζ^-1 and hence q | d and d | 3q.  If d ∤ q then since d ≠ 3 (becasue ζ^-1≠ 1), and d | q (p, 3) we must have that (p, 3) = 3.
		This gives us two cases: 
		
			
  * d = q
			
  * d = 3q and (p, 3) = 3
		
		Recall that by Theorem <ref>, for a reflection set X to generate G(m, p, 3), we need (pb, m) = p and (pb, δ(X), m) = 1.  
		
		In the first case d  = q so we can write λ = ζ_q^a = ζ_m^pa and ζ = ζ_q^b = ζ_m^pb with (b, q)= 1 = (a, q). By Lemma <ref>, we see that 
    (δ(X), m) = (pa, m) 
    ((a+b)p, m)

		Since neither of these satisfy (pb, δ(X), m) = 1, it is not possible for a nice 4-tuple in G(m, p, 3) to have d = q. 
		
		Suppose now that p = 3k so m = 3kq. Again we can write ζ = ζ_q^b = ζ_m^pb with (b, q) = 1, and this time since d= 3q we write λ = ζ_m^ka with (a, m) = 1. In this case the possible options for δ(X) gives us
		
    (δ(X), m) = (ka, m) 
    (ka + pb, m)

		In either case we get (pb, δ(X), m) = k so we conclude k = 1. This also gives us that λ must be of order m. 
		Thus, we conclude nice 4-tuples can only exist in G(m, 3, 3). In this case, we can write λ = ζ_m^a with (a, m) = 1 and we still have ζ = ζ_m^3b with (b, m) = 1 for q = p^-1m. We can take our 4-tuple to be [s(1, 2; 0), s(2, 3; 1), s(2, 3; -1), s(3; m-3)], noting that since 3 | m we also have that (m-3, m) =3. Here δ(X) = 2 so (m-3, 2, m) = 1 and we conclude this tuple is indeed nice.  
	
	
	
		Note that in all 3 cases described above, we will have to take middle convolution with parameter λ^-1 since our equations use the tuple product and not the inverse product.
	

	

 §.§ Nice 4-Tuples in G(m, p, 4)
 
	
	We are seeking nice tuples satisfying 
    r_1r_2r_3r_4 = A
 with r_i a reflection in G(m, p,4) and A has a nontrivial eigenvalue of multiplicity 2. 		Again the only possibilities are p = 1 and p = m since the group must be generated by 4 reflections.
	
		In G = G(m, 1, 4) there are no nice tuples of 4 reflections.
	
	
		Suppose we had r_1r_2r_3r_4 = A such that r_i generate G and A has eigenvalue λ≠ 1 of multiplicity 2. Since {r_i} generate G, we know exactly three of them are of Type 1 and exactly one is of Type 2 with nontrivial eigenvalue an m-th root of unity, say ζ_m'. Note that since the three Type 1 reflections should correspond to transpositions that generate all of S_4, they must be three distinct transpositions with each of {1, 2, 3, 4} showing up at least once. This means their product cannot be a transposition. Using : G ⟶ S_4 we get that (A) = σ for some 4-cycle σ since the permutation must be odd and cannot be a transposition. 
		Writing A = [a_1, a_2, a_3, a_4 |σ] we can compute that the character polynomial of A p_A(x) must be 
    p_A(x) = x^4 - ζ_m^a_1+ a_2 + a_3 + a_4

		This gives us that all the eigenvalues of A are the distinct 4th roots of ζ_m^a_1 + a_2 + a_3 + a_4 which means A cannot have an eigenvalue of multiplicity 2. 
	
	
	In the case of G(m, m, 4), we know for any nice 4-tuple, we must have that all the reflections are of Type 1. This means they all map to transpositions in S_4 and hence the permutation associated to A must be even. In S_4 the only options are e, the disjoint product of transpositions and a three cycle. We would need at least 6 transpositions to write e as the transitive product of transpositions, so we only have two choices. Without loss of generality, if A corresponds to disjoint transpositions, we take it to be (12)(34) and if A corresponds to a 3-cycle we take it to be (132).
	
	 
		
		The number of transitive ordered factorizations of (12)(34) into four transpositions in S_4 is 96 <cit.>. Using a computer, we find that there are 12 factorizations up to simultaneous conjugation. We can reduce these 12 even further by eliminating factorizations that are related by the action of one of the generators of B_4, since this means they lie in the same braid group orbit. 
		These preliminary computations give us three equivalence classes of factorizations, (which may still lie in the same braid group orbit) which we write with representatives: 
		
			
  * [(12), (34), (13), (13)]
			
  * [(13), (13), (12), (34)]
			
  * [(13), (24), (23), (14)]
		
		Choosing general reflections in G(m, m, 4) associated to these factorization and computing the products we get:
		
			
  * s(1,2; b_1) · s(3, 4; b_2) · s(1, 3; b_3) · s(1, 3; b_4) = [                0          ζ_m^b_1                0                0; ζ_m^-b_1+b_3-b_4                0                0                0;                0                0                0          ζ_m^b_2;                0                0 ζ_m^-b_2-b_3+b_4                0 ]
			
  * s(1,3; b_1) · s(1,3; b_2) · s(1, 2; b_3) · s(3, 4; b_4) = [                0  ζ_m^b_1-b_2+b_3                0                0;         ζ_m^-b_3                0                0                0;                0                0                0 ζ_m^-b_1+b_2+b_4;                0                0         ζ_m^-b_4                0 ]
			
  * s(1, 3; b_1) · s(2,4; b_2) · s(2, 3; b_3) · s(1, 4; b_4) = [             0   ζ_m^b_1-b_3             0             0;   ζ_m^b_2-b_4             0             0             0;             0             0             0  ζ_m^-b_1+b_4;             0             0 ζ_m^-b_2 +b_3             0 ]
		
	
	
	 
		Similarly, we can consider transitive ordered factorizations of (132) into four transpositions in S_4. Up to simultaneous conjugation and the action of a generator of B_4, we again get three equivalence classes of factorizations (which may not be distinct): 
		
			
  * [(14), (14), (12), (23)]
			
  * [(14), (14), (13), (12)]
			
  * [(14), (14), (23), (13)]
		
		These factorizations give us the following matrices: 
		
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(1, 2; b_3) · s(2, 3; b_4) = [                   0                   0 ζ_m^b_1-b_2+b_3+b_4                   0;            ζ_m^-b_3                   0                   0                   0;                   0            ζ_m^-b_4                   0                   0;                   0                   0                   0       ζ_m^-b_1 +b_2 ] 
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(1, 3; b_3) · s(1,2; b_4) = [                0                0 ζ_m^b_1 -b_2+b_3                0;         ζ_m^-b_4                0                0                0;                0    ζ_m^-b_3 +b_4                0                0;                0                0                0     ζ_m^-b_1+b_2 ]
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(2, 3; b_3) · s(1, 3; b_4) = [               0               0 ζ_m^b_1-b_2+b_4               0;    ζ_m^b_3 -b_4               0               0               0;               0        ζ_m^-b_3               0               0;               0               0               0   ζ_m^-b_1 +b_2 ]
		
	
		 
			In G = G(m, m, 4) if we had r_1r_2r_3r_4 = A with r_i reflections generating G and A has a nontrivial eigenvalue λ of multiplicity 2 then the eigenvalues of A are one of the following: 
			
				
  * {λ, λ, -λ, -λ} for λ a 4-th root of unity
				
  * {λ, λ, λ_1, λ_2} for λ a 4-th root of unity and λ_i ≠λ are the two distinct cube roots of λ^-1 other than λ.  
			
		
		
			We know all the r_i are of Type 1, so using : G ⟶ S_4 we know that (A) must be an even permutation, so it is either a double transposition or a 3-cycle. Let the eigenvalues of A be {λ, λ, λ_1, λ_2}. 
			If (A) is a double transposition then we see that 
			
    0    = (A) = 2λ + λ_1 + λ_2 
    
    				1    = (A) = λ^2 λ_1 λ_2

			Rearranging the equation for trace gives us that 
    1 = |λ| = | λ_1 + λ_2|/2 = |λ_1| + |λ_2|/2 = 1

			By properties of the triangle inequality, we get that λ_1 and λ_2 are colinear, hence λ_1 = ±λ_2. However, if λ_1 = -λ_2 then the trace equation means λ = 0 which is not possible, hence λ_1 = λ_2. Now the trace equation tells us that 2(λ + λ_1) = 0, hence λ_1 = -λ. 
			Next, we can use the determinant equation to see that 1 = λ^4 and so λ must be a 4th root of unity. This gives us that the eigenvalues of A are as described in case 1). 
			
			Further, when (A) is a double transposition, say A = [a_1, a_2, a_3, a_4 | (12)(34)], the characteristic polynomial is p_A(x) = (x^2 -ζ_m^a_1 + a_2)(x^2 - ζ_m^a_3 + a_4). However, since we know the roots of these polynomials agree, we see that ζ_m^a_1 + a_2 = ζ_m^a_3 + a_4, but we already knew that ζ_m^a_1 + a_2 = ζ_m^-a_3 - a_4 since m | a_1 + a_2 + a_3 + a_4. Thus we conclude that ζ_m^a_1 + a_2 = ± 1 and so ζ_m^a_1 = ±ζ_m^-a_2 and similarly, ζ_m^a_3 = ±ζ_m^-a_4 where the signs must agree for both pairs. 
			If (A) is a 3-cycle, we write A = [a_1, a_2, a_3, a_4 | (132)], since taking a different 3-cycle just changes the indices. The determinant condition stays the same as above, however our equation for trace is now: 
			
    ζ_m^a_4 = (A) = 2λ + λ_1 + λ_2

			Computing the characteristic polynomial we get p_A(x) = (x-ζ_m^a_4)(x^3 - ζ_m^a_1+a_2+a_3) and since the cube roots of ζ_m^a_1 + a_2 + a_3 are all distinct, we know that for λ to have multiplicity two, it must be one of the cube roots and λ = ζ_m^a_4. Since ζ_m^a_1 + a_2 + a_3 = ζ_m^-a_4 = λ^-1, we see that λ, λ_1 and λ_2 must be the three distinct cube roots of λ^-1. Now we have that λ^3 = λ^-1 and hence λ^4 = 1, proving that the eigenvalues of A are as described in case 2).
		
	
	 
		Nice 4-tuples exist in G(m, m, 4) iff  m = 2 or m = 4. In either case, λ is of order m. 
	
	
		Say we had a nice tuple in G(m, m, 4) such that r_1r_2r_3r_4 = A. 
		Suppose first that A corresponds to a three-cycle, then our nice 4-tuple is equivalent to one of the factorizations in Remark <ref>.  For all three factorizations, by the previous Lemma, we know that ζ_m^-b_1 +b_2 = λ and by Theorem <ref>, we know that δ(X) = |b_1 - b_2| must be coprime to m. This means λ is a primitive m-th root and by Lemma <ref> we know m = 4 or m = 2 in this case. 
		
		If A corresponds to a double transposition, then our nice 4-tuple is equivalent to one of the factorizations in Remark <ref> and we may write A = [a_1, a_2, a_3, a_4 | (12)(34)]. From the first factorization and the proof of Lemma <ref>, we see that ± 1 = ζ_m^a_1 + a_2 = ζ_m^b_3 -b_4 = λ^2. In this case δ(X) = |b_3 -b_4| which we know must be coprime to m by Theorem <ref>, we get that λ^2 is a primitive m-th root so m = 2. The same argument will also give that m must be 2 if our nice tuple is equivalent to the second factorization in Remark <ref>. For the third possible factorization, we see that the cycle has four edges and so δ(X) = |b_1 -b_3 +b_2 -b_4 since the cycle corresponds to the permutation (1324). But again we have that ± 1 = ζ_m^a_1 + a_2 = ζ_m^b_1 + -b_3 + b_2 -b_4 so λ^2 is a primitive m-th root so m = 2. 
			If m = 4, then we can take r_1 = s(1, 4; -1), r_2 = s(1, 4; 0), r_3 = s(1, 2; -1), r_4 = s(2, 3; 0). Then δ(X) = 1 so these reflections generate G(4, 4, 4). Further, we have 
    A = [  0  0 -1  0;  i  0  0  0;  0  1  0  0;  0  0  0  i ]
 which has eigenvalue i with multiplicity 2.	
		If m = 2, then we can take r_1 = s(1, 4; 1), r_2 = s(1, 4; 0), r_3 = s(1, 2; 0), r_4 = s(2, 3; 0) so again δ(X) = 1 and generates G(2, 2, 4). In this case 
    A = [  0  0 -1  0;  1  0  0  0;  0  1  0  0;  0  0  0 -1 ]
 which has eigenvalue -1 with multiplicity 2.
	
	Note that when taking the middle convolution for such tuples, the parameter should be λ^-1 since these equations are for the tuple product and not the inverse product.
	
		

 §.§ Nice 5-Tuples in G(m, p, 4)

	Searching for 5-tuples G(m, p, 4) so that the middle convolution will be rank 2 means we want the product to have an eigenvalue of multiplicity 3. A 4 × 4 matrix with an eigenvalue of multiplicity 3 can be viewed as a scalar times a reflection, so we want to explore the existence of [r_1, r_2, r_3, r_4, r_5] such that
	
    r_1r_2r_3r_4r_5 = λ r_6^-1
 where λ≠ 1 and r_6 is some reflection, not necessarily in G(m, p, 4). 
	
	
		If we have 5 reflections generating G = G(m , p, 4) such that 
		
    r_1r_2r_3r_4r_5 = λ r_6^-1
 
		then it must be that
		
			
  * r_6 ∈ G and G = G(m, m, 4) with order(λ) | m 
			
  * λ^4 = 1 
		
	
	
	
		Note first that  G(m, m, 4) ≤ G(m, p, 4) ≤ G(m, 1, 4) so r_i ∈ G(m, 1, 4) for i < 6 and so λ r_6^-1∈ G(m, 1, 4). Since this matrix has an eigenvalue of multiplicity 3, we know d = order(λ) is such that d divides 3 of the degrees in G(m, 1, 4) = {m, 2m , 3m, 4m}. This tells us that d | m as if it did not then d | 3m -2m = m. Since λ has order dividing m, λ I is in the center of G(m, 1, 4) and thus we can conclude r_6 ∈ G(m, 1, 4), though not yet necessarily in G(m, p, 4). This changes our problem into studying 6 reflections in G(m, 1, 4) satisfying
		
    r_1r_2r_3r_4r_5r_6 = λ I

		
		Next we use the homomorphism : G(m, 1, 4) ⟶ S_4. By Theorem <ref>, we know (r_1r_2r_3r_4r_5r_6) = (λ I) = e and since e is even, we know an even number of (r_i) must be transpositions.
		
		To conclude all 6 are transpositions, note that {r_i}_i < 6 are supposed to generate G(m, p, 4) which means {(r_i)}_i<6 must generate a transitive subgroup of S_4, hence S_4 itself. This removes the possibility of just 2 transpositions since we need at least 3 distinct transpositions to generate S_4.
		
		While it is possible to generate S_4 with 4 transpositions, our equation also requires that their product is the identity which is not possible with at least 3 distinct transpositions (as can be verified by hand). 
		
		Thus we know all 6 reflections must be Type 1 and so {r_i}_i < 6 can only generate G(m , m, 4), we conclude G = G(m , m ,4).
		
		Using that all reflections of Type 1 have nontrivial eigenvalue -1, taking determinant gives us that 
    1 = (-1)^ 6 = (r_1r_2r_3r_4r_5r_6) =  (λ I) = λ^4
 so we know lambda must be a 4th root of unity. Further, using that order(λ) ∈{2, 4 } must divide at least 3 degrees in G(m, m , 4) = {m, 2m, 3m, 4} we conclude that order(λ) | m. 
	
	
	 
		To study such tuples in G(m, m, 4) we first find the ways to decompose e in S_4 into a transitive product of 6 transpositions up to conjugacy. To make such a list, note that we can assume the first two transpositions are not disjoint, as if they are we can swap them around. This tells us their product is a 3-cycle and up to conjugation we may assume it is (132). We then found all the 4-tuples of transpositions that multiple to (123) up to simultaneous conjugation and generate all of S_4. Adjoining these options to the (12), (13) on the left, we get 27 options for transitive factorizations of the identity , and using computer code to partially compute the braid group orbit we were able to reduce the list further to two options (not necessarily distinct). 
		The tuples in S_4 are: 
		
			
  * [(12), (13), (12), (23), (24), (24)]
			
  * [(12), (13), (24), (24), (12), (23)]
		
		Using the Type 1 notation for the reflections in G(m, m, 4) as r_i =s(j_i, k_i; b_i) we can take the matrix product. The first tuple gives us the equation
		
    [               ζ_m^b_1 -b_3                          0                          0                          0;                          0 ζ_m^-b_1 +b_2-b_4 +b_5-b_6                          0                          0;                          0                          0         ζ_m^-b_2 +b_3 +b_4                          0;                          0                          0                          0              ζ_m^-b_5 +b_6 ]   = [ λ 0 0 0; 0 λ 0 0; 0 0 λ 0; 0 0 0 λ ]

		and the second tuple gives us the equation 
		
    [ ζ_m^b_1 +b_3 -b_4 -b_5                      0                      0                      0;                      0      ζ_m^-b_1 +b_2-b_6                      0                      0;                      0                      0     ζ_m^-b_2 +b_5 +b_6                      0;                      0                      0                      0          ζ_m^-b_3 +b_4 ]   = [ λ 0 0 0; 0 λ 0 0; 0 0 λ 0; 0 0 0 λ ]

	

	
		Nice 5-tuples exist in G(m, m, 4) if and only if m = 2 or m = 4 and in either case λ is of order m. 
	
	
		Suppose we had a nice 5-tuple in G(m,m, 4), so by the previous lemma we know 2 | m. We know it must be equivalent to one of the two factorizations in Remark <ref>. Considering the first factorization and looking at all transitive subsets of 4 reflections we can choose from the 6, we get the following possibilities for δ(X): 
		
    δ(X) = 
    				|b_1 -b_3| 
    
    				|b_6 -b_5| 
    
    				|b_4 +b_3-b_2| 
    
    				|b_4 +b_1 -b_2|

		The first three of these are such that ζ_m^δ(X) equals λ up to the absolute value on δ(X), so if any of those are coprime to m then λ is of order m. So the only way m ≠ 2, 4 is if the only option for δ(X) that is coprime to m is |b_4 +b_1-b_2|. 
		Writing m = 2^sk with (k, 2) =1 and s ≥ 1, we can write λ = ζ_m^k (s-j) for j = 1, 2 depending on the order of λ. The equations for b_i  modulo m then becomes: 
		
    b_1-b_3 ≡ k2^s-j
    
    			-b_1 + b_2-b_4+b_5 -b_6 ≡ k2^s-j
    
    			-b_2 +b_3 +b_4 ≡ k2^s-j
    
    			-b_5 +b_6 ≡ k2^s-j

		The second and fourth equation give us -b_1 +b_2-b_4 ≡ 2 k2^s-j which is not coprime to m. Thus, this option for δ(X) does not work so one of the first three options are coprime to m and hence λ is of order m. Since λ is a 4-th root of unity, we know m = 2 or m = 4. 
		
		A similar argument will give the same conclusion for the second factorization. 
		For m = 4, consider the tuple [s(1, 2; 1), s(1, 3; 0), s(1, 2; 0), s(2,3; 1), s(2, 4; 1)]. Then δ(X) = 1 and so these reflections will generate G(4, 4, 4). The product of this tuple will be i · s(2, 4; 2) as this corresponds to the first factorization.
		For m = 2, we can take the tuple [s(1, 2; 1), s(1, 3; 0), s(1, 2; 0), s(2,3; 1), s(2, 4; 0)]. Then δ(X) = 1 and so these reflections will generate G(2, 2, 4).  The product of this tuple will be -1 · s(2, 4; 1) as this corresponds to the first factorization.
	
		Note that with the examples above, we will have to take middle convolution with parameter λ^-1 since our equations are for the tuple product and not the inverse product.
	
	

 §.§ Nice n-Tuples in G(m, p, n) for n ≥ 5
 
	Recall that the only possible groups when nice n-tuples can exist are when p = 1 and p = m since these are the only well-generated imprimitive groups (See Definition <ref>).
	Fact: A connected graph of n vertices and n-1 edges must be a tree.
	
	
	 The product of n-1 transpositions is an n-cycle if and only if 
		the graph associated to a set of transposition in S_n is a tree, where the edge (a,b) is only present if (ab) is one of the n-1 transpositions. 
	
	
		By induction: if the product of n-1 transpositions is the product of a disjoint k-cycle and an (n-k)-cycle, then each of these correspond to  disjoint trees with k-1 and (n-k-1) edges respectively, so the remaining edge must connect the two trees, resulting in a tree with n-1 edges, hence the product is an n-cycle.
	
	
	
		There are no nice n-tuples in G(m, 1, n) for n ≥ 5. 
	
	
	
		In G(m, 1, n) any nice n-tuple consists of n-1 reflections of Type 1 and one reflection of Type 2, say r_n. Suppose we had r_1r_2⋯ r_n = A with [r_1, …, r_n] a nice tuple. Then by the claim(s) above, the graph associated to the transpositions {(r_i) }, with (r_n) = e must be a tree since the graph is connected by transitivity and therefore (A) is an n-cycle. Thus the characteristic polynomial 
    char_A(x) = (-1)^n (x^n - ζ)
 for some m-th root of unity ζ and therefore has n distinct roots so no A has no eigenvalues of multiplicity n-2 for n > 3.
	
	For G(m, m, n) we need the additional observations provided below.
	
	
		Let σ = (x_1x_2, · x_n) be an n-cycle in S_n and τ = (y_1y_2) be some transposition in S_n with y_1 <y_2. Letting c denote the distance between y_1 and y_2 in σ modulo the length of σ, we have that the product σ·τ is the product of a disjoint c-cycle and a disjoint (n-c)-cycle. 
	
	
		Since σ fixes no points, it is enough to show that multiplying with τ cannot fix two points in {1, …, n}. Letting y_1 = x_i and y_2 = x_j with i <j then we get 
    y_1  ↦ x_j+1↦ x_j+2↦…↦ x_i-1↦ x_i = y_1
 where the subscripts are read modulo n. The length of this cycle is precisely j-i  n. Doing a similar process for y_2 we get a cycle of length n -(j-i)  n. 
	
	Note that since σ is written in disjoint cycle notation the `distance' between, i.e. number of digits between y_1 and y_2 read in the order of multiplication (e.g. left to right if multiplication is left to right) is well defined. 
	
		In the notation of the claim above, if n > 2, then the product στ has at most one fixed point, with this occurring only if y_1 and y_2 are adjacent in the cycle σ. 
	
	
		If y_1 and y_2 are adjacent then c  = 1, so we are left with just an n-1 cycle which has exactly 1 fixed point when n-1 > 1 and 2 fixed points when n = 2. 
	
	
	
		There are no nice n-tuples in G(m, m, n) for n ≥ 5.
	
	
		In G(m, m, n), any nice tuple [r_1, r_2, …, r_n] consists of n reflections of Type 1. Again by transitivity, we know the graph associated to {(r_i)} is connected and since we only need n-1 edges to make a connected graph on n vertices, we know there is some transposition we can remove so that the associated graph is still connected. By cyclic permutation of the r_i via simultaneous conjugation, we can assume the redundant transposition corresponds to r_n, so (A · r_n^-1) is an n-cycle by the claim above. This tells us that (A) is the product of an n cycle and a transposition. 
		By the proofs above, we know (A) is either an (n-1)-cycle or the product of a c-cycle and an (n-c)-cycle. So either the characteristic polynomial is 
    char_A(x) = (-1)^n(x^n-1-ζ_1)(x-ζ_2)
 or 
    char_A(x) = (-1)^n (x^c -ζ_1)(x^n-c-ζ_2)

		We see that in both cases an eigenvalue of A can have atmost multiplicity 2, so when n > 4, there are no eigenvalues of multiplicity n-2, hence no such nice tuples can exist in G(m, m, n). 
	
	
	

§ PRIMITIVE REFLECTION GROUPS

	
	For each rank r ∈{3, 4}, we investigate the existence of nice T-tuples for T = r and T = r+1 in the primitive reflection groups, denoted G_i for i = 23, 24, …, 32. 
	For each group and each choice of T, we provide a table of the different types (up to inverse pairs), including the label for the type, the residues of the eigenvalues, as well as the nontrivial eigenvalues of the exemplar tuple under `Tuple Eigenvalues'.
	We also list the chosen exemplars 𝐀 = [A_1, A_2, …, A_T] for each type. For each exemplar we compute the middle convolution with the suitable choice(s) of parameter λ and this data is available in the appendices <ref>, <ref>, <ref>. These tables also include, for each choice of eigenvalue λ, the residue of λ in the `ξ' column as well as the notation used in Magma to identify the root of unity under `λ'.  We list the 2 × 2 matrices composing MC(𝐀) = [M_1, …, M_T] and record the order of the GL_2() subgroup generated by the these matrices under `S. Size'. Note that a zero in this column means the subgroup is infinite. 
	
	To get the induced tuple Â from MC_λ(𝐀) many choices of character will work, so we provide the one we use under the column `Character' or sometimes abbreviated as `Char.'. 
	In some cases, we may need to go to a larger field to construct a suitable character. We then compute and record the braid group orbit size of the induced tuple (under action of the positive powers of the braid group generators) and record it under `O. Size'. More information about how we compute the braid group orbit is provided in Section <ref>.
	For examples where the braid group orbits are the same size, we check if the signature of one lies in the orbit of the other.
	
		We say two braid group orbits are 𝐞𝐪𝐮𝐢𝐯𝐚𝐥𝐞𝐧𝐭 if the signature of one of the induced tuples lies in the orbit of another induced tuple.
	  
	This is one reason why our braid group code optionally can return the list of signatures in each orbit. 
	Note that this differs from the broader notion of equivalence in other literature such as Tykhyys's classification <cit.>. Note that the work of Lisovy-Tykhyy <cit.> and Tykhyy <cit.> only consider the case where matrices in the tuple do not have a shared eigenvector. In our case, since nice tuples give an irreducible representation and middle convolution preserves irreducibility, we know all our tuples cannot share a common eigenvector. Thus, we expect to identify which tuples in these classifications come from the complex reflection groups via middle convolution. Details on what parameters we use to compare our orbits with these classifications is provided below at the start of Section <ref> and <ref>.   
	
	For each distinct orbit we find, we provide data in the comparison tables on how our orbits relate to other classifications, as well as identify the finite SL_2() subgroups generated by the tuples. We list the subgroups of SL_2() in the Magma notation, given by order, number where order is the order of the finite group and number is the numbering of the group in the Small Groups Database. The numbering used by Magma is the same as that in the Small Group database implemented in GAP (see <https://magma.maths.usyd.edu.au/magma/handbook/text/779>, accessed on July 7th, 2024) and can be readily found online for all the groups we see. For infinite groups, the order is listed as `0', following the notation used in Magma. The notation SL(2, k) means 2× 2 matrices of determinant 1 over the finite field of _k.
	
		

 §.§ Nice 3-Tuples in Rank 3 Groups
 
	Nice tuples in this case will be tuples of 3 reflections generating the whole group such that their inverse product has some nontrivial eigenvalue of multiplicity 1. 
	After middle convolution, our the induced tuple will correspond to SL_2() local systems on the 4-punctured sphere and we can compare our braid group orbits to those in the literature. We compute the parameters (ω_X, ω_Y, ω_Z, ω_4) in the notation of <cit.> and found that the ω parameters for each of our orbits match (up to permutation) with one listed in the work of Lisovy-Tykhyy. Since Tykhyy's classification <cit.> further refines the original classification by Lisovy-Tykhy <cit.>, each tuple of ω parameters gives multiple orbits that Tykhyy identifies uniquely in terms of residues of the trace coordinates  (θ_1, θ_2, θ_3, θ_4) and (σ_12, σ_23, σ_13, σ_24) for some representative. In the nice 3-tuple case, using the ω parameters, we are able to easily identify how our orbit size relates the orbit size listed by Tykhyy for each ℳ^(4) (in Tykhyy's notation) and narrow down the possibilities for which of Tykhyy's orbits our tuple corresponds to. 
	In Tykhyy's classification,  tuples are considered equivalent up to:  multiplying any two matrices by -1, complex conjugation of all elements of all matrices, cyclically permuting the matrices in the tuple, and up to taking inverses and reversing the order of the tuple <cit.>. We perform all these operations and compute the residue signatures for the matrices in each orbit to find one that matches the example provided in Tykhyy's work. In most cases we found exact matches, and in other cases we provide the closest match, marking the differing coordinate with an asterisk (see Section <ref> and <ref>).   
	
	Note that we only list this data for orbits that are considered different by our definition and these may end up being the same orbit in the Lisovy-Tykhyy or Tykhyy classifications (<cit.>, <cit.>). 
	For each orbit we compute, we provide the identifying label used in both the classifications, as well as our orbit size and the orbit size they provided  (which is the same in both works). 
	
	
	

  §.§.§ G_23
 
	The group G_23 of order 120 is also denoted W(H_3) with degrees (G_23) = {2, 6, 10} <cit.>. All reflections in this group are conjugate with nontrivial eigenvalue -1. 
	For nice 3-tuples in G_23, the computation returned three possible sets of eigenvalues for the inverse product. 
	
	With q = ζ_5, we provide one example of a nice tuple such that the inverse product has eigenvalues corresponding to each type. 
	
	For Type A in G_23 one exemplar is:
	
    A_1    = [  1  0  0;  0  1  1;  0  0 -1 ]   A_2 = [            0           -1            0;           -1            0            0; -q^3-q^2 + 1   -q^3-q^2+1            1 ]   A_3 = [       1       0       0; q^3+q^2      -1      -1;       0       0      -1 ]

	Similarly for Type B, an exemplar is: 
	
    B_1 = [  1  0  0;  0  1  1;  0  0 -1 ]   B_2 =[        1 -q^3-q^2 -q^3-q^2;        0        0       -1;        0       -1        0 ]   B_3 = [       1       0       0; q^3+q^2      -1      -1;       0       0       1 ]
 	
	For Type C an exemplar is: 	
    C_1 = [  1  0  0;  0  1  1;  0  0 -1 ]   C_2 = [        1 -q^3-q^2        0;        0       -1        0;        0        1        1 ]   C_3 = [          0         -1          0;         -1          0          0; -q^3-q^3+1 -q^3-q^2+1          1 ]

	The data for all three types for G_23 is compiled in Table <ref>. For all three types, choosing a different exemplar would not have resulted in a new orbit. The two orbits of size 10 arising from Type A are the same, as are the two orbits of size 10 from Type B. However, these two orbits are distinct across the two types in the sense that their signatures did not show up in the others' orbit. For Type C, the two orbits of size 18 are the same. Information about how these 6 distinct orbits relate to the other classifications is provided in Table <ref>. In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
		

  §.§.§ G_24
 
	The group G_24 of order 336 is also denoted W(J_3(4)) with degrees (G_24) = {4, 6, 14} <cit.>. All reflections in this group are conjugate with nontrivial eigenvalue -1. 
	Looking at the degrees, we first expand the base field to be the cyclotomic field of degree 84. There is one inverse pair of nice 3-tuples in this group, so we only provide the details for one in Table <ref>.
	
	
	Letting z = ζ_7, an exemplar of Type A is: 
	
    A_1 = [          1 -z^4-z^2-z -z^4-z^2-z;          0          0         -1;          0         -1          0 ]   A_2 = [  1  1  0;  0 -1  0;  0  1  1 ]   A_3 = [        -1         0         0;         1         1         0; z^4+z^2+z         0         1 ]

	The middle convolution data for G_24 is provided in Table <ref>. These three distinct orbits are the only orbits from Type A and we checked that in all three cases, the inverse of the exemplars do not produce any new orbits. The comparison data for G_24 is provided in <ref>.
	
	
	

  §.§.§ G_25
 
	The group G_25 of order 648 is also denoted W(L_3) with degrees (G_25) = {6, 9, 12} <cit.>. 
	G_25 has two conjugacy classes, consisting of matrices with nontrivial eigenvalue w or w^2 respectively, for w = ζ_3.
	Expanding the base field to be the cyclotomic field of order 36, we get 8 possible types, which actually split into 4 pairs of inverses. We only list one set of inverse eigenvalues for each pair in Table <ref>.
	
	
	With w = ζ_3 the exemplars for each type are provided below. 
	Exemplar of Type A: 
	
    A_1 = [   1   0   0;   0   1 w^2;   0   0   w ]   A_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   A_3 = [   1   w   0;   0 w^2   0;   0   w   1 ]

	Exemplar of Type B: 
	
    B_1 = [   1   0   0;   0   1 w^2;   0   0   w ]   B_2 = [   -w    1  w^2;    1 -w^2    1;  w^2    1   -w ]   B_3 = [   1   w   0;   0 w^2   0;   0   w   1 ]

	Exemplar of Type C: 
	
    C_1 = [    1 -w^2    0;    0    w    0;    0 -w^2    1 ]   C_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   C_3 = [    1    0    0; -w^2    w -w^2;    0    0    1 ]

	Note that for Type D, there is only two choices of eigenvalue to take midlde convolution with and only one of them will result in 2× 2 matrices. Exemplar of Type D: 
	
    D_1 = [    1 -w^2   -w;    0 -w^2    1;    0   -w    0 ]   D_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   D_3 = [    0   -w  w^2;    0    1    0; -w^2   -1 -w^2 ]

	The details for the middle convolution of each type is provided in Table <ref>. For all three types, different exemplars do not produce new orbits nor do their inverses. For Type A, the two orbits of size 12 are distinct, as are all three orbits of size 36 from Type B. For Type C, we had to extend to a field containing ζ_72 to construct the character. The two orbits of size 4 from Type C are the same. Including the one from Type D, we get 8 distinct orbits from G_25. The comparison data is provided in Table <ref>. Comparing with the Lisovy-Tykhyy paper, all the ω parameters seemed to correspond to the family of orbits listed in Lemma 39 in <cit.>, so we have provided the specific values for the variables that work in each case. When comparing with Tykhyy's classification, there was only one orbit (from Type B) that we could not find an exact match for. We listed the closest match, which seemed to be Orbit 5 in <cit.> and distinguish the different coordinate with an asterisk. 
	In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
	
	

  §.§.§ G_26
 
	The group G_26 of order 1296 is also denoted W(M_3) with degrees (G_26) = {6, 12, 18} <cit.>.
	In G_26 there are three conjugacy classes of reflections, corresponding to nontrivial eigenvalues in {ζ_3, ζ_3^2, -1}. In this group the are 6 sets of eigenvalues for the inverse product, which correspond to 3 inverse pairs, so we only list one from each pair.  
	
	The exemplars for this group are all provided over (ζ_36) with z = ζ_36.
	
	Here is an exemplar of Type A:
	
    A_1 = [       1       0       0;      -1  2· z^6     z^6;  -z^6+2 -3· z^6    -z^6 ]   A_2 = [      0    z^6      0;      1 -z^6+1      0;  z^6-2  z^6+1      1 ]   A_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	An exemplar for Type B: 
	
    B_1 = [  z^6-1 2· z^6    z^6;      0      1      0;      0      0      1 ]   	B_2 = [      0  z^6-1  z^6-1;    z^6      2      1; -z^6-1  z^6-2  z^6-1 ]   B_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	An exemplar for Type C: 
	
    C_1= [          1          0          0;       -z^6      z^6-1      z^6-1;      z^6+1 -3·(z^6+1) -2·(z^6-1) ]   C_2 = [      1 -z^6+1 -z^6+1;      0      1      0;      0 -z^6-1   -z^6 ]   C_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	
	In G_26 choice of different exemplars nor the inverses for any type produce new orbits. The middle convolution data for this group can be found in Table 
	<ref>. Further, all three orbits from Type A are distinct, as are all three orbits from Type B. When comparing with Tykhyy's classification, there was only one orbit (from Type B) that we could not find an exact match for. We listed the closest match, which seemed to be Orbit 5 in <cit.> and distinguish the differing coordinate with an asterisk. 
	
	In this group we also find nice 4-tuples, details provided in Section <ref>
	
		 
			λ     ω_X, Y, Z, 4      Lisovy-Tykhyy #     θ_1, 2, 3, 4     σ_12, 23, 13, 24       -90 Tykhyy #       -90 Their O. Size      -90 Our O. Size       -90 SL_2() Subgroup 
  [0pt]0pt0ex
 [0ex] 
			9|c|G_26 Type A Comparisons [0pt]0pt3ex
 
			z^3     (0, √(3), 0, 0)    Type II, X' = 0, X” = √(3)     ( 5/12, 1/4, 5/12, 3/4)     (1/2, 1/2, 5/6, 1/2)      4      2     24     0 [0pt]0pt4ex 
 [1ex] 
			-z^3     (0, -√(3), 0, 0)     Type II, X' = 0, X” = -√(3)     (1/4, 1/12, 1/4, 11/12)     (1/2, 1/2, 1/2 , 5/6)     4       2     24     0 [0pt]0pt4ex
 [1ex] -z^6+1     (0, -z^9+2z^3, 0, 0)     Type II, X' = 0, X” = -z^9+2z^3     (1/6, 1/2, 2/3, 1/2)     (1/2, 1/2, , 5/6, 1/2 )     3      2     24     24, 4 [0pt]0pt4ex
 [1ex] 9|c|G_26 Type B Comparisons [0pt]0pt3ex
 
			z^10     (-z^9+2z^3, -z^9+2z^3, 2, -1)     Type III, ω = -z^9+2z^3     (7/9,7/18, 7/18, 2/3)     ( 1/2, 1/3, 1/2, 5/6^*)     5^*     3     36     0 [0ex]0ex4ex 
 [1ex]
			-z^4     (z^9-2z^3, z^9-2z^3, 2, -1)     Type III, ω = z^9-2z^3     (1/9,1/18, 1/18, 2/3)     (1/2, 1/3, 1/2, 1/6)     5     3     36     0 [0ex]0ex4ex 
 [1ex]
			-z^16     (-z^9+2z^3, -z^9+2z^3, 2, -1)     Type III, ω = -z^9+2z^3     (5/9, 5/18, 5/18, 2/3)     (1/2, 1/3, 1/2, 1/6)     5     3     36     0 [0ex]0ex4ex 
 [1ex]
			9|c|G_26 Type C Comparisons [0pt]0pt3ex
 
			-z^6+1     (0, 0, 2, -1)     Type III, ω = 0     (1, 1/2, 1/2, 2/3)     ( 1/2, 1/3, 1/2, 1/2)     5     3     12     12, 1 [0ex]0ex4ex 
 [1ex]   Here w = ζ_3, z = ζ_36. The correct signature for Tykhyy's orbit 5 would have 3(7/18) ≡1/6 as σ_24 but such a signature was not in the orbit so we provide the closest match. The subgroups 24, 4 and 12, 1 are the dicyclic groups of order 24 and 12 respectively.
	
	

  §.§.§ G_27
	
	This group is of order 2160 and is also denoted W(J_3(5)).
	This group only has one conjugacy class of reflections with nontrivial eigenvalue -1 and (G_27) = {6, 12, 30}. The original base ring of this group in Magma is the cyclotomic field of order 15, and we expanded the base ring to the cyclotomic filed of order 60 to ensure all the eigenvalues are found. We found three distinct inverse pair types in this group.
	
	
	Writing z = ζ_60 and setting u = -z^10-z^8+z^2 and v = z^16 + z^4, we get that z^50v = -z^14+ z^6+ z^4 so we can write the exemplar for Type A as: 
	
    A_1 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   A_2 = [        0        0    -z^20; u+z^10+2        1   v+z^10;     z^10        0        0 ]   A_3 = [  1  1  0;  0 -1  0;  0  1  1 ]

	Using the same notation, we can write the Type B exemplar as: 
	
    B_1 = [   -1    u    u;  v+1   -u -u-1; -v-1  u+1  u+2 ]   B_2 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   B_3 = [  1  1  0;  0 -1  0;  0  1  1 ]

	Similarly, the Type C exemplar is:
	
    C_1 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   C_2= [  1  1  0;  0 -1  0;  0  1  1 ]    C_3 = [    1    0    0; -v-1  u+1  u+2;  v+1   -u -u-1 ]

	In G_27 for all three types, different exemplars do not produce new orbits. Moreover the inverse types do not produce new orbits and we have checked that these 9 orbits are distinct. The middle convolution details for Type A, B and C are listed in Table <ref>, <ref>, and <ref> respectively.  The comparison data is provided in Table <ref>. In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
	

 §.§ Nice 4-Tuples in Rank 3 Groups

	Nice tuples in this group are 4-tuples of reflections generating the whole group such that the inverse product has a nontrivial eigenvalye of multiplicity 2. If (ζ) denotes the trace field for some FCRG W, and we have a matrix A with eigenvalues {λ, λ, λ_1}, then we know (A) = 2λ + λ_1 ∈(ζ). If we further knew that λ∈(ζ) then we can conclude that λ_1 ∈(ζ) and so we do not need to extend the base field to ensure all characteristic polynomials of such matrices A split. By Theorem <ref> we know that any such λ of multiplicity 2 must have order dividing at least two of the degrees of W. Studying degrees, we see that in G_23 and G_24, λ must have order dividing 2, i.e. λ = -1 and so we do not need to expand the base ring. Similarly, for G_25, G_26, G_27 such  λ must have order dividing 6, which means we do not have to expand the base ring in these cases either. 
	Our search found nice 4-tuples in all the primitive rank 3 groups except for G_24. 
	
	In comparing with the existing classifications, we can only compare with Tykhyy's ℳ^(5) orbits <cit.> since our 4-tuples gives us local systems on the 5-punctured sphere. Unfortunately, without the ω parameters from the work of Lisovy-Tykhyy <cit.> like in the 3-tuple cases, it is more difficult to relate the our orbit sizes to Tykhyy's. We still computed all the signatures for the different possible equivalent representatives for each orbit and attempted to find a match, which was successful. 
	
	

  §.§.§ G_23
 
	We only found one type of nice tuple. 
	

	Letting q = ζ_5, here is an exemplar for Type A: 
	
    A_1 = [    q^3+q^2    q^3+q^2    q^3+q^2; -q^3-q^2+1 -q^3-q^2+1   -q^3-q^2;    q^3+q^2         -1          0 ]     A_2 = [ -q^3-q^2+1 -q^3-q^2+1   -q^3-q^2;    q^3+q^2    q^3+q^2    q^3+q^2;         -1    q^3+q^2          0 ]
     A_3 = [   -q^3-q^2          1          0;    q^3+q^2    q^3+q^2          0;   -q^3-q^2 -q^3-q^2+1          1 ]      A_4 = [       -1        0        0; -q^3-q^2        1        0;        0        0        1 ]

	The middle convolution details can be found in Table <ref>. This type is its own inverse. 
	For this tuple we could not compute the orbit size as it was too large, but we were able to find a signature in the orbit (up to Tykhyy's equivalence relations) that agreed with one listed in Tykhyy's paper. Because we cannot compute the full braid group orbit, we also cannot check if different exemplars would result in a different orbit after middle convolution. We provide the comparison details in Table <ref>. 
	
	
	

  §.§.§ G_25

	We found 6 different types which split into 3 inverse pairs, so we only provide the details for one type from each pair in Table <ref>. 
	
	
	For all the exemplars listed below, z = ζ_3. 
	Here is an exemplar for Type A: 
	
    A_1 = [   z   0   0; z^2   1   0;   0   0   1 ]   A_2 = [  -z   1  -z;   0   1   0;   z z^2   0 ]   A_3 = [   1   z   0;   0 z^2   0;   0   z   1 ]   A_4 = [  -z   1 z^2;   1 z+1   1; z^2   1  -z ]

	Here is an exemplar for Type B: 
	
    B_1 = [  -z   1   0; z+1   0   0;   1   z   1 ]   B_2 = [  -z   1  -z;   0   1   0;   z z^2   0 ]   B_3 = [   0 z^2   z;   0   1   0;  -z   1  -z ]   B_4 = [   1   z   0;   0 z^2   0;   0   z   1 ]

	Here is an exemplar for Type C:
	
    C_1 = [   1   z   1;   0   0 z+1;   0   1  -z ]   C_2 = [   z   0   0; z^2   1   0;   0   0   1 ]   C_3 = [ z+1  -1 z+1;   0   1   0; z^2  -z   0 ]   C_4 = [  -z   1   0; z+1   0   0;   1   z   1 ]

	The middle convolution details for G_25 can be found in Table <ref>.
	The orbits for Type A and Type B are the same. All three types do not split into new orbits and their inverses do not produce new orbits. The comparison details for the two distinct orbits can be found in Table <ref>.
	
	

  §.§.§ G_26

	In this group we found two inverse pairs of nice tuples and we provide the details for one type from each pair in Table <ref>. 
	
	With the notation z = ζ_3, here is an examplar for Type A: 
	
    A_1 = [     z   z+2     1;     0     1     0;   z+2 z^2-z    -z ]   A_2 = [   z   0   0; z^2   1   0;   0   0   1 ]   A_3 = [ -z+1 2z^2  z^2;  z^2   -1   -1;    0    0    1 ]   A_4 = [    -z    -1    -1;     z   z+2   z+1; z^2-z z^2-1   z^2 ]

	The exemplar for Type B: 
	
    B_1 = [    1  z^2    0;    0    z    0;    0 -z+1    1 ]   B_2 = [   -z   -1    0;  z^2    0    0; 2z+1  z+2    1 ]   B_3 = [   z   0   0; z+1   1   0;   0   0   1 ]   B_4 = [     1 z^2-z    -z;     0   z^2   z^2;     0  2z+1   z+1 ]

	The middle convolution details for G_26 can be found in Table <ref>. 
	Neither of these types split into multiple orbits nor do their inverse types produce new orbits. The finite group generated by the matrices in SL_2() are the same for both groups, Magma identifies this as SmallGroup<24, 4> which corresponds to the dicyclic group of order 24.
	
	
	

  §.§.§ G_27
 
	In this group we found one inverse pair, and we provide the details for one type of the pair. 
	
	
	Here is an exemplar of Type A, where z = ζ_15, u = z^4 + z, v = z^7-z^3+z^2-1: 
	
    A_1    = [    u  u-1   -1; -u-1    u -v-u;    0    0    1 ]   A_2 = [ -1  0  0;  1  1  0; -u  0  1 ]   A_3 = [          u+2v        v+z^10 -u+z^5(z^5-1);             0             1             0;          -u-1        -v-u-1         -2v-u ]
     A_4    = [ -v-u-z^5   -v-z^5      u+1;  v+u+z^5        v  -u+z^10;   -v-u+1  -v-z^10   u-z^10 ]

	
	The middle convolution details can be found in Table <ref>. In this case we could not compute the orbit size since it was too long. Similar to G_23, we were able to compute some signatures for tuples in the orbit and we found a match with a signature listed in Tykhyy's classification.
	
	
	

 §.§ Nice 4-Tuples in Rank 4 Groups

	In this case we are looking for nice tuples of 4 reflections generating the whole group such that their inverse product has a nontrivial eigenvalue of T-2 = 2. Of the five rank 4 reflection groups, we found nice 4-tuples in G_28, G_30 and G_32. In the case of G_32, the field containing all the eigenvalues would be (ζ_360) which is much too large to perform the computations over. For G_32, in Section <ref> we justify why it is enough to search for such tuples over (ζ_12). 
	
	After taking the middle convolution, multiplying with a suitable character and adjoining the inverse product, our tuples become SL_2() local systems on the 5-punctured sphere. We found that in some cases, one of the 5 matrices in the local system ends up being the identity, which means our tuple comes from some local system on the 4-punctured sphere. In the case where we have 4 nontrivial matrices, we can compute the ω parameters in the notation of Lisovy-Tykhyy <cit.> and identify the corresponding ℳ^(4) orbit in Tykhyy's paper <cit.>. When all 5 of the matrices are nontrivial, we can only compare with Tykhyy's ℳ^(5) orbit signatures from <cit.>.
	
	

  §.§.§ G_28
 
	The group G_28 of order 1152 is also denoted W(F_4) with degrees (G_28) = {2, 6, 8, 12} <cit.>. 
	All reflections in G_28 have nontrivial eigenvalue -1, and form two conjugacy classes of reflections. We only found one type of nice 4-tuples in this group, with two possible choices of λ parameters for middle convolution. 
	
	An exemplar for such a tuple of Type A is: 
	
    A_1 = [  2  3  4  2; -1 -2 -4 -2;  0  0  1  0;  0  0  0  1 ]   A_2 = [  1  2  2  0;  0 -1 -2  0;  0  0  1  0;  0  1  1  1 ]   A_3 = [ -1 -2 -4 -2;  2  3  4  2; -1 -1 -1 -1;  0  0  0  1 ]   A_4 = [ -1  0  0  0;  1  1  0  0;  0  0  1  0;  0  0  0  1 ]
	
	A different choice of exemplar does not produce a new orbit and the two orbits computed above are the same, so we only get one new orbit from this group. The middle convolution details can be found in Table <ref>. After putting the matrices in SL_2(), we see that the inverse product of the middle convolution of this tuple is the identity, so this tuple comes from a local system on the 4-punctured sphere. As in the case of 3-tuples, we provide the details comparing with the results of both Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) in Table <ref>. The ω parameters appear to correspond to Orbit 31, but in fact this orbit is of Type III, as described in Lemma 39 in <cit.>.
	

	
	

  §.§.§ G_30
 
	The group G_30 of order 14400 is also denoted W(H_4) with degrees (G_30) = {2, 12, 20, 30} <cit.>.
	All reflections in G_30 have nontrivial eigenvalue -1 and are conjugate to each other. Looking at the degrees, we extend the base field to (ζ_60). We found three distinct types of nice 4-tuples, the details are in Table <ref>.
	
	For all of the exemplars below z = ζ_60 and u = z^14-z^6-z^4. 
	An exemplar of type A is given by: 
	
    A_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   A_2 = [     1     0     0     0;  2u-1  2u-1   u-1    -1; -2u+1 -2u+2  -u+2     1;  -u+1   -2u    -u    -u ]
     A_3 = [     u   u-1   u-1    -1; -2u+1 -2u+2 -2u+1    -u;  2u-1  2u-1    2u     u;    -u    -u    -u    -u ]   A_4 = [ -u+1 -u+1 -u+1   -u;    0    1    0    0;    u  u-1    u    u;   -1    u    u    0 ]
	
	An exemplar of type B is given by:
	
    B_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   B_2 = [    0   -1   -1   -1;   -u -u+1   -u   -u;    0    0    1    0;  u-1  u-1  u-1    u ]
     B_3 = [  1 -u -u -u;  0  0 -1 -1;  0  0  1  0;  0 -1 -1  0 ]   B_4 = [ -2u+2 -3u+2 -2u+1  -u+1;  2u-1  3u-1  2u-1   u-1;    -u  -u+1  -u+1     1;     u   u-1     u     0 ]

	An exemplar of type C is given by: 
	
    C_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   C_2 = [     1     0     0     0;  -u+1 -2u+1  -u+1    -u;   u-1    2u     u     u;  2u-1  2u-2  2u-1     u ]
     C_3 = [   u-1  2u-1    2u     u; -3u+1 -3u+3 -2u+2  -u+1;  3u-1  3u-2  2u-1   u-1;     0     0     0     1 ]   C_4 = [    0   -1   -1   -1;   -u -u+1   -u   -u;    0    0    1    0;  u-1  u-1  u-1    u ]

	The middle convolution details for each type is presented in Table <ref>. For all three types, different exemplars do not produce new orbits and since each type is their own inverse type, these are all the orbits from this group.  Further, the two orbits produced from Type A are distinct, as are the two from Type C. The two orbits from Type B are actually the same. In total G_30 gives us 5 different orbits. 
	
	For all 5 orbits, once we put the matrices in SL_2(), we see that the last matrix is the identity, hence these actually corresponds to orbits of 4 matrices multiplying to the identity. We compared these to the existing classification(s) by Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) as we did for the rank 3 complex reflection groups. Within each type, the two orbits are equivalent to the same orbit of ℳ^(4) in Tykhyy's classification, so we only provide the details once in Table <ref>.
	
	

  §.§.§ G_32
 
	The group G_32 of order 155520 is also denoted W(L_4) <cit.>.
	In this group, the nontrivial eigenvalues of reflections are in {ζ_3, ζ_3^2 } corresponding to two conjugacy classes. 
	For G_32, (G_32) = {12, 18, 24, 30} so the eigenvalues of multiplicity 2 must be 12-th roots of unity. Expanding the field to (ζ_12) we found 5 distinct eigenvalue sets for the inverse product. Note that to ensure we found all examples, we would have to run the code over (ζ_360) since there may be matrices with one or two eigenvalues other than the 12-th roots which would not be found, however we claim it is enough to check over (ζ_12). 
	
		If A = A_1A_2A_3A_4, with [A_1, A_2, A_3, A_4] a nice tuple in G_32 so that A has a nontrivial eigenvalue of multiplicity 2, then the other two eigenvalues of A are both m-th roots of unity (not necessarily primitive) for the same m ∈{12, 18, 24, 30}.
	
	
		Let the eigenvalues of A be {λ, λ, λ_1, λ_2}. Taking residues, we write 
    Res(λ) = x/12,    Res(λ_1) = p_1/u_1· 6,    Res(λ_2) = p_2/u_2 · 6
 where u_i ∈{2, 3, 4, 5} which we know must be the case by the degrees of G_32. Suppose λ_i are not 24-th roots of unity. So u_i must be 3 or 5 and (p_i, u_i) = 1. 
		
		Using that (A_i) is a 3-rd root of unity, we know that (A) ∈{1, ζ_3, ζ_3^2}. This tells us that λ_1λ_2 = (λ^2)^-1(A) and so λ_1λ_2 must be a 6-th root of unity since λ^2 is a 6-th root.  
		In terms of the residues, this means 
		
    p_1/u_1 · 6 + p_2/u_2 · 6 = u_2p_1 + u_1p_2/6 · u_1u_2 = y/6
 for some 1≤ y ≤ 6.
		
		This equation tells us that u_1 | u_2 p_1 and since (p_1, u_1) = 1 we know u_1 | u_2. Similar arguemnt shows u_2 | u_1, so u_1 = u_2 if one of them is 3 or 5.
		
		Suppose now that Res(λ_1) = p_1/4 · 6, is a primitive 24-th root of unity, i.e. u_1 = 4, so (p_1, 24) = 1. We know u_2 is not 3 or 5, so it can only be 2 or 4. If u_2 = 4 and (p_2, 24) = 1 then we are done since they are both primitive 24th roots. Else λ_2 is a 12-th root of unity, which means λ_1 = (λ^2)^-1(A) λ_2^-1. Since the right hand side is a product of 12th roots, λ_1 must also be a 12th root, completing the proof.
	
	In the notation of the lemma above, we know that for nice 4-tuples, if the eigenvalues of the inverse product A is {λ, λ, λ_1, λ_2} then λ_1 + λ_2 = -2λ + (A) ∈(ζ_12) and similarly λ_1λ_2 = λ^2 (A) ∈(ζ_12). 
	With the result from the lemma, we do computer search in each of the fields (ζ_18), (ζ_24), (ζ_30) to find a pair of roots of unity (s, t) other than 12-th roots such that s +t, s· t ∈(ζ_12). For (ζ_18) and (ζ_30) no such pair could be found. In (ζ_24) one example that works is the pair (ζ_24, -ζ_24), however no pairs are found if we further require that s ≠ -t. This tells us the only cases we might be missing when searching over (ζ_12) are tuples whose inverse product has eigenvalues {λ, λ, λ_1, -λ_1} with λ a 12th-root (not nec. primitive) and λ_1 a primitive 24th root. In the notation of the lemma above, using that (A) is a 3rd root, we know -λ^2 ·λ_1^2 is also a 3rd root. Note that if λ_1 is a primitive 24th-root then -λ_1^2 is a primitive 12th-root. Again using a computer (or by hand) we can check that no primitive 12th root multiplied with the square of a 12th-root can be a 3rd root. Thus, we conclude it is enough to search for tuples over (ζ_12).
	
	Searching for nice tuples of in G_32 we find 3 inverse pairs. We list only one from each pair, noting that Type B is its own inverse.
	
	
	For all the exemplars listed below, w = ζ_3. 
	An exemplar of type A is given by: 
	
    A_1 = [    1    0    0    0; -w^2    w  w+2    w;   -w  w+2  -2w    1;    0    0    0    1 ]   A_2 = [   1   0   0   0;   0   1 w^2   0;   0   0   w   0;   0   0 w^2   1 ]
     A_3 = [    1    0    0    0; -w^2    w -w^2    0;    0    0    1    0;  w^2 -w+1  w^2    1 ]   A_4 = [      0      w  w^2-1     -w;   -w^2      2 -w+w^2      1;      w   -w^2      w    w^2;      0      0      0      1 ]
	
	An exemplar of type B is given by: 
	
    B_1 = [  -w   1  -w   0;   0   1   0   0;   w w^2   0   0; w^2   1  -w   1 ]   B_2 = [   1   0   0   0;   0   1 w^2   0;   0   0   w   0;   0   0 w^2   1 ]
     B_3 = [    1    w -w^2    w;    0   -w   -1  w^2;    0   -1 -w^2   -1;    0  w^2   -1   -w ]   B_4 = [    0   -w  w^2   -w;    0    1    0    0;    1    w  w+2    w; -w^2   -1    w    0 ]
	
	An exemplar of type C is given by: 
	
    C_1 = [      1      0      0      0;      0      1      0      0;     -1 -w+w^2     -1    w^2;    w^2   -w+1   2w^2   -w+1 ]   C_2 = [      0    w^2    w-1    w^2;     -w      2 -w+w^2      1;    w^2     -w    w^2     -w;      0      0      0      1 ]
     C_3 = [    1    w    1 -w^2;    0    0 -w^2    w;    0    0    1    0;    0   -w   -1   -w ]   C_4 = [   1   w   0   0;   0 w^2   0   0;   0   w   1   0;   0   0   0   1 ]
	
	
	The middle convolution details can be found in Table <ref>. Different exemplars nor the inverse types produce new orbits. The two orbits from Type B are distinct as are the two orbits from Type C.
	Of the five middle convolutions computed in G_32, after putting the matrices into SL_2() three of them (one from Type B and both orbits from Type C) produce tuples where the 5-th matrix is the identity. For these three, we compare with the classification(s) by Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) to see which ℳ^(4) orbits they correspond. The orbits from Type C correspond to the same orbit in Tykhyy's classification so we only provide the details for it once. 
	The tuples that correspond to nontrivial local systems on the 5-punctured sphere are those coming from Type A and the other middle convolution from Type B. For the Type B tuple, after putting the matrices into SL_2(), we get that the product of the third and fourth matrix is not of finite order, hence we cannot compute its signature to compare with Tykhyy's table of ℳ^(5).
	The matrices in this tuple MC_-z^2+1(𝐁) are: 
	
    B̂_̂1̂   = [  z^3    z;    0 -z^3 ], B̂_̂2̂ = [  z^3-z      0; -z^3+z     -z ],
    		B̂_̂3̂ = [ -z^3   -z;    0  z^3 ],
    		B̂_̂4̂ = [      z      0;     -z -z^3+z ],
    		B̂_̂5̂ = [ z^2-1     0;     0  -z^2 ]

	
	
		
	
	
	

 §.§ Nice 5-Tuples in Rank 4 Groups

	In this case we are searching for tuples of 5 reflections generating the group such that their inverse product has a nontrivial eigenvalue of multiplicity 3. Searching all the rank 4 groups, we found that such tuples only existed in G_32. 
	

  §.§.§ G_32

	Since (G_32) = {12, 18, 24, 30}, we know that any eigenvalue of multiplicity three must be a 6-th root of unity. Using the trace, we then get that the 4-th eigenvalue must also be a 6-th root, so we do not need to expand the field to compute all the examples. 
	In G_32, there were too many possible tuples to perform an exhaustive search for all the nice tuples. We were able to find at least 4 types, which correspond to two inverse pairs, so we only provide the details for one type from each pair. 
	
	
	An exemplar of Type A is provided below with w = ζ_3: 
	
    A_1    = [    1    0    0    0;  w+1    w  w+1    0;    0    0    1    0; -w-1 -w+1 -w-1    1 ]   A_2 = [    1    0    0    0;    0    1 -w-1   -w;    0    0  w+1   -1;    0    0    w    0 ]   A_3 = [   1   w   1 w+1;   0   0 w+1   w;   0   0   1   0;   0  -w  -1  -w ]
     A_4    = [    w    0    0    0; -w-1    1    0    0;    0    0    1    0;    0    0    0    1 ]   A_5 = [    1    0    0    0;    1 2w+2 -w+1  w+1; -w-1 -w+1 -w-1   -w;    0    0    0    1 ]

	An exemplar of Type A is provided below with w = ζ_3:
	
    B_1    = [    1    w    1    w;    0    0  w+1   -1;    0  w+1 -w+1  w+1;    0   -1  w+1    0 ]   B_2 = [    w    0    0    0; -w-1    1    0    0;    0    0    1    0;    0    0    0    0 ]   B_3 = [   1   w   1   0;   0   0 w+1   0;   0   1  -w   0;   0 w+1  -w   1 ]
    
    		B_4    = [    1    0    0    0;    0    1 -w-1    0;    0    0    w    0;    0    0 -w-1    1 ]   B_5 = [    1  w+1   -w    1;    0  w+1    1  w+1;    0    0    1    0;    0 -w-1    w    0 ]

	The details of the middle convolution for both types is presented in Table <ref>. The SL_2() subgroups generated by both tuple types is the dicyclic group of order 24. Inspecting the 6 SL_2() matrices we get for both of these types, both tuples satisfy the description of `TYPE' A of ℳ^(6) belonging to the dihedral group in Tykhyy's classification <cit.>. In these cases we also could not check if different examplars/ inverses produce different orbits as we could not generate the list of all nice tuples. 
	plain
	 
	
	The notation for all the appendices are the same.
	The `ξ' column gives the residue of the root of unity in the `λ' column used to compute middle convolution. The `Character' column is the character we tensor our tuple with to get matrices in SL_2(). The column `O. Size' is the size of the SL_2() character variety orbit of the tuple (i.e after it is tensored with the character). In these tables `S. Size' denotes the size of the GL_2() subgroup generated by the 2 × 2  matrices  we get after middle convolution but before we tensor with the character. The SL_2() subgroup size after tensoring with the character is provided in each section for each group as part of the comparison tables. For infinite subgroups, the order is listed as `0', following the notation used in Magma.
	

§ MIDDLE CONVOLUTION TABLES FOR NICE 3-TUPLES
 
	

		 
			8|c|G_23 Middle Convolutions	[0pt]0pt3ex

			ξ     λ     M_1     M_2     M_3     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
			8|c|G_23 Type A Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [            1            0; -q^3-q^2 + 1            1 ]     [  1 -1;  0  1 ]     [ 1 0; 1 1 ]     (1, 1, 1, 1)     40     0 [0pt]0pt4ex 
 
			1/10     -q^3     [  q^3    0; -q^3    1 ]     [ q^3  -1;   0   1 ]     [   1   0; q^3 q^3 ]      (q, q, q, q^2)     10     600 [0pt]0pt4ex
 9/10     -q^2     [  q^2    0; -q^2    1 ]     [ q^2  -1;   0   1 ]     [   1   0; q^2 q^2 ]     (q^4, q^4, q^4, q^3)     10     600 [0pt]0pt4ex
 8|c| G_23 Type B Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [ 1 0; 1 1 ]     [  1 -1;  0  1 ]     [         1         0; q^3+q^2+2         1 ]     (1, 1, 1, 1)     40     0 [0pt]0pt4ex 
 
			3/10     -q^4     [        q^4          0; -q^4+q^3+1          1 ]     [ q^4  -1;   0   1 ]     [         1         0; q^4-q^3-1       q^4 ]     (q^3, q^3, q^3, q)     10     600 [0pt]0pt4ex
 7/10     -q     [       q       0; q^2-q+1       1 ]     [  q -1;  0  1 ]     [        1        0; -q^2+q-1        q ]     (q^2, q^2, q^2, q^4)     10     600 [0pt]0pt4ex
 8|c| G_23 Type C Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [  u-1 3u-5;    1 -u+3 ]     [  1 -1;  0  1 ]     [   1   0; u+1   1 ]     (1, 1, 1, 1)     72     0 [0pt]0pt4ex 
 
			1/6     z^5     [ -u-v-z^10      -z^5; u+2v+z^10       u+v ]     [ -z^5   -1;    0    1 ]     [        1        0; -u-v-z^5     -z^5 ]     -(z^5, z^5, z^5, -1)     18     360 [0pt]0pt4ex
 5/6     -z^10     [     z^5+v      z^10; -u-2v-z^5         v ]     [ z^10   -1;    0    1 ]     [      1      0; v+z^10   z^10 ]     (z^10, z^10, z^10, 1)     18     360 [0pt]0pt4ex
  tableHere q = ζ_5, z = ζ_30, u = z^7-z^3-z^2+1 and v = -z^4+z
	

		

			
				8|c|G_24 Middle Convolutions
				[0pt]0pt3ex
 ξ     λ     M_1     M_2     M_3     Character     O.Size     S. Size [0pt]0pt3ex
 [0.5ex]
				8|c|G_24 Type A Middle Convolutions [0pt]0pt3ex
 
				1/14     -z^4     [    z^5+z^4+z+1 -z^5+z^4+z^3+1;           -z^2         -z^5-z ]     [ z^4   1;   0   1 ]     [    1    0; -z^4  z^4 ]     (z^5, z^5, z^5, z^6)     28     0 [0pt]0pt4ex
 9/14     -z     [ -z^6-z^5-z^4    z^6-z^3+1;         -z^4     -z^3-z^2 ]     [ z 1; 0 1 ]     [  1  0; -z  z ]     (z^3, z^3, z^3, z^5)     28     0 [0pt]0pt4ex
 11/14     -z^2     [        -z^5-z^3-z -z^6 +z^5 +z^2 +1;                -z          -z^6-z^4 ]     [ z^2   1;   0   1 ]     [    1    0; -z^2  z^2 ]     -(z^6, z^6, z^6, z^3)     28     0 [0pt]0pt4ex 
 
				 tableHere z = ζ_7 
		

		
		
		
		
		
		
			 
				8|c|G_27 Type A Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				2/60     z^2     [                  z^6u-z^2 2(z^50+z^4)+z^22+z^18-z^6;             (u+1)(z^12+1)                   -z^6u+1 ]     [ -z^2 z^50;    0    1 ]     [            1            0; z^24+2z^12+1         -z^2 ]     (z^14, z^14, z^14, z^18)     60      0 [0pt]0pt4ex 
 
				38/60     -z^8     [                z^24-z^20 (z^28+2z^4)u +2(z^8-z^2);             z^14+u-z^2-1              -z^24-u+z^2 ]     [   z^8 -z^20;     0     1 ]     [          1          0; -2z^18-z^6        z^8 ]     (z^26, z^26, z^26, z^42)      60     0 [0pt]0pt4ex
 50/60     -z^20     [   -u-z^20         u; -u-z^20-1       u+1 ]     [ -z^20  z^20;     0     1 ]     [           1           0; -z^24+z^6+1       -z^20 ]     (z^20, z^20, z^20, 1)     60     360 [0pt]0pt4ex
  Here z = ζ_60 and u = -z^10-z^8+z^2
			

			 	
				8|c|G_27 Type B Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     B̃_̃1̃     B̃_̃2̃     B̃_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				5/60     z^5     1/3[       z^5f_1-2z^10+1  f_1+2z^15+3z^10-z^5;              f_1-g_1 -z^5f_1+2z^10-3z^5+2 ]     [         -z^5 -z^20f_1-u+2;            0            1 ]     [          1          0; z^25u+z^15       -z^5 ]      (q^25, q^25, q^25, q^45)     96     0 [0pt]0pt4ex 
 
				35/60     -z^5     1/3[     -z^5f_1-2z^10+1 f_1-2z^15+3z^10+z^5;             f_1+g_1 z^5f_1+2z^10+3z^5+2 ]     [          z^5 -z^20f_1-u+2;            0            1 ]     [          1          0; z^25u-z^15        z^5 ]     (q^55, q^55, q^55, q^75)     96     0 [0pt]-5pt4ex
 50/60     -z^20      [ f_1-u-1+2z^10     -f_1+2u+1;   f_1-u+2z^10   -f_1+u-z^20 ]     [      z^20 -f_1+2u+2;         0         1 ]     [       1       0; f_1-u+1    z^20 ]     (z^20, z^20, z^20, 1)      96     360 [0pt]-5pt4ex
  Here z = ζ_60, q = ζ_120, f_1 = -z^14-2z^10-z^8+z^6+z^4+z^2 and g_1 = 4z^15+3z^13-3z^11-3z^9-3z^7-5z^5+3z.
			

			 
				8|c|G_27 Type C Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     C̃_̃1̃     C̃_̃2̃     C̃_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				10/60     z^10      [ z^50v-z^10-1 z^50v-u+z^10;     2z^50v-3     -z^50v+2 ]     [ -z^10   u+2;     0     1 ]     [                     1                     0; 1/2(3z^20v+u+2z^10+4)                 -z^10 ]     (z^10, z^10, z^10, -1)     60     360 [0pt]0pt4ex 
 
				34/60     -z^4     [           -z^8-z^6    z^18f_1+2z^12+4; z^14-z^50v+2z^10+u          u-z^50v+1 ]     [ z^4 u+2;   0   1 ]     [                          1                          0; 1/2(z^4f_1+3z^24-z^12+z^6)                        z^4 ]     (z^28, z^28, z^28, z^36)     60     0 [0pt]0pt4ex
 46/60     -z^16      [      z^14-z^4+z^2               f_2; -v+z^28v+z^10-z^8    (z^50+z^28)v-u ]     [ z^16  u+2;    0    1 ]     [                          1                          0; 1/2(z^22+2z^16u+z^10-3z^6)                       z^16 ]     (z^22, z^22, z^22, z^54)     60     0 [0pt]0pt4ex
  Here z = ζ_60, v = z^16+z^4, u = -z^10-z^8+z^2, f_1 = -z^14-2z^10-z^8+z^6+z^4+z^2 and f_2 = -4z^14-z^12+z^10+2z^6+3z^4-2z^2
		
		
		

§ MIDDLE CONVOLUTION TABLES OF NICE 4-TUPLES
 
		In these sections the notation is the same as above.
		
		
		

			
			 
				10|c|G_23 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Ã_̃4̃      Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				A     1/2     -1     [    u -u+1;  u-1 -u+2 ]     [  0  1; -1  2 ]     [   1 u+2;   0   1 ]     [  1  0; -1  1 ]     (1, 1, 1, 1, 1)     -     0 [0pt]0pt4ex 
 
				 tableHere q = ζ_5 and u = q^3 +q^2
		

		

			 
				10|c|G_25 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				A      1/3     z     [  0 -z;  z -z ]     [ 1 1; 0 1 ]     [ 1 1; 0 1 ]     [  1  0; -1  1 ]     (z^2, 1, 1, 1, z)       45     0 [0pt]0pt4ex 
 
				B      1/3     z     [   1 z^2;   0   1 ]     [  1  0; -z  1 ]     [  1  0; -z  1 ]     [  1  0; -z  1 ]     (1, 1, 1, 1, 1)      45       0[0pt]0pt4ex 
 
				C     1/6     z+1     [ -z  0; -1  1 ]     [    -z    -1; z^2-1     z ]     [  -1 z+1;   0   1 ]     [  1  0;  1 -z ]     (u^7, u^3, u^3, u^7, u^4)      120      72 [0pt]0pt4ex 
 
				 tableHere z = ζ_3 and u = ζ_12 
		

		
			 
				10|c|G_26 4-Tuple Middle Convolutions  [0pt]0pt4ex

				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex 

				A     1/6     z+1     [ z^2   0; z+2   1 ]     [ -1 -1;  0  1 ]     [   1   0; z^2  -z ]     [   1   0; z^2  -z ]     (u^8, -u^3, u^7, u^7, u^5)      120      72 [0pt]0pt4ex 
 
				B     1/6     z+1     [   -z  z+1; -z+1    z ]     [  1  0;  z -z ]     [ -1 -z;  0  1 ]     [    1    0; 2z+1  z^2 ]     (u^3, u^7, u^3, u^8, u^3)      240       72 [0pt]0pt4ex 
 
				 Here z = ζ_3 and u = ζ_12
			 
				10|c|G_27 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. S.     S. Size [0pt]0pt4ex
 [0.5ex] 
				A     5/6     -z^5     [   -f_2+1   1/4g_1;      g_2 z^5 +f_2 ]     [  f_1+2 1/2f_2;    g_3    g_4 ]     [    z^5 1/4g_5;      0      1 ]     [   1   0; g_6 z^5 ]     (z^5, z^5, z^5, z^5, z^10)      -     360 [0pt]0pt4ex 
 
				 Here z = ζ_15 f_1 = z^7-z^3+z^2-1, f_2 = z^5-z^4-z +2, g_1 = 2z^7-3z^5+3z^4-2z^3+2z^2+3z-2, g_2 = 2z^7-z^4-2z^3+2z^2-z+1, g_3 = -3z^7+3z^5-2z^4+3z^3-3z^2-2z^2-2z+3, g_4 = -z^7+z^5+z^3-z^2, g_5 =  -z^7-z^5+z^3-z^2-2, g_6 = -z^7+3z^5-z^4+z^3-z^2-z+2
			 
				9|c|G_28 Type A Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Ã_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				1/6     z^4     [    1    1;    0 -z^4 ]     [ -z^4+2 -z^4+1;     -2     -1 ]     [ -z^4   -1;    0    1 ]     [     1     0; -2z^4  -z^4 ]     (-z^4, -z^4, -z^4, -z^4, (-z^4)^2)       45     72 [0pt]0pt4ex 
 
				5/6     -z^4+1     [   1   1;   0 z^8 ]     [ z^4+1   z^4;    -2    -1 ]     [ z^8  -1;   0   1 ]     [    1    0; 2z^8  z^8 ]     (z^8, z^8, z^8, z^8,-z^4)       45     72 [0pt]0pt4ex 
 
				 Here z = ζ_24
		
		
			 
				ξ     λ     M_1     M_2     M_3     M_4     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				9|c|G_30 Type A Middle Convolutions [0pt]0pt3ex
 3/10     z^18     [        -z^6v          z^6; -z^24+z^12-1        -z^24 ]     [ -z^18+z^6     -z^18;       v^2        -v ]    [ -z^18    -1;     0     1 ]     [       1       0; -z^18+v   -z^18 ]     (z^6, z^6, z^6, z^6, z^36)      50     600 [0pt]0pt4ex 
 
				7/10     -z^12     [  z^12-v   -z^24; -z^18+v     z^6 ]     [ -z^24+z^12       z^12;  z^24-z^6v     z^24+1 ]     [ z^12   -1;    0    1 ]     [            1            0; -z^24+z^12-1         z^12 ]     (z^24, z^24, z^24, z^24, z^24)    50     600 [0pt]0pt4ex
 9|c|G_30 Type B Middle Convolutions [0pt]0pt3ex
1/6     z^10      [ -z^20-z^16 -z^16-z^10;     z^16-1       z^16 ]    [ -z^10     0;  z^10     1 ]     [ -z^10   u-1;     0     1 ]     [     1     0; -z^10 -z^10 ]     -(z^10, z^10, z^10, z^10, -z^20)     90     360 [0pt]0pt4ex 
 
				5/6     -z^10+1       [   y+z^20 y+z^10-2;       -y     -y+1 ]     [  z^20     0; -z^20     1 ]     [ z^20  u-1;    0    1 ]     [    1    0; z^20 z^20 ]     (z^20, z^20, z^20, z^20, -z^10)     90     360 [0pt]-5pt4ex
 9|c|G_30 Type C Middle Convolutions [0pt]0pt3ex

				1/10     z^6      [    1 -u-1;    0 -z^6 ]     [      -v -z^12+v;       u       0 ]     [ -z^6  u+1;    0    1 ]     [       1       0; -z^12-1    -z^6 ]     (z^12, z^12, z^12, z^12, z^12)     50     600 [0pt]0pt4ex 
 
				9/10     -z^24       [    1 -u-1;    0 z^24 ]     [ z^24+1   z^6v;      u      0 ]     [ z^24  u+1;    0    1 ]     [      1      0; u+z^12   z^24 ]     (z^18, z^18, z^18, z^18, z^48)     50     600 [0pt]-5pt4ex
  Here z denotes a 60th root of unity, u = z^14-z^6-z^4 and v = z^6 -1 and y = z^10 +z^8-z^2.
			 
				ξ     λ     M_1     M_2     M_3     M_4     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				9|c|G_32 Type A Middle Convolutions [0pt]0pt3ex
 5/6     -z^2 +1     [  0 -1; -1  0 ]     [   1   0;   0 z^2 ]     [ z^2   0;   0   1 ]     [   1   0;   0 z^2 ]      (-z^3, z^5, z^5, z^5, 1)      40     72 [0pt]0pt4ex 

				9|c|G_32 Type B Middle Convolutions [0pt]0pt3ex
 1/6     z^2     [  1 -w;  0 -w ]     [  w^2  w-1;   -1 -w^2 ]     [ -w  w;  0  1 ]     [  1  0; -w -1 ]      (z, z^3, z, z^3, z^2-1)      60     72 [0pt]0pt4ex 
 
				5/6     -z^2 +1     [  1 -w;  0 -1 ]     [    1    0;   -1 -w^2 ]     [ -1  w;  0  1 ]     [  1  0; -1 -w ]     (z^3, z^5, z^3, z, 1)         0 [0pt]0pt4ex
 9|c|G_32 Type C Middle Convolutions [0pt]0pt3ex
 1/12     z     [ -z^3    0; z^11    1 ]     [    1  z^4;    0 -z^3 ]     [ -z^3 -z^4;    0    1 ]     [     1     0; -z^11  -z^3 ]      (q^3, q^3, q^3, q^3, -1)      20     96 [0pt]0pt4ex 
 
				7/12     -z     [   z^3     0; -z^11     1 ]     [   1 z^4;   0 z^3 ]     [  z^3 -z^4;    0    1 ]     [    1    0; z^11  z^3 ]     (q^9, q^9, q^9, q^9, -1)    20     96 [0pt]0pt4ex
  Here z = ζ_12, w = ζ_3 and q = ζ_24
		
		
		

§ MIDDLE CONVOLUTION TABLES FOR NICE 5-TUPLES
 
		The notation here is the same as above.

		
		


	
	
	
	
	
	


 Paper Title: A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences.



Slices of Schwarz reflections, and B-involutions]A general dynamical theory
 of Schwarz reflections, B-involutions,
 and algebraic correspondences




[Y. Luo]Yusheng Luo
Department of Mathematics, Cornell University, 212 Garden Ave, Ithaca, NY 14853, USA
yusheng.s.luo@gmail.com

Y.L. was partially supported by NSF Grant DMS-2349929.

[M. Lyubich]Mikhail Lyubich
Institute for Mathematical Sciences, Stony Brook University, 100 Nicolls Rd, Stony Brook, NY 11794-3660, USA
mlyubich@math.stonybrook.edu

M.L. was partially supported by NSF grants DMS-2247613 and 1901357, the Clay Fellowship, and the Centre for Nonlinear Analysis and Modeling (CNAM)

[S. Mukherjee]Sabyasachi Mukherjee
School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India
sabya@math.tifr.res.in
S.M. was supported by the Department of Atomic Energy, Government of India, under project no.12-R&D-TFR-5.01-0500, an endowment of the Infosys Foundation, and SERB research project grants MTR/2022/000248.


§ ABSTRACT 
In this paper, we study matings of (anti-)polynomials and Fuchsian/reflection groups as Schwarz reflections/B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces.
We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups.
These matings emerge naturally as degenerate (anti-)polynomial-like
maps, and we show that the corresponding parameter space slices for such matings bear strong resemblance with parameter spaces of polynomial maps.
Furthermore, we provide algebraic descriptions for these matings, and construct algebraic correspondences that combine generic (anti-)polynomials and genus zero orbifolds in a common dynamical plane, providing a new concrete evidence to Fatou’s vision of a unified theory of groups and maps.


[
    Amy Navarathna,1,2 Benjamin J. Carey,1,2 James S. Bennett,3 Soroush Khademi,1,2 and Warwick P. Bowen1,2,4*
    August 5, 2024
==============================================================================================================














§ INTRODUCTION







A dynamical theory of Schwarz reflection maps has emerged in the past decade
that provided a general framework for matings between anti-polynomial  maps and Fuchsian reflection groups as (anti-)algebraic correspondences, leading to relations between the corresponding parameter spaces (which also gave a new insight into the holomorphic case). Many special families have been thoroughly explored <cit.>. In this paper we will unify these various pieces into
a general theory.

A prototype of the theory we develop is the classical theory of polynomial-like maps.
Polynomial-like maps were first defined by Douady and Hubbard as a tool to study similarities between parameter spaces of complex polynomials and those of more general families of holomorphic maps <cit.>. This led to great deal of work on the understanding of parameter spaces of complex dynamical systems. In the process, intimate connections between polynomial-like maps and renormalization theory emerged (see <cit.>). 

Inou and Kiwi developed a more general theory of polynomial-like maps and the associated straightening maps (that relate parameter spaces of holomorphic maps to appropriate standard models) which address the additional complexity furnished by higher degree renormalizations and critical orbit interactions <cit.>. 

However, the above theory has some limitations: over the years, various situations related to parabolic external dynamics started to emerge which necessitated a more general set-up of pinched polynomial-like maps <cit.>. 

The presence of parabolic elements is also an important trait of all the groups that appear in our mating theory, which brings us to the general theory of pinched polynomial-like maps (that generalizes a theory that has been recently developed in <cit.>). 


The principal results of this paper can be summarized as follows.


  *  We prove the existence of conformal matings between `generic' polynomials (with connected Julia set) and large classes of Fuchsian/reflection groups. Such matings are realized both as maps in one complex variable as well as algebraic correspondences (Theorems <ref>, <ref>, <ref>, <ref>).


  * We introduce the class of degenerate polynomial-like maps that unifies the above mating constructions. This framework allows us

    
  * to interpret the mating structure of algebraic correspondences in terms of the classical description of polynomial-like maps as matings of `hybrid classes' and `external maps' (Sections <ref>, <ref>, <ref>, <ref>), and

    
  * to prove general bijection/homeomorphism statements between connectedness loci of polynomials and various parameter spaces of matings (Theorems <ref>, <ref>, <ref>, <ref>, <ref>).


In item (<ref>) above, the term `generic' refers to the collection of at most finitely renormalizable, periodically repelling polynomials and geometrically finite polynomials.

Before elaborating on the main theorems, let us briefly mention some of the technical novelties involved in carrying out the tasks mentioned above. 



  * We use the structure of degenerate polynomial-like maps to adapt classical `puzzle piece' machinery for our setting. This plays a pivotal role in establishing various combinatorial continuity and rigidity results, which lie at the heart of our conformal mateability theorems (Sections <ref> and <ref>).


  * The construction of algebraic correspondences (global objects) from one variable matings between polynomials and groups (i.e., certain degenerate polynomial-like maps that are semi-global objects) requires algebraic descriptions of such matings. This is straightforward for Schwarz reflection maps, which are known to be `rationally semi-conjugate' to reflection in the unit circle. To deal with the holomorphic situation, we introduce a class of meromorphic maps called B-involutions, and give an explicit algebraic description to facilitate the task of globalization (see Section <ref>, cf. <cit.>).


  * In order to manufacture algebraic correspondences that are matings between generic polynomials and groups with multiple parabolics, we need to define such correspondences on nodal spheres or cacti of spheres. This extends previously
known mating frameworks for correspondences; and in particular, gives first examples of correspondences associated with piecewise Schwarz reflection maps (Sections <ref>, <ref>). Even for the simplest examples of piecewise Schwarz reflection maps (namely, the Circle-and-Cardioid Schwarz reflections studied in <cit.>), such a correspondence description is new.


In Appendix <ref>, we investigate canonical product structures in certain families of degenerate polynomial-like maps. This product structure is based on the classical idea of decomposing a polynomial-like map into a hybrid class and an external map <cit.>. Loosely speaking, these families bind connectedness loci of polynomials (parameter space of hybrid classes) and Teichmüller spaces of certain surfaces (parameter space of external maps) together in a dynamically natural fashion. A more detailed analysis of the boundary of such families of degenerate polynomial-like maps and the associated degeneration phenomena will be presented in a subsequent work.




 §.§ Antiholomorphic matings: Schwarz reflections

A domain Ω in the Riemann sphere is called a quadrature domain if it admits an anti-meromorphic map S:Ω→ that continuously extends to the identity map on the boundary ∂Ω. Such a map  S:Ω→ is called the Schwarz reflection map of Ω. A simply connected domain Ω⊊ is a quadrature domain if and only if its Riemann uniformization ϕ:→Ω extends as a rational map of the Riemann sphere. 

A quadrature multi-domain  is the union of finitely many disjoint simply connected domains {Ω_j}_j=1^k admitting a continuous map S:→ which is anti-meromorphic on  and which fixes ∂ pointwise. Each component of a quadrature multi-domain is a quadrature domain. The map S, which we call a Schwarz reflection of a quadrature multi-domain, acts as the Schwarz reflection of Ω_j on each Ω_j, for j∈{1,⋯,k}.


The compact set T(S):=∖ is called a droplet. Its boundary consists of real-analytic curves possibly with finitely many cusp and double point singularities. Removing these finitely many singular points from the droplet yields the desingularized droplet or the fundamental tile of S. We denote the fundamental tile of S by T^0(S).

The dynamical plane of S admits an invariant partition into the escaping/tiling set T^∞(S) and the non-escaping set K(S). The former set  comprises all points that eventually escape to the fundamental tile T^0(S), and the latter set is its complement (see Section <ref> for precise definitions).


The exploration of Schwarz reflection dynamical systems was initiated in <cit.>, and it was observed that in many cases, the Schwarz reflection map S of a quadrature multi-domain is a combination/mating of an anti-holomorphic polynomial (anti-polynomial for short) and an ideal polygon reflection group. Roughly, a Schwarz reflection S is called a mating of an anti-polynomial and  an ideal polygon reflection group if the dynamics of S on its non-escaping set is topologically conjugate (conformally conjugate on the interior) to that of an anti-polynomial on its filled Julia set, while the action of S on the tiling set is conjugate to a piecewise anti-Möbius map, called the Nielsen map, that is orbit equivalent to the ideal polygon reflection group (see Definition <ref> for the precise notion of conformal mating and Figure <ref> for an illustration of the Nielsen map of the ideal triangle reflection group). The Nielsen map of an ideal (d+1)-gon reflection group is also compatible with degree d anti-polynomials as the action of the Nielsen map on the unit circle is topologically conjugate to the map z^d.
In <cit.>, the so-called Circle-and-Cardioid (or C&C) family of Schwarz reflections was studied in details, and it was shown that many maps in the C&C family arise as matings of quadratic anti-polynomials and the ideal triangle reflection group.

In the following, we will discuss the general theory of two natural classes of Schwarz reflections:


  * polygonal Schwarz reflections; and

  * anti-Farey Schwarz reflections.




  §.§.§ Polygonal Schwarz reflections


We say that  is a tree-like quadrature multi-domain if  is connected and simply connected. In this case, each Ω_j is a Jordan domain, and their touching structure can be recorded by a plane tree (see Figure <ref>).
A Schwarz reflection S:→ associated with a tree-like quadrature multi-domain is polygonal if the fundamental tile T^0(S) is conformally equivalent to an ideal (d+1)-gon in the hyperbolic plane (where d is the degree of the Schwarz reflection), and S has no critical value in T^0(S). The terminology `polygonal' is motivated by the fact that a polygonal Schwarz reflection is conformally conjugate to the Nielsen map of an ideal polygon reflection group near its droplet (see Section <ref> for more details).
We remark that Schwarz reflections in the C&C family are prototypical examples of polygonal Schwarz reflections. 


As in the degree two case, a degree d polygonal Schwarz reflection map S has a connected non-escaping set if and only if no critical point of S escapes to the fundamental tile. This property is also equivalent to the fact that the tiling set dynamics of S is conformally conjugate to the Nielsen map of an ideal (d+1)-gon reflection group. 



We denote the connectedness locus of (normalized) degree d polygonal Schwarz reflections by 𝒮__d. Similarly, we denote by 𝒞^-_d the connectedness locus of monic and centered anti-polynomials of degree d. We say that two maps f, g∈𝒞^-_d (respectively, f, g∈𝒮__d) are combinatorially equivalent, denoted by f∼ g, if they have the same rational laminations (in analogy with polynomial dynamics, the rational lamination of a polygonal Schwarz reflection captures the co-landing patterns of its pre-periodic dynamical rays, see Definition <ref>).

Let 𝒞^-_d, r⊂𝒞^-_d be the subset of periodically repelling anti-polynomials. For polygonal Schwarz reflections, the set 𝔖 of singular (fixed) points on the droplet boundary is special as these points always have parabolic dynamics. Hence, we define the analogous subspace 𝒮__d, r for polygonal Schwarz reflections as the collection of those maps S in 𝒮__d for which no point of 𝔖 has an attracting direction in K(S) and each periodic point of S not in 𝔖 is repelling. Finally, let 𝒞^-_d, r = 𝒞^-_d, r/∼ (respectively, 𝒮__d, r = 𝒮__d, r/∼) be the space of corresponding combinatorial classes.

The following theorems generalize and sharpen the main results of <cit.>. We remark that in <cit.>, only the space of geometrically finite degree two polygonal Schwarz reflections were studied.


There is a dynamically natural homeomorphism 

    Φ: 𝒞^-_d, r⟶𝒮__d, r.

Here `dynamically natural' means that the circle homeomorphism conjugating the Nielsen map to the power map z^d carries the rational lamination of a Schwarz reflection to the rational lamination of an anti-polynomial.


(See Section <ref> for the proof of Theorem <ref>.)

Let 𝒮__d, fr⊂𝒮__d, r be the subset of at most finitely renormalizable periodically repelling Schwarz reflections, and let 𝒞^-_d, fr⊂𝒞^-_d,r be its counterpart for anti-polynomials.
Adapting similar methods as in the polynomial setting, we show that each map in 𝒮__d, fr (or 𝒞^-_d, fr) is combinatorially rigid; i.e., its combinatorial class consists of a single map.
As an immediate corollary of this rigidity, we have the following result, which is proved in Section <ref>.

There is a dynamically natural homeomorphism 

    Φ: 𝒞^-_d, fr⟶𝒮__d, fr.

Here dynamically natural means that for any f∈𝒞^-_d, fr, the Schwarz reflection Φ(f) is the unique conformal mating of f and the Nielsen map of the ideal (d+1)-gon reflection group.









It is natural to ask if the above natural homeomorphism can be extended to a homeomorphism between 𝒞^-_d and 𝒮__d.
The next result shows that it is not the case.

There is a dynamically natural bijection 

    Φ: 𝒞^-_d, gf⟶𝒮__d, gf
 
between geometrically finite anti-polynomials and polygonal Schwarz reflections with connected Julia/limit set, such that for any f∈𝒞^-_d, gf, the Schwarz reflection Φ(f) is the unique conformal mating of f and the Nielsen map of the ideal (d+1)-gon reflection group.
However, both Φ and Φ^-1 are discontinuous.


The existence of the bijection Φ appearing in Theorem <ref> is established in Section <ref>, and the discontinuity of Φ, Φ^-1 is proved in Section <ref>.


The above theorem is analogous to a famous theorem of Kerckhoff-Thurston which states that the natural identification between two different Bers embeddings fails to extend to a continuous map on their boundaries.

It is worth emphasizing that the map Φ that relates the connectedness loci 𝒞^-_d and 𝒮__d (appearing in the previous two theorems) replaces the external dynamics of an anti-polynomial f (i.e., the dynamics of f on its basin of infinity) with the action of the Nielsen map of an ideal (d+1)-gon reflection group, or equivalently, mates the filled Julia set dynamics of f with the dynamics of the Nielsen map. The following implication of Theorems <ref>, <ref> and <ref> is important in its own right (see Definition <ref> for the definition of combinatorial mating).


Let f be a degree d anti-polynomial with connected Julia set.
Suppose that f is either 


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.

Then, f can be conformally mated with the Nielsen map of an ideal (d+1)-gon reflection group, and the resulting Schwarz reflection map is unique up to Möbius conjugacy.

Further, each periodically repelling map in 𝒞_d^- can be combinatorially mated with the Nielsen map of an ideal (d+1)-gon reflection group.





We remark that notions of mating without assuming local connectivity of Julia sets also appeared <cit.>.



  §.§.§ Anti-Farey Schwarz reflections

In the antiholomorphic world, there is another important expansive piecewise real-analytic circle endomorphism, called the anti-Farey map, that is closely related to reflection groups. The anti-Farey map, which was first formally introduced in <cit.> (and had its precursors in <cit.>), is a factor of the Nielsen map of a regular ideal polygon reflection group (see Section <ref>). Schwarz reflection maps that admit the anti-Farey map as the conformal model of their escaping set dynamics will be referred to as anti-Farey Schwarz reflections.


The proofs of Theorems <ref>–<ref> can be adapted for the family of anti-Farey Schwarz reflections, demonstrating close parameter space relations between the space of anti-Farey Schwarz reflections and the connectedness loci of anti-polynomials. We refer the reader to Theorem <ref> for the precise statements. As a consequence of these results, we have the following generalization of <cit.>.

Let f be a degree d anti-polynomial with connected Julia set.
Suppose that f is either 


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.

Then, f can be conformally mated with the degree d anti-Farey map, and the resulting Schwarz reflection map is unique up to Möbius conjugacy.

Further, each periodically repelling map in 𝒞_d^- can be combinatorially mated with the degree d anti-Farey map.




 §.§ Holomorphic matings: B-involutions

We now pass to the holomorphic analog of Schwarz reflection maps that we will study in this paper. 
In order to develop a mating framework for complex polynomials and Fuchsian groups, we define a suitable class of holomorphic maps which we call B-involutions. 
Special cases of B-involutions were introduced in <cit.> in the context of mating parabolic rational maps with Hecke groups (cf. Section <ref>).

An inversive multi-domain  is the union of finitely many disjoint simply connected domains {Ω_j}_j=1^k admitting a continuous map S:→ which is meromorphic on  and which restricts to an orientation-reversing order two self-homeomorphism of the boundary ∂. The map S is called a B-involution of an inversive multi-domain (see Section <ref> for a precise definition). 
We use a conformal welding argument to give an algebraic characterization of inversive multi-domains and B-involutions in terms of uniformizing rational maps. This parallels the characterization of quadrature domains as univalent images of disks under global rational maps (see Theorem <ref>).






To study the dynamics of a B-involution of an inversive multi-domain, we partition its dynamical plane into escaping and non-escaping sets (see Section <ref>). The action of a B-involution of an inversive multi-domain on its escaping set will be referred to as its external dynamics.

Natural examples of such B-involutions are provided by matings of complex polynomials with connected Julia set and certain orientation-preserving external maps, called factor Bowen-Series maps (see Theorem <ref> and Proposition <ref>). Factor Bowen-Series maps, which first appeared in <cit.>, can be thought of as holomorphic cousins of Nielsen or anti-Farey maps. These maps are semi-conjugates of classical Bowen-Series maps associated with certain genus zero orbifolds (more precisely, with the Fuchsian models of such orbifolds); in particular, they are piecewise analytic, expansive, circle coverings of degree greater than one. A formal description of factor Bowen-Series maps is given in Section <ref>.

Our analysis shows that spaces of B-involutions having factor Bowen-Series maps as the conformal model of their external dynamics are closely related to the connectedness loci of complex polynomials (see Theorem <ref>). As a consequence of  Theorem <ref>, we have an analog of Theorem <ref> that confirms the existence of conformal matings of generic complex polynomials (with connected Julia set) and appropriate factor Bowen-Series maps.




Let f be a degree d polynomial with connected Julia set.
Suppose that f is either 


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.

Then, f can be conformally mated with any degree d factor Bowen-Series map, and the resulting B-involution is unique up to Möbius conjugacy. 

Further, each periodically repelling degree d polynomial with connected Julia set can be combinatorially mated with any degree d factor Bowen-Series map.





 §.§ Matings as degenerate (anti-)polynomial-like maps


To provide a common language for all the mating results stated above, we now put forward an appropriate generalization of polynomial-like maps, called degenerate (anti-)polynomial-like maps. Pinched versions of polynomial-like maps were introduced in <cit.>, and more general notions (still not sufficient for the purpose of the current paper) appeared in the papers mentioned above.

A polygon is a closed Jordan disk in  with a piecewise smooth boundary. The points where two smooth local boundary arcs meet are called the corners of the polygon (note that cusps are also considered to be corners).

A pinched polygon is a set in  which is homeomorphic to a closed disk quotiented by a finite geodesic lamination, and which has a piecewise smooth boundary. The cut-points of a pinched polygon will be called its pinched points, and the non-cut points where two smooth local boundary arcs meet are called the corners of the pinched polygon. 


Let P_1, P_2⊂ be pinched polygons such that 


  * P_1⊂ P_2,

  * the pinched points (respectively, corners) of P_2 are also pinched points (respectively, corners) of P_1, and

  * ∂ P_1∩∂ P_2 consists of the corners and pinched points of P_2.

Suppose that there is a degree d continuous map g:P_1 → P_2 which is a proper holomorphic/anti-holomorphic map from each component of P_1 onto some component of P_2, and such that the singular points of ∂ P_1 are the g-preimages of the singular points of ∂ P_2.



We then call the triple (g, P_1, P_2) a degenerate (anti-)polynomial-like map of degree d.



As with polynomial-like maps, we define the filled Julia set or non-escaping set of a degenerate (anti-)polynomial-like map to be 

    K(g)=⋂_n≥ 0 g^-n( P_1).
 
Analogous to classical polynomial-like maps, the filled Julia set K(g) of a degenerate (anti-)polynomial-like map is connected if and only if it contains all of the critical values of g.

For a degree d degenerate (anti-)polynomial-like map (g, P_1, P_2) with connected filled Julia set, one can define the notion of an external map following the classical construction of external maps of polynomial-like maps (cf. <cit.>). Note that the external map of such a g is a piecewise real-analytic, degree d, covering map of 𝕊^1. Since the fundamental domain P_2∖P_1 of g is a pinched annulus, the external map of g is in general not expanding on 𝕊^1. This causes significant difficulty in the study of degenerate polynomial-like maps.
In fact, the external maps of all degenerate polynomial-like maps that we will encounter in this paper have parabolic cycles and hence are not expanding. However, they enjoy the weaker property of being expansive on 𝕊^1.


Polygonal or anti-Farey Schwarz reflections and B-involutions are some natural examples of degenerate (anti-)polynomial-like maps.
For concreteness, consider S:→ as a polygonal Schwarz reflection map.
The restriction S:S^-1()→ is a finite degree proper anti-holomorphic map from a pinched polygon to a larger pinched polygon (see Figure <ref>). We note that S^-1() is not compactly contained in . Thus, the map S:S^-1()→ is a degenerate anti-polynomial-like maps as in Definition <ref>.

The description of S as a mating of an anti-polynomial and an ideal polygon reflection group can be recast in the classical language of polynomial-like maps: the internal class of S:S^-1()→ is that of a degree d anti-polynomial with connected Julia set and its external class is represented by the Nielsen map of an ideal (d+1)-gon reflection group.
The parameter space homeomorphisms appearing in the theorems stated in the previous subsections can be regarded as straightening theorems between slices of degenerate (anti-)polynomial-like maps and connectedness loci of (anti-)polynomials. 






 §.§ Matings as (anti-)holomorphic correspondences


The first obstruction to mating a polynomial map with a reflection/Fuchsian group is the fundamental mismatch between invertible and non-invertible dynamics. 
The passage from a group to its Nielsen map, anti-Farey map or factor Bowen-Series map (i.e., reduction from group dynamics to semi-group dynamics) allowed us to address this discord and combine the two dynamical systems into a single plane. However, as the conformal mating is a non-invertible map, its dynamics does not showcase the whole structure of the underlying group.

It was originally proposed by Fatou in the 1920s that the category of dynamical systems which truly encompasses rational dynamics as well as actions of Kleinian group is given by algebraic correspondences <cit.>. The first precise formulation of this proposal was given by Bullett and Penrose <cit.>, who studied certain algebraic correspondences of bi-degree 2:2 on the Riemann sphere that combine the actions of quadratic polynomials with the modular group in a precise sense. 


Let f be a degree d (anti-)polynomial with connected Julia set and let G be a discrete subgroup of (anti-)conformal automorphism group of the unit disk .
Let ℭ be an (anti-)holomorphic correspondence on a compact, simply connected (possibly noded) Riemann surface 𝔚.

We say ℭ a mating of f and G if there is a ℭ-invariant partition 𝔚=⊔ such that the following hold.

	
  * On , the dynamics of ℭ is equivalent to the action of a group of (anti-)conformal automorphisms acting properly discontinuously. Further, /ℭ is biholomorphic to /G.
	
	
  *  can be written as the union of two copies _1, _2 of (f) (where (f) is the filled Julia set of f), such that _1 and _2 intersect in finitely many points. Furthermore, ℭ has a forward (respectively, backward) branch carrying _1 (respectively, _2) onto itself with degree d, and this branch is conformally (respectively, anti-conformally) conjugate to f|_𝒦(f). 



It turns out that in all three cases we considered in this paper, we can define (anti-)holomorphic correspondences that combine actions of (anti-)polynomials with the corresponding groups.
Specifically, such a correspondence is manufactured by pulling back the dynamics of the mating via the rational maps that uniformize the associated simply connected quadrature/inversive domains.
The first instance of this construction appeared in <cit.>, where matings of z^d with regular ideal polygon reflection groups were realized as antiholomorphic correspondences. 
Our next theorem provides a systematic way of producing such examples.


Let G be an ideal (d+1)-gon reflection group or the anti-Hecke group.

Let f be a degree d anti-polynomial with connected Julia set which is either


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.


Then there exists an antiholomorphic correspondence ℭ which is a mating of f and G.

Similarly, let Σ be a hyperbolic orbifold of genus zero with arbitrarily many (at least one) punctures, at most one order two orbifold point, and at most one order ν≥ 3 orbifold point, with the corresponding Fuchsian group G.
Let 

    d(Σ)=
    1-2ν·χ_orb(Σ)  if Σ has an order ν≥ 3 orbifold point,
    
    1-2χ_orb(Σ)  otherwise.
 
Let f be a degree d(Σ) polynomial with connected Julia set which is either


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.

Then there exists a holomorphic correspondence ℭ which is a mating of f and G.


The antiholomorphic parts of Theorem <ref> are proved in Sections <ref>, <ref>, and the holomorphic part of it is proved in Section <ref>.

We remark that the correspondences produced by Theorem <ref> are reversible; i.e., their forward branches are (anti-)conformally conjugate to their backward branches. This property facilitates the dynamical study of such correspondences.



 §.§ Connections with known combination theorems for correspondences




  §.§.§ Special cases of Theorem <ref>

We now list several known results in the literature that are generalized by Theorem <ref>.


  * The algebraic correspondences that arise as matings of quadratic polynomials and the modular group are precisely the bi-degree 2:2 correspondences introduced by Bullett and Penrose <cit.> (cf. <cit.>). Bullett and Haïssinsky showed in <cit.> that this space of correspondences contains matings of Collet-Eckmann quadratic polynomials (with connected Julia set) and the modular group.



  * The antiholomorphic correspondences that arise from Theorem <ref> as matings of quadratic anti-polynomials and the anti-Hecke group isomorphic to /2∗/3 were investigated in <cit.>.


  * Some examples of correspondences realizing matings of degree d complex polynomials with the Hecke group (isomorphic to /2∗/(d+1)) were furnished in <cit.>. In <cit.>, it was conjectured that all complex polynomials with connected Julia set can be mated with the corresponding Hecke group. Theorem <ref> proves this conjecture for all  geometrically finite polynomials and finitely renormalizable, periodically repelling polynomials.


  * In <cit.>, correspondences were constructed as matings of degree d semi-hyperbolic anti-polynomials (i.e., anti-polynomials with no parabolic cycles and no recurrent Julia critical points) with connected Julia set and the anti-Hecke group (isomorphic to /2∗/(d+1)).


  * In <cit.>, correspondences were constructed as matings of semi-hyperbolic polynomials with connected Julia set and the corresponding Fuchsian group under an additionally imposed real-symmetry condition. These examples also become special cases of Theorem <ref>.





  §.§.§ (Anti-)polynomials vs parabolic (anti-)rational maps


We now compare Theorem <ref> with related (but technically different) constructions of algebraic correspondences as matings of parabolic (anti-)rational maps and (anti-)Hecke groups.

The external maps arising from reflection/Fuchsian groups (namely, Nielsen, anti-Farey, and factor Bowen-Series maps) have at least one parabolic fixed point on the circle, and hence are not quasisymmetrically conjugate to the power map z^d or z^d. This discrepancy prohibits one from applying standard quasiconformal surgery tools to mate Nielsen/anti-Farey/factor Bowen-Series maps with (anti-)polynomials. In other words, the conformal matings described in Theorems <ref>, <ref>, and <ref> cannot be constructed by quasiconformal surgery.

However, it turns out that the anti-Farey maps and the factor Bowen-Series maps associated with Hecke groups are somewhat special as they have a unique parabolic fixed point on the circle. Equivalently, the (anti-)Hecke groups have a unique conjugacy class of parabolic elements. This implies that the above maps are quasisymmetrically conjugate to parabolic (anti-)Blaschke products since the latter class of maps also have a unique parabolic fixed point on the circle. This quasisymmetric compatibility property was utilized


  * to prove that every quadratic parabolic rational map (with connected Julia set) can be mated with the modular group as a holomorphic correspondence <cit.>,


  * to construct anti-holomorphic correspondences as matings of arbitrary degree d parabolic anti-rational maps (with connected Julia set) with anti-Hecke groups <cit.>, and 


  * to construct holomorphic correspondences as matings of arbitrary degree d parabolic rational maps (with connected Julia set) with Hecke groups <cit.>.


To conclude, let us note that for semi-hyperbolic (anti-)polynomials with connected Julia set, the conformal matings furnished by Theorems <ref>, <ref>, and <ref> have David regularity. Specifically, for a semi-hyperbolic (anti-)polynomial f with connected Julia set, the topological conjugacy (conformal on the interior) from the filled Julia set dynamics of f to the non-escaping set dynamics of the conformal mating between f and the Nielsen/anti-Farey/factor Bowen-Series map is the restriction of a global David homeomorphism (cf. <cit.>, <cit.>). In general, we ask the following question:


What is the regularity of the mating between a general periodically repelling, finitely renormalizable (anti-)polynomial  and the Nielsen/anti-Farey/factor Bowen-Series map?




 §.§ Proof techniques

Here we present an overview of the techniques employed in this paper:

    
  * Utilizing the David surgery method, we construct matings of post-critically finite (anti-)polynomials / geometrically finite (anti-)polynomials with appropriate ideal polygon reflection groups / anti-Hecke groups / Fuchsian genus zero orbifold groups.
    It is important to note that while David surgery is a powerful tool, there are analytic and geometric constraints, as the current techniques only apply to John domains.
    
  * To extend our constructions to more general setting, we take limits of post-critically finite matings.
    These matings emerge naturally as degenerate (anti-)polynomial-like maps.
    As a preliminary step, we prove a boundedness result for the connectivity locus of such degenerate (anti-)polynomial-like maps.
    
  * The structure of degenerate (anti-)polynomial-like maps enables us to adapt puzzle techniques, which allow us to establish combinatorial continuity of the association between periodically repelling (anti-)polynomials and their corresponding matings.
    
  * By dividing the plane into pieces using puzzles, we apply generalized renormalization methods to establish combinatorial rigidity for periodically repelling, finitely renormalizable matings.

These results permit us to conclude the existence of conformal matings and combinatorial matings, as well as continuity and combinatorial continuity of `straightening maps' between the parameter spaces.
Notably, the fixed ray puzzles give a natural stratification of the space of Schwarz reflections. 
This stratification leads to discontinuity of the straightening map between geometrically finite parameters.
Furthermore, we derive algebraic descriptions for these matings, which enable us to transfer the previous mating results to the framework of correspondences.






Acknowledgements. Part of the work on this project was carried out during the authors' visit to The Fields Institute, Urgench State University, and Simons Laufer Mathematical Sciences Institute (MSRI).



Notation: 


  * For X⊂, the sets ∂ X, X^c stand for the boundary and the complement of X in .


  * η^+(z)=1/z.


  * η^-(z)=1/z.


  * ^*=∖.


  * B(a,r):={| z-a|<r}, where a∈ and r>0.


  * B(a,r):={| z-a|≤ r}, where a∈ and r>0.





  * Aut() = the group of all Möbius automorphisms of ,
 
Aut^±() = the group of all Möbius and anti-Möbius automorphisms of .






PART: 
Antiholomorphic world





§ REFLECTION GROUPS AND ASSOCIATED EXTERNAL MAPS


We denote the regular (d+1)-gon reflection group by G_d. It is generated by reflections ρ_1,⋯,ρ_d+1 in the circles C_1,⋯,C_d+1 such that C_i∩ is a hyperbolic geodesic of  connecting ω^i-1 and ω^i, where ω:=e^2π i/d+1. Let Π be the ideal polygon in  with vertices at the (d+1)-st roots of unity. Then Π is a fundamental domain for the G_d-action on . 

The group of conformal and anti-conformal automorphisms of  generated by G_d and ⟨ M_ω⟩, where M_ω(z)=ω z and ω:=e^2π i/d+1, is called the anti-Hecke group. We denote the anti-Hecke group by G_d. It is an index d+1 extension of G_d.





 §.§ Nielsen map
 The Nielsen map _d associated with the regular ideal polygon group G_d is defined as

    _d:∖Π⟶,   z↦ρ_i(z)  if   z∈H_i,

where H_i is the hyperbolic half-plane cut out by the bi-infinite geodesic C_i∩ and the counter-clockwise circular arc ω^i-1,ω^i, for i∈{1,⋯,d+1} (see Figure <ref>). The Nielsen map _d fixes ∂Π pointwise, and restricts to an orientation-reversing, C^1, expansive, degree d covering map of 𝕊^1. Hence, it is topologically conjugate to the map z^d: 𝕊^1 ⟶𝕊^1. There is a unique such conjugacy that sends 1 to 1. We denote this map by ℰ_d:𝕊^1→𝕊^1, and call it the d-th Minkowski circle homeomorphism.



 §.§ Anti-Farey map


Let us now associate a circle endomorphism with the anti-Hecke group G_d.
Roughly speaking, we first pass to the index d subgroup G_d⩽G_d, look at its Nielsen map _d, and then pass to a semi-conjugate (or factor) of the Nielsen map using the torsion element M_ω.

More precisely, observe that due to the rotational symmetry of the fundamental domain Π, the Nielsen map _d of commutes with M_ω. This allows one to construct a factor dynamical system

    _d:(∖Π)⟨ M_ω⟩⟶⟨ M_ω⟩

that is semi-conjugate to _d:∖Π→ via the quotient map → /⟨ M_ω⟩.


Finally, as z↦ z^d+1 defines a conformal isomorphism ξ:/⟨ M_ω⟩⟶, we have the so-called degree d anti-Farey map

    _d:= ξ∘_d∘ξ^-1:∖ℋ→,

where ℋ:=ξ(Π/⟨ M_ω⟩) (see Figure <ref>).

The fact that _d fixes ∂Π pointwise, implies that _d fixes ∂ℋ pointwise. Moreover, the map _d restricts to an expansive, orientation-reversing, degree d, piecewise anti-analytic covering of 𝕊^1. Hence, _d|_𝕊^1 is topologically conjugate to z^d_𝕊^1.

The anti-Farey map _d has a couple of key differences from the Nielsen map _d. These dissimilarities come from passing to a factor dynamical system. Note that the piecewise Möbius map _d has no critical point, but the anti-Farey map has a unique critical point (of multiplicity d) at ξ(ρ_1(0)) with associated critical value at 0. Further, while the Nielsen map has d+1 parabolic fixed points on the circle, the anti-Farey map has a unique parabolic fixed point on 𝕊^1 (at the point 1).


We refer the reader to <cit.> for a detailed discussion of the anti-Farey map and the anti-Hecke group.


The idea of constructing an external map as a factor of another will also be used in Section <ref> to define factor Bowen-Series maps.




§ SCHWARZ REFLECTION MAPS




 §.§ Simply connected quadrature domains



A domain Ω⊊ with ∞∉∂Ω and int(Ω)=Ω is called a quadrature domain if there exists a continuous function S:Ω→ satisfying the following two properties:


  * S=id on ∂Ω, and


  * S is anti-meromorphic on Ω.


The map S is called the Schwarz reflection map of Ω.


The boundary of a quadrature domain is real-analytic with conformal cusps and double points as only possible singularities <cit.>. Thus, the map S can be regarded as a semi-global anti-meromorphic extension of the local Schwarz reflection maps with respect to the non-singular points of ∂Ω.




<cit.>,<cit.>
A simply connected domain Ω⊊ (with ∞∉∂Ω and int(Ω)=Ω) is a quadrature domain if and only if the Riemann map ϕ:𝔻→Ω is rational.
In this case, the Schwarz reflection map S of Ω is given by ϕ∘η^-∘(ϕ|_)^-1. 

Moreover, if the degree of the rational map ϕ is d, then S:S^-1(Ω)→Ω is a (branched) covering of degree (d-1), and S:S^-1(Ω^c)→Ω^c is a (branched) covering of degree d.


    d[swap]η^-rϕ   ΩdS
    rϕ   ℂ
 

We will call the degree of the rational map ϕ the degree of the associated quadrature domain Ω.



 §.§ Schwarz reflections in quadrature multi-domains



Let {Ω_1,⋯,Ω_k} be a finite collection of disjoint simply connected domains with Ω_j⊊, ∞∉∂Ω_j and int(Ω_j)=Ω_j, for j=1,⋯,k. We call =_j=1^k Ω_j a quadrature multi-domain if there exists a continuous function S:→ satisfying the following two properties:


  * S=id on ∂, and


  * S is anti-meromorphic on .


The map S is called the Schwarz reflection map of the quadrature multi-domain .


Note that for a quadrature multi-domain :=_j=1^kΩ_j, each component Ω_j is a quadrature domain and S|_Ω_j agrees with the Schwarz reflection map S_j of Ω_j. 

Let ϕ_j:→Ω_j be the Riemann uniformizations of the simply connected quadrature domains Ω_j such that each ϕ_j extends as a rational map of  of degree d_j. It follows that S_j:S_j^-1(Ω_j)→Ω_j is a branched covering of degree (d_j-1), and S_j:S_j^-1(Ω_j^c)→Ω_j^c is a branched covering of degree d_j. Therefore, S:S^-1()→ is a branched covering of degree 

    d:=∑_j=1^k d_j-1.

On the other hand, S:S^-1(^c)→^c is a branched covering of degree d+1.
We refer to S as a degree d Schwarz reflection map.

Set 

    T(S):=∖,  and   T^0(S):=T∖𝔖,

where 𝔖 is the set of singular points (cusps and double points) on ∂ T(S). We call the components of T^0(S) the fundamental tile and the components of S^-n(T^0(S)) the tiles of rank n.
The escaping/tiling set T^∞(S) of S is defined as the union of all tiles; i.e., 

    T^∞(S):=⋃_n=0^∞ S^-n(T^0(S)).
 
The non-escaping set of S is defined as K(S):=∖ T^∞(S). A connected component of K(S) is called a Fatou component of S. The common boundary of K(S) and T^∞(S) is called the limit set of S, and is denoted by Λ(S).



 §.§ Conformal mating and Schwarz reflections



Let f be a monic, centered complex polynomial of degree d with a connected and locally connected Julia set. We denote the Böttcher coordinate of f by

    ψ_f:∖→ℬ_∞(f):=∖𝒦(f).
 
This map, which conjugates z^d to f, is normalized to be tangent to the identity map near infinity.
As 𝒥(f) is locally connected, ψ_f extends continuously to 𝕊^1 to yield a semi-conjugacy between z^d|_𝕊^1 and f|_𝒥(f). 
By Section <ref>, there exists a homeomorphism 𝔥:𝕊^1→𝕊^1 that conjugates z^d to _d (respectively, _d). We normalize 𝔥 so that it sends the fixed point 1 of z^d to the fixed point 1 of _d (respectively, _d).

We define an equivalence relation ∼ on 𝒦(f) generated by 

    ψ_f(ζ)∼𝔥(ζ), for all z∈𝕊^1.
 

We now precise what it means to conformally mate f with the Nielsen map _d (the case of the anti-Farey map _d is exactly the same).


The maps f and _d are said to be conformally mateable if there exist a continuous map S: Dom(S)⊊→ (called a conformal mating of _d and f) that is complex-analytic in the interior of Dom(S) and continuous maps 
	
    𝔛_f:𝒦(f)→ and 𝔛__d: →,

	conformal on 𝒦(f) and  (respectively), satisfying
	
		
  *  𝔛_f(𝒦(f))∪𝔛_Σ() =,
		
		
  *  Dom(S)= 𝔛_f(𝒦(f))∪𝔛__d(∖Π),
		
		
  * 𝔛_f∘ f(z) = S∘𝔛_f(z),  for z∈𝒦(f),
		
		
  * 𝔛__d∘_d (w) = F∘𝔛__d(w),  for w∈∖Π,  and 
		
		
  *  𝔛_f(z)=𝔛__d(w) if and only if z∼ w where ∼ is the equivalence relation on 𝒦(f)⊔ defined by Relation (<ref>).
	


Using David surgery techniques, it was proved in <cit.> that many anti-polynomials with connected Julia set can be conformally mated with the Nielsen/anti-Farey map. Crucially, the property of _d:∖Π⟶ (respectively, of _d:∖ℋ⟶) that it fixes ∂Π (respectively, ∂ℋ) pointwise, translates to the fact that the resulting conformal matings are Schwarz reflection maps.

<cit.><cit.>
Let f be a degree d hyperbolic anti-polynomial with connected Julia set.
Then, there exists a conformal mating S of f:𝒦(f)→𝒦(f) and _d:∖Π⟶ (respectively, _d:∖ℋ⟶). Moreover, S is a Schwarz reflection map in a quadrature multi-domain.







§ POLYGONAL SCHWARZ REFLECTION MAPS




 §.§ Basic definitions and properties


We continue to use the notation of the previous section.
Suppose further that  is connected and simply connected. Since the boundary of a quadrature domain consists of real-analytic curves possibly with cusps and double point singularities, no more than two distinct components of  can touch at a point. Thus, the above condition is equivalent to following ones.


  * Each Ω_j is a Jordan domain, and the contact graph of the quadrature multi-domain (i.e., a graph having a vertex for each Ω_j and an edge connecting two vertices if the corresponding quadrature domains touch) is a tree.

  *  is a pinched polygon, and the associated finite geodesic lamination contains no polygon.

We will refer to such a  as a tree-like quadrature multi-domain, and call the Schwarz reflection of the tree-like quadrature multi-domain  a Schwarz reflection of a pinched polygon.


For a tree-like quadrature multi-domain, the desingularized droplet T^0(S) is homeomorphic to an ideal polygon in the hyperbolic plane.


A degree d Schwarz reflection map S:→ of a pinched polygon 
is said to be polygonal if  T^0(S) is homeomorphic to an ideal (d+1)-gon in  and S has no critical values in T^0(S).
 
A degree d polygonal (piecewise) Schwarz reflection map is called regular polygonal if T^0(S) is conformally equivalent to the regular ideal (d+1)-gon Π.  
We denote by 𝒮_d the space of degree d regular polygonal Schwarz reflections.

We will often omit the word `regular' for brevity. We normalize a polygonal Schwarz reflection S so that the conformal map between Π and T^0(S) sends the origin to ∞ and has asymptotics z↦ 1/z+O(z) as z→ 0.


In the above definition, the condition that T^0(S) is homeomorphic to an ideal (d+1)-gon in  is equivalent to requiring that the sum of the number of cusps and twice the number of double points on ∂ is equal to d+1.


The following result justifies the nomenclature `polygonal Schwarz reflection maps'. Indeed, it shows that such maps are characterized by having the Nielsen map of a polygonal reflection group as the conformal model of their dynamics near the droplet.


Let S:→ be a degree d Schwarz reflection map of a pinched polygon. Then, S is regular polygonal if and only if S:S^-1(T^0(S))→ T^0(S) is conformally conjugate to _d:_d^-1(Π)→Π.


It is clear that if S:S^-1(T^0(S))→ T^0(S) is conformally conjugate to _d:_d^-1(Π)→Π, then S is regular polygonal.

For the converse, let us assume that S is regular polygonal. Then, there exists a conformal map ψ_S:Π→ T^0(S) that preserves the ideal vertices. The fact that T^0(S) is simply connected and S has no critical point in S^-1(T^0(S)) together imply that S^-1(T^0(S)) consists of d+1 components, each of which maps diffeomorphically onto T^0(S) under S. Similarly, _d^-1(Π) consists of d+1 components, each of which maps diffeomorphically onto Π under _d.
Moreover, S and _d act as identity maps on ∂ T^0(S) and ∂Π, respectively. Hence, ψ_S can be lifted to a conformal isomorphism from _d^-1(Π) onto S^-1(T^0(S)). Note that the trivial actions of S and _d on ∂ T^0(S) and ∂Π (respectively) ensure that the lifts match on the boundaries of Π, T^0(S). By construction, the extended map conjugates S:S^-1(T^0(S))→ T^0(S) to _d:_d^-1(Π)→Π.


An important consequence of Lemma <ref> is that for a degree d polygonal (piecewise) Schwarz reflection S:→, the restriction S:S^-1()→ is a degree d degenerate anti-polynomial-like map. Many of the following results can be regarded as analogs of basic facts about classical polynomial-like maps in the current setting.




Let S∈𝒮_d.

  * T^∞(S) and K(S) are completely invariant under S.


  * The escaping set T^∞(S) is an open and connected. The non-escaping set K(S) is a full compact subset of . 


  * Each Fatou component of S is simply connected.



1) This is immediate from the definitions.

2) Let us denote the union of the tiles of rank 0 through k by E^k. Since every tile of rank k≥1 is attached to a tile of rank (k-1) along a boundary curve and E^0=T^0(S) is connected, it follows that E^k is connected. Moreover, E^k⊊E^k+1 for each k≥ 0. 

Note that if z∈ T^∞(S) belongs to the interior of a tile of rank k, then it lies in the interior of E^k. On the other hand, if z∈ T^∞(S) belongs to the boundary of a tile of rank k, then it lies in the interior of E^k+1. Hence, T^∞(S)=⋃_k≥ 0E^k. 
Thus, T^∞(S) is an increasing union of open, connected sets, and hence itself is such. Consequently, its complement K(S) is a full compact subset of . 

3) This follows from the fact that K(S) is a full compact.




Let (S:→)∈𝒮_d, μ be an S-invariant Beltrami coefficient supported on K(S), and Φ be a quasiconformal map solving the Beltrami equation with coefficient μ. Then Φ∘ S∘Φ^-1∈𝒮_d.


The assumption that μ is S-invariant implies that Φ∘ S∘Φ^-1 is anti-meromorphic on Φ() that continuously extends to the identity map on ∂Φ(). It follows that each component of Φ() is a simply connected quadrature domain, and hence Φ∘ S∘Φ^-1 is a degree d Schwarz reflection map with connected non-escaping set. Since μ is trivial on T^∞(S), the quasiconformal map Φ is conformal on T^∞(S). Therefore, T^0(Φ∘ S∘Φ^-1)=Φ(T^0(S)) is conformally equivalent to Π; i.e., Φ∘ S∘Φ^-1∈𝒮_d.





 §.§ Connectedness locus of polygonal Schwarz reflections




Let (S:→)∈𝒮_d. Then the following are equivalent.


  * K(S) is connected.

  * S^∘ n(c)∉ T^∞(S), ∀ c∈crit(S).

  * S:T^∞(S)∖T^0(S)⟶ T^∞(S) is conformally conjugate to the Nielsen map _d:∖Π→.



(1)  (2) If some critical point of S lies in the interior of a tile, then this tile would be mapped forward with degree larger than one and its closure would have at least two complementary components. Each of these complementary components would intersect non-escaping set of S, providing a disconnection of K(S).

Next suppose that some critical point of S lies on the boundary of a tile. Then this critical point is a common boundary point of two tiles of equal rank, and the closure of the union of these two tiles has at least two complementary components. Again, each of these components would intersect K(S), proving that K(S) is disconnected. 


(2)  (3) Let ψ_S:Π⟶ T^0(S) be a homeomorphism that is conformal on the interior. Since T^0(S) is simply connected and none of the critical points of S escapes to T^0(S), it follows that each tile of S maps diffeomorphically onto T^0(S) under some iterate of S. Similarly, the tiles of the tessellation of  arising from the regular (d+1)-gon reflection group G_d map diffeomorphically onto Π under iterates of _d. Furthermore, S and _d act as identity maps on ∂ T^0(S) and ∂Π, respectively. This allows us to lift ψ_S to a conformal isomorphism from  (which is the union of all iterated preimages of Π under _d) onto T^∞(S) (which is the union of all iterated preimages of T^0(S) under S). Note that the trivial actions of S and _d on ∂ T^0(S) and ∂Π (respectively) ensure that the iterated lifts match on the boundaries of the tiles. By construction, the extended map conjugates _d:∖Π→ to S:T^∞(S)∖T^0(S)→ T^∞(S).

(3)  (1) The assumption that T^∞(S) is a topological disk implies that K(S) is connected.



We define the  connectedness locus as

    𝒮__d:= {S ∈𝒮_d: K(S)  is connected.}⊆𝒮_d.
 



Note that by Proposition <ref>, the connectedness locus 𝒮__d consists precisely of maps having the Nielsen map _d as the conformal model of their tiling set dynamics.





 §.§ Fatou components and critical points


Let (S:→)∈𝒮_d.


  * Every Fatou component of S is eventually periodic.


  * Every periodic Fatou component U of S is of one of the following types.


  * U is an immediate basin of attraction of an attracting cycle.

  * U is an immediate basin of attraction of a parabolic cycle.

  * U is a Siegel disk.

  * U is an immediate basin of attraction of some w∈𝔖.




(1) Since the space 𝒮_d is parametrized by finitely many rational maps the sum of whose degrees is d+1, it follows that 𝒮_d is finite-dimensional. Moreover, the family 𝒮_d of maps is closed under quasiconformal conjugation in the sense of Lemma <ref>. Hence, the classical proof of non-existence of wandering Fatou components for rational maps can be adapted for the space 𝒮_d (cf. <cit.>).
 
(2) Let U be a periodic Fatou component of period k of S. By Proposition <ref>, U is simply connected. According to <cit.>, the Fatou component U is either the (immediate) basin of attraction of an attracting cycle, or a Siegel disk, or all orbits in U converge to a unique boundary point w. If w lies in , then the classical Snail Lemma implies that w must lie on a parabolic cycle. Otherwise, w∈∂∩ K(S), and hence w is a singular point of ∂ T(S).



Let (S:→)∈𝒮_d.


  * If S has an attracting or parabolic cycle, then the forward orbit of a critical point of S converges to this cycle.  


  * Every Cremer point (i.e. an irrationally neutral, non-linearizable periodic point) and Siegel disk boundary (of S) is contained in the ω-limit set of a recurrent the critical point of S. 


  * If w∈𝔖 has an attracting direction in K(S), then some critical orbit of S converges to w.



(1) and (2) The proof of <cit.> applies to the current setting (cf. <cit.>).

(3) Let us first assume that w∈𝔖 is a cusp and has an attracting direction in K(S). By <cit.>, the asymptotics of S^∘ 2 near w is of the form 

    ζ↦ζ+c_1(ζ-w)^n/2+O((ζ-w)^(n+1)/2),
 
for some c_1∈^*, and n≥ 5.
Standard arguments used in the theory of parabolic germs allow one to count the number of attracting/repelling directions at w (cf. <cit.>). These directions occur in an alternating fashion. Let U be a Fatou component of S such that the S^∘ 2-orbit of every point in U converges to w asymptotic to some attracting direction. A change of coordinate of the form ζ↦ 1/(ζ-w)^n/2-1 that sends w to ∞ can now be applied to a sector containing this attracting direction to bring the map S^∘ 2 to the form ξ↦ξ+1+O(1/ξ) on a right-half plane. This enables one to argue as in <cit.> to prove the existence of an attracting petal in U on which the action of S^∘ 2 is conformally conjugate to translation by +1 on a right half-plane. Finally, the proof of <cit.>, applied to the current setting, shows that the boundary of the maximal domain of definition of this conjugacy contains a critical point of S^∘ 2. It follows that U contains a critical point of S.

Let us now consider a double point w∈𝔖 that has an attracting direction in K(S). After possibly renumbering the quadrature domains in , we may assume that ∂Ω_1∩∂Ω_2={w}. Then, S^∘ 2 is given by the two tangent-to-identity parabolic germs S_2∘ S_1, S_1∘ S_2 near w. These two germs are inverses of each other, and hence have the same number of attracting/repelling directions. Let U be a Fatou component of S such that the S^∘ 2-orbit of every point in U converges to w asymptotic to some attracting direction. Since S^∘ 2 is given by a parabolic germ, it follows that there exists an attracting petal in U on which the action of S^∘ 2 is conformally conjugate to translation by +1 on a right half-plane, and that the boundary of the maximal domain of definition of this conjugacy contains a critical point of S^∘ 2 (cf. <cit.>. Thus, U contains a critical point of S.




 §.§ Dynamical rays and lamination


We now define G_d-rays in  arising from the Cayley graph of G_d with respect to the generating set ρ_1,⋯, ρ_d+1. 


Let (i_1, i_2, ⋯)∈{1,⋯, d+1}^ with i_j≠ i_j+1 for all j. The corresponding infinite sequence of tiles {Π,ρ_i_1(Π),ρ_i_1∘ρ_i_2(Π),⋯} shrinks to a single point of 𝕊^1≅/, which we denote by θ(i_1, i_2, ⋯). We define a G_d-ray at angle θ(i_1, i_2, ⋯) to be the concatenation of hyperbolic geodesics (in ) connecting the consecutive points of the sequence {0,ρ_i_1(0),ρ_i_1∘ρ_i_2(0),⋯}. 


We note that in general there may be more than one G_d-rays at a given angle, but all of them land at the point θ(i_1, i_2, ⋯)∈𝕊^1.
For instance, both the symbol sequences (1,d+1,1,d+1,⋯) and (d+1,1,d+1,1,⋯) yield G_d-rays at angle 0, and both land at the point 1.
Moreover, the image of the tail of a ray at angle θ under _d is the tail of a ray at angle _d(θ).





For S∈𝒮__d, the image of a G_d-ray at angle θ under the map ψ_S:⟶ T^∞(S) (that conjugates _d to S) is called a θ-dynamical ray of S.

Evidently, the image of the tail of a dynamical θ-ray (of S) under S is the tail of a dynamical ray at angle _d(θ).
We denote the set of all (pre)periodic points of _d:𝕊^1→𝕊^1 by Per(_d).


Let S∈𝒮__d. Then the following statements hold true.


  * Let θ∈Per(_d). Then a dynamical ray of S at angle θ lands on Λ(S). 


  * The dynamical rays of S at fixed angles land at points of 𝔖. Conversely, each point of 𝔖 is the landing point of a fixed ray.


  * The dynamical rays of S at periodic angles of period larger than two land at repelling or parabolic periodic points of Λ(S). 



1) After possibly applying some iterate of S, we may assume that θ is periodic of period k under _d. Then, the angle θ corresponds to a sequence in {1,⋯, d+1}^ that is periodic under the left-shift map; i.e., θ=θ(i_1, i_2, ⋯, i_k) (see Definition <ref>). Consider the sequence of points w_n:=ψ_S((ρ_i_1∘⋯∘ρ_i_k)^∘ n(0)), n≥1, on the θ-dynamical ray of S. 
Note that if γ_n is the sequence of arcs on this θ-dynamical ray bounded by w_n and w_n+1, then S^∘ k(γ_n)=γ_n-1, and ℓ_hyp(γ_n)=c for all n≥1, where c is a fixed positive constant, and ℓ_hyp denotes the hyperbolic length in T^∞(S). We now consider two cases.

Case a. (The accumulation set of the θ-ray is contained in ∪_n≥ 0 S^-n(𝔖).) Since the accumulation set of a ray is connected, and the set of iterated preimages of 𝔖 is totally disconnected, it follows that the θ-ray lands. 

Case b. (The accumulation set of the θ-ray intersects Γ(S)∖∪_n≥ 0 S^-n(𝔖).) Let w∈Γ(S)∖∪_n≥ 0 S^-n(𝔖) be an accumulation point of the θ-ray. By our assumption, the map S^∘ k admits an (anti-)holomorphic extension in a neighborhood of w. Moreover, since the arcs γ_n of fixed hyperbolic length accumulate on the limit set Γ(S), their Euclidean lengths must shrink to 0. One can now apply the arguments of <cit.> to the current setting, and deduce that S^∘ k(w)=w. But the set of fixed points of S^∘ k in Γ(S) is totally disconnected (as the set of fixed points of the holomorphic second iterate S^∘ 2k in Γ(S) is discrete). As before, connectedness of the accumulation set of a ray implies that the θ-ray lands.

2) Let θ be fixed under _d; i.e., θ=j/d+1, for some j∈{0,⋯,d}. Since ∂ T^0(S) is locally connected, the image of the radial line at angle θ under ψ_S lands at some point w∈Λ(S). The fact that _d does not admit an anti-conformal extension to a neighborhood of an ideal boundary point of Π implies that S does not have an anti-conformal extension to a neighborhood of w. Hence, w∈𝔖. We will show that the θ-dynamical ray of S lands at w.

By the asymptotic development of S^∘ 2 (which is given by parabolic germs if w is a double point and by Equation (<ref>) if w is a cusp), the map S^∘ 2 admits an inverse branch g in a relative neighborhood of w with the following properties:

i) g(w)=w, and

ii) for w'∈ S^-2(T^0(S))∩Domain(g) is sufficiently close to w, the sequence g^∘ n(w') converges to w horocyclically (in the simply connected domain T^∞(S)).

It is easy to see that the sequence g^∘ n(w') is within a bounded distance from the dynamical θ-ray of S, and hence the θ-ray lands at w.

For the converse statement, note that each w∈𝔖 is the landing point of the curve {ψ_S(re^2π iθ):r∈[0,1)}, for some angle θ fixed under _d. By the previous paragraph, the dynamical ray of S at angle θ lands at w.
 
3) We claim that the dynamical rays of S of period larger than two do not land on 𝔖. Once this claim is established, the result follow by the classical Snail lemma.

Let us fix w∈𝔖. Once again, by the asymptotic development of S^∘ 2 near w, the map S^∘ 2|_S^-2() extends to an orientation-preserving local homeomorphism in a neighborhood of w. One can apply the arguments of <cit.> to this map to conclude that all rays landing at w have the same period under S^∘ 2 (cf. <cit.>). Since w is the landing point of some fixed ray of S (by the previous part), it follows that all rays landing at w are fixed by S^∘ 2. Therefore, the dynamical rays of S of period larger than two cannot land on 𝔖




Let S∈𝒮__d. Then, every repelling and parabolic periodic point of S is the landing point of finitely many (at least one) dynamical rays. Moreover, all these rays are periodic of the same period under S^∘ 2.


The proof of the corresponding statement for polynomials with connected Julia set (cf. <cit.>) can be easily adapted for the current setting.



Using the same techniques as in <cit.>, we can prove stability of dynamical rays landing at pre-repelling points whose orbit avoids critical points.

Let S∈𝒮__d. Assume that a dynamical ray lands at a repelling pre-periodic point z such that the orbit of z avoids the critical points of S.
Then, the corresponding dynamical ray lands at the real-analytic continuation z' (of the pre-periodic point z) for all S'∈𝒮__d sufficiently close to S.





In light of Proposition <ref>, one can define a landing map L:Per(_d)→Λ(S) that associates to every (pre)periodic angle θ  the landing point of the θ-dynamical ray of S.


For S∈𝒮__d, the rational lamination of S is defined as the equivalence relation on Per(_d)⊂/ such that θ, θ'∈Per(_d) are related if and only if L(θ)=L(θ'). We denote the rational lamination of S by λ(S).


Rational laminations of Schwarz reflections in 𝒮__d are analogous to rational laminations of anti-polynomials with connected Julia set (cf. <cit.>). In particular, λ(S) is a _d-invariant equivalence relation.


The push-forward (ℰ_d)_∗(λ(S)) of the rational lamination of S is defined as the image of λ(S)⊂Per(_d)×Per(_d) under ℰ_d×ℰ_d. Clearly, (ℰ_d)_∗(λ(S)) is an equivalence relation on /. Similarly, the pullback (ℰ_d)^∗(λ(f)) of the rational lamination of an anti-polynomial f with connected Julia set is defined as the preimage of λ(f)⊂/×/ under ℰ_d×ℰ_d.



For Schwarz reflections as well as anti-polynomials, we will denote by λ(S) (respectively, λ(f)) the smallest equivalence relation on / that contains the closed set λ(S)⊂/×/ (respectively, λ(f)⊂/×/).
Note that the push-forward operation defined in Definition <ref> can be applied to λ(S) as well.




 §.§ Combinatorial mating of _d with periodically repelling anti-polynomials


Recall that the filled Julia set of a periodically repelling anti-polynomial f∈𝒞_d^- has empty interior. By Proposition <ref>, the same statement holds for the non-escaping set of a periodically repelling Schwarz reflection S∈𝒮__d.
The proof of <cit.> implies that if the Julia/limit set of such a map is locally connected, then its filled Julia set/non-escaping set dynamics is topologically modeled by the induced map

    m_-d:/λ(f)⟶/λ(f)

or

    _d:/λ(S)⟶/λ(S).

Further, if 

    (ℰ_d)_*(λ(S))=λ(f),

then the above induced maps are topologically conjugate via the homeomorphism

    ℰ_d: /λ(S)⟶/λ(f)

induced by the conjugacy ℰ_d:𝕊^1→𝕊^1 between _d and z^d. In particular, local connectedness of the Julia (respectively, limit) set of f (respectively, S) and the condition (ℰ_d)_*(λ(S))=λ(f) together imply that f|_𝒦(f) and S_|K(S) are topologically conjugate.

The above discussion motivates the following weaker definition of mating.


We say that a periodically repelling Schwarz reflection S∈𝒮__d is a combinatorial mating of _d with a periodically repelling anti-polynomial f∈𝒞_d^- if (ℰ_d)_*(λ(S))=λ(f).







§ GEOMETRICALLY FINITE POLYGONAL SCHWARZ REFLECTIONS



A Schwarz reflection map S∈𝒮_d is said to be geometrically finite if every critical point of S in the
limit set Λ(S) is pre-periodic.


By Propositions <ref> and <ref>, each periodic Fatou component of a geometrically finite Schwarz reflection is an attracting/parabolic basin, or a basin of attraction of some singular point x∈𝔖. Moreover, S has no Cremer cycle.



Let S∈𝒮__d,gf. Then the following hold true.

  * Λ(S) is locally connected.

  * Λ(S) has zero area.



1) The proof is analogous to the proof of local connectivity of connected Julia sets of geometrically finite polynomials (cf. <cit.>, <cit.>).

2) We say that a point w∈Γ(S) is a radial point if there exists δ > 0, an infinite sequence of positive integers { n_k }, and well-defined inverse branches of S^∘ n_k defined on B(S^∘ n_k(w), δ) sending S^∘ n_k(w) to w for all k. The set of all radial points of Γ(S) is called the radial limit set of S. Arguments used in <cit.> show that the radial limit set of S is given by Γ(S)∖⋃_j=0^∞ S^-j(𝔖∪𝒫∪ C_preper), where 𝒫 is the union of the parabolic cycles and C_preper is the set of strictly pre-periodic critical points of S. Finally, the nowhere dense structure of Λ(S) can be transferred from moderate scales to microscopic scales at radial points (using the inverse branches of iterates of S), and this implies that no radial limit point is a point of Lebesgue density for Γ(S) (cf. <cit.> or <cit.>). The above facts, together with the Lebesgue density theorem, imply that Λ(S) has zero area. 


Local connectedness of limit sets of maps in 𝒮__d,gf, combined with the arguments of <cit.>, implies that for S∈𝒮__d,gf, the dynamics of S on its limit set Λ(S) is topologically conjugate to 

    _d: (/)λ(S)⟶(/)λ(S) .

In particular, λ(S) is generated by co-landing of the dynamical rays of S. A straightforward adaptation of <cit.> for Schwarz reflections yields the following properties of λ(S) for S∈𝒮__d,gf.




Let S∈𝒮__d,gf. Then, λ(S) satisfies the following properties.

  * λ(S) is closed in /×/.


  * Each equivalence class A of λ(S) is a finite subset of /.


  * If A is a λ(S)-equivalence class, then _d(A) is also a λ(S)-equivalence class.


  * If A is a λ(S)-equivalence class, then A↦_d(A) is consecutive reversing; i.e., for every connected component (s,t) of /∖ A, we have that (_d(s),_d(t)) is a connected component of / ∖ _d(A).


  * λ(S)-equivalence classes are pairwise unlinked; i.e., if A and B are two distinct equivalence classes of λ(S), then there exist disjoint intervals I_A, I_B⊂/ such that A⊂ I_A and B⊂ I_B.


  * If γ⊂(/)λ(S) is a periodic simple closed curve, then the corresponding return map is not a homeomorphism.


  * If c is a critical point of P on the limit set Λ(S), then τ_S^-1(c)⊂Per(_d), and τ_S^-1(c)↦_d(τ_S^-1(c)) has some degree δ>1. Every other equivalence class of λ(S) maps injectively onto its image equivalence class under _d. Here, τ_S:→ T^∞(S) is the normalized conformal map that conjugates _d to S.





To prove Theorem <ref>, we need the following technical lemma that guarantees the existence of continuous extensions of circle homeomorphisms as quasiconformal/David homeomorphisms of . 




  * The homeomorphism ℰ_d^-1:𝕊^1→𝕊^1, that conjugates z^d to _d, admits a continuous extension to  as a David homeomorphism.



  * Let B:→ be a degree d Blaschke (respectively, anti-Blaschke product) with a parabolic fixed point on 𝕊^1 that has an attracting direction in . Then there exists a homeomorphism H:𝕊^1→𝕊^1 that conjugates z^d (respectively, z^d) to B and admits a continuous extension to  as a David homeomorphism.



  * Let n≥ 2, and 

    B:𝕊^1×/n⟶𝕊^1×/n, B(z,i)=(B_i(z),i+1)

be such that each B_i is a degree d_i anti-Blaschke product. Assume further that each 

    B_i+n-1∘⋯∘ B_i+1∘ B_i:𝕊^1→𝕊^1
 
is a degree d Blaschke or anti-Blaschke product with a parabolic fixed point on 𝕊^1 that has an attracting direction in , depending on whether n is even or odd, where d=d_0 d_1⋯ d_n-1, i∈/n.
Then, there exists a homeomorphism H:𝕊^1×/n⟶𝕊^1×/n conjugating 

    P:𝕊^1×/n⟶𝕊^1×/n, P(z,i)=(z^d_i,i+1)

to B.
Moreover, H continuously extend as a David homeomorphism of ×/n.


  * Let n≥ 2, and 

    B:𝕊^1×/n⟶𝕊^1×/n, B(z,i)=(B_i(z),i+1)

be such that each B_i is a degree d_i anti-Blaschke product. Assume further that each 

    B_i+n-1∘⋯∘ B_i+1∘ B_i:𝕊^1→𝕊^1
 
is a degree d Blaschke or anti-Blaschke product with an attracting fixed point in , depending on whether n is even or odd, where d=d_0 d_1 ⋯ d_n-1, i∈/n.
Then, there exists a homeomorphism H:𝕊^1×/n⟶𝕊^1×/n conjugating 

    P:𝕊^1×/n⟶𝕊^1×/n, P(z,i)=(z^d_i,i+1)

to B.
Moreover, H continuously extends as a quasiconformal homeomorphisms of ×/n.



1) Follows from <cit.>.

2) Note that the parabolic (anti-)Blaschke product B_d^-(z) :=(d+1)z^d+(d-1)/(d+1)+(d-1)z^d (or the parabolic Blaschke product B_d^+(z)=(d+1)z^d+(d-1)/(d+1)+(d-1)z^d) is conjugate to B via a quasisymmetric homeomorphism H_1:𝕊^1→𝕊^1 (cf. <cit.>). We extend H_1 to a quasiconformal homeomorphism of  (also called H_1).

By <cit.>, there exists a circle homeomorphism H_2:𝕊^1→𝕊^1 that conjugates z^d or z^d to B_d^±, such that H_2 admits a continuous extension to  as a David homeomorphism. By <cit.>, the desired map H is given by H_1∘ H_2.



	
3) Let us assume that n is odd. The proof of the even case is completely analogous.

By the previous part, there exists a topological conjugacy H_n-1:𝕊^1→𝕊^1 between z^d and B_n-2∘⋯∘ B_0∘ B_n-1. Since B_j and z^j are circle coverings of the same degree for j∈/n, we can succesively lift H_n to obtain circle homeomorphisms H_n-2,⋯,H_0, H_n-1 such that the following diagram commutes:


    𝕊^1 rz^d_n-1[swap]dH_n-1   𝕊^1 rz^d_0[swap]dH_0   𝕊^1 dH_1rz^d_1   𝕊^1 dH_2rz^d_2   ⋯rz^d_n-3d   𝕊^1 [swap]dH_n-2rz^d_n-2   𝕊^1 dH_n-1
    𝕊^1 [swap]rB_n-1   𝕊^1 [swap]rB_0   𝕊^1 [swap]rB_1   𝕊^1 r[swap]B_2   ⋯[swap]rB_n-3   𝕊^1 [swap]rB_n-2   𝕊^1.

It now follows that

    (B_n-2∘⋯∘ B_0∘ B_n-1) ∘H_n-1 = (B_n-2∘⋯∘ B_0∘ B_n-1) ∘ H_n-1,

and hence H_n-1∘ H_n-1^-1 is a deck transformation for the circle covering B_n-2∘⋯∘ B_0∘ B_n-1. 
Hence, possibly after post-composing H_n-2,⋯,H_0, H_n-1 with deck transformations of appropriate B_js, we can assume that H_n-1=H_n-1. Hence, 

    H:𝕊^1×/n→𝕊^1×/n, (z,i)↦ (H_i(z), i)

is the desired topological conjugacy between P and B.

Finally, as each H_i conjugates z^d to a degree d anti-Blaschke product with a parabolic fixed point on 𝕊^1 that has an attracting direction in , the first part (of this proposition) allows us to conclude that H_i can be continuously extended to David homeomorphisms of . 

4) The proof, which uses the fact that any expanding, analytic, orientation-reversing covering of 𝕊^1 is quasisymmetrically conjugate to z^n (where n is the degree of the map), is similar to the previous part.







The proof will be divided into three parts.


Constructing a map Φ:𝒞^-_d,gf⟶𝒮__d,gf. Let f∈𝒞^-_d,gf be geometrically finite. Our goal is to construct a Schwarz reflection in 𝒮__d,gf that is a conformal mating of f and _d.

By an antiholomorphic version of <cit.> (the proof of the orientation-preserving case applies mutatis mutandis to the orientation-reversing case thanks to the orientation-reversing version of Thurston Realization Theorem, cf. <cit.>, <cit.>), there exists a postcritically finite map f_0∈𝒞^-_d such that f|_𝒥(f) is topologically conjugate to f_0|_𝒥(f_0). We first construct a partially defined continuous map on the sphere from f_0 as follows.


∙ Let us denote the David extension of ℰ_d^-1:𝕊^1→𝕊^1 (that conjugates z^d to _d) to  by ξ. Further, let τ_f_0:→ℬ_∞(f_0) be a conformal map that conjugates z^d to f_0 (where ℬ_∞(f_0) is the basin of infinity of f_0).  We replace the action of f_0 on ℬ_∞(f_0) by τ_f_0∘ξ^-1∘_d∘ξ∘τ_f_0^-1. 

Note that the map τ_f_0∘ξ^-1∘_d∘ξ∘τ_f_0^-1 preserves the pullback of the standard complex structure on  by ξ∘τ_f_0^-1.


∙ Suppose that U_0,⋯,U_n-1 is a cycle of bounded Fatou components of f, and U_0,⋯,U_n-1 is the corresponding cycle of bounded Fatou components of f_0. One can choose conformal maps κ:×/n→_i∈/n U_i and κ:×/n→_i∈/nU_i such that κ^-1∘ f∘κ:×/n→×/n is a fiberwise anti-Blaschke product B satisfying the hypotheses of part (3) or part (4) of Proposition <ref>, and κ^-1∘ f_0∘κ:×/n→×/n is a fiberwise power map P as in part (3) or part (4) of Proposition <ref>. Hence, there exists a homeomorphism H:𝕊^1×/n⟶𝕊^1×/n that conjugates P to B, and admits a David/quasiconformal extension ×/n.
We replace the action of f_0 on _i∈/nU_i by the map κ∘H^-1∘B∘H∘κ^-1.

Again, the map κ∘H^-1∘B∘H∘κ^-1 preserves the pullback of the standard complex structure on ×/n by H∘κ^-1.



∙ Finally, let U be a strictly pre-periodic Fatou component of f with f(U)=V, and U, V be the corresponding components of f_0. The maps f:U→ V and f_0:U→V induce, via Riemann maps of the Fatou components, anti-Blaschke actions of , and these anti-Blaschke actions are quasisymmetrically conjugate on 𝕊^1 (cf. <cit.>). This allows us to replace the action of f_0 on U by the conformal model of the action of f on U as in the previous steps. We note that this step is only necessary for Fatou components U containing a critical point.

The above recipe defines a continuous map S on a subset of the sphere and an invariant Beltrami coefficient on the union of the periodic Fatou components of f_0. We use S to spread this Beltrami coefficient out to all the pre-periodic Fatou components of f_0, and set it equal to zero on the Julia set of f_0. Since f_0 is subhyperbolic, one can now argue as in <cit.> to conclude that this invariant Beltrami coefficient μ satiesfies the David condition and that the conjugate of S by the corresponding integrating map Ψ is antiholomorphic. We denote the conjugated map Ψ∘S∘Ψ^-1 by S.

By construction, S is defined on the complement of the simply connected domain Ψ(τ_f_0(ξ^-1(Π))), and fixes the boundary of its domain of definition pointwise. Thus, S is a Schwarz reflection map whose escaping and non-escaping sets are given by Ψ(ℬ_∞(f_0) and Ψ(K(f_0)), respectively. The tiling set dynamics of S is conformally conjugate to _d via ξ∘τ_f_0^-1∘Ψ^-1; i.e., S∈𝒮__d,gf. It also follows from the construction of S that its non-escaping set dynamics is topologically conjugate (conformally on the interior) to the filled Julia set dynamics of f. It is now easily checked that S is a conformal mating of f and _d. 


We also note that by <cit.> and the fact that ℬ_∞(f_0) is  John domain, we have that the limit set of S is conformally removable.

We claim that such an S∈𝒮__d,gf is unique. Indeed, if S_1 were another conformal mating of f and _d, then the tiling set dynamics and non-escaping set dynamics of S and S_1 are conformally conjugate. Further, these conjugacies match continuously along the limit sets to yield a topological conjugacy between S and S_1 that is conformal away from Λ(S). By conformal removability of Λ(S), this conjugacy is a Möbius map; i.e., S and S_1 are Möbius conjugate. The normalization of the maps in 𝒮__d now implies that S=S_1. 
Alternatively, one can employ classical pullback arguments to deduce uniqueness of the map S (see <cit.> for an implementation of this principle to prove rigidity of geometrically finite Schwarz reflections in the C&C family).

Thus, we have a well-defined map Φ:𝒞^-_d,gf⟶𝒮__d,gf that sends f to the unique conformal mating S of f and _d.


Injectivity of Φ. Suppose that Φ(f)=Φ(f_1)=S. Then, the basin of infinity dynamics and filled Julia set dynamics of f and f_1 are conformally conjugate. Further, these conjugacies match continuously along the Julia sets to yield a topological conjugacy between f and f_1 that is conformal away from 𝒥(f). By conformal removability of 𝒥(f) (see <cit.>), this conjugacy is a Möbius map. (Equivalently, injectivity of Φ follows from well-known results about rigidity of geometrically finite polynomials, cf. <cit.>.) By the normalization of Böttcher coordinates for maps in 𝒞^-_d, this Möbius conjugacy between the monic, centered anti-poynomials f, f_1 is tangent to the identity at ∞. Hence, the Möbius conjugacy must be the identity map; i.e., f=f_1.


Surjectivity of Φ. Let S∈𝒮__d,gf and λ(S) be the rational lamination of S. By Proposition <ref> and the orientation-reversing version of <cit.>, there exists a postcritically finite anti-polynomial f_0∈𝒞^-_d with λ(f_0)= (ℰ_d)_∗(λ(S)). Having f_0 at our disposal, we can now produce a geometrically finite anti-polynomial f∈𝒞^-_d by performing quasiconformal/David surgeries on the bounded Fatou components of f_0 (as in the first step of the proof) so that the filled Julia set dynamics of f is topologically conjugate (conformally on the interior) to the non-escaping set dynamics of S. Evidently, the map f is the desired preimage of S under Φ.



























§ STRATA OF THE CONNECTEDNESS LOCUS 𝒮__D




 §.§ Fixed ray lamination and a coarse partition


In this section, we will explain that the connectedness locus of the space of degree d regular polygonal Schwarz reflections admits a finite partition.
This coarse partition gives a natural stratification of the connectedness locus.



 §.§ Fixed ray lamination



The Nielsen map _d:𝕊^1→𝕊^1 has exactly d+1 fixed points 0,⋯,d/d+1. We denote the set of these fixed points by Fix(_d).






We define a fixed ray lamination as a geodesic lamination ℒ⊂ so that


  * each leaf ℓ = (a,b) of ℒ has end points a, b ∈Fix(_d) ⊂≅∂;

  * two leaves in ℒ have disjoint closures in .


Given a fixed ray lamination ℒ, we call each complementary component G ⊂∖ℒ a gap.
Let G be a gap. Then ∂ G ∩ has length equal to k ·1/d+1 for some multiple k.
We shall call this number k the degree of the gap, and is denoted by (G).
From the definition, we have

    d+1 = ∑_G(G),

where the sum is over all gaps of ℒ.


Let us denote the number of fixed points of _d in (∂ G ∩) by (G), and call it the cusp number of the gap.
We also denote the number of leaves on ∂ G by tan(G), and call it the tangent number of the gap.
An easy computation shows that

    (G) = (G) + tan(G).


Let G_1, G_2 be two gaps.
We say that they are adjacent if ∂ G_1 ∩∂ G_2 ≠∅.
Note that if two gaps are adjacent, then ∂ G_1 ∩∂ G_2 is a leaf of ℒ.
It is easy to see that there are exactly tan(G) number of gaps adjacent to G.



 §.§ Fixed ray laminations associated with quadrature domains



Recall that a polygonal Schwarz reflection map S∈𝒮__d (normalized as in Section <ref>) induces the action of the Nielsen map _d on the ideal boundary I(T^∞(σ))≅/ of the escaping set, and hence has exactly d+1 fixed points on its ideal boundary.






Let ℒ be a fixed ray lamination.
We say it is associated with a marked degree d polygonal Schwarz reflection S if


  * ℓ = (a, b) is a leaf of ℒ if and only a, b ∈ I(T^∞(S)) correspond to the same point on ∂ T^∞(S).



Recall that 𝒮__d is the connectedness locus of degree d regular polygonal Schwarz reflections. As mentioned in Subsection <ref>, the domain of a Schwarz reflection S∈𝒮__d is a pinched polygon, and each cut-point of this pinched polygon is the touching point of precisely two Jordan quadrature domains defining S. 
Thus, we have a coarse partition

    𝒮__d = ⋃_ℒ𝒮__d, ℒ,

where 𝒮__d, ℒ consists of all marked degree d polygonal Schwarz reflections with connected limit set and fixed ray lamination ℒ, and the union is over all possible fixed ray laminations.

The following theorem gives the topological structure of the quadrature domains for S ∈𝒮__d, ℒ.

Let ℒ be a fixed ray lamination. Let S : ⟶∈𝒮__d, ℒ.
Then 


  * the components of  are in one-to-one correspondence with the gaps of ℒ:

    = ⋃_GΩ_G;


  * two quadrature domains Ω_G, Ω_G' share at most one boundary point; moreover, they share a boundary point if and only if G, G' are adjacent;

  * the touching point of two quadrature domains Ω_G, Ω_G' is a non-singular point for both ∂Ω_G and ∂Ω_G' (i.e., it is a double point of ∂);

  * the quadrature domain Ω_G has (G) number of cusps on its boundary; and

  * the uniformizing rational map of the quadrature domain Ω_G has global degree (G).


 
Let ψ:→ T^∞(S) be a conformal conjugacy between _d and S. Also recall that Π denotes the ideal (d+1)-gon in  with ideal vertices at the (d+1)-st roots of unity.
By definition, =∖ψ(Π), where the closure is taken in . By injectivity of ψ on , the only points on Π that may be identified by ψ are the (d+1)-st roots of unity. By design, each such identification is recorded by a leaf of the fixed ray lamination of ℒ. 

1) and 2) It follows from the above discussion that the cut-points of the pinched polygon  correspond bijectively to the leaves of  and hence the components of  correspond bijectively to the gaps of . The touching structure of the components of  also follow from above.

3) As each Ω_G is a Jordan quadrature domain, the only singularities on ∂Ω_G are conformal cusps. Let ϕ_G:→Ω_G be a uniformizing rational map. Then, each conformal cusp on ∂Ω_G is a critical value of ϕ_G with an associated critical point on 𝕊^1. By the injectivity of ϕ_G on , such a cusp is an inward pointing cusp for Ω_G. This shows that the touching point of Ω_G and Ω_G' cannot be a conformal cusp for the boundary of either domain. The result follows.

4) Note that _d does not admit an anti-conformal extension in a neighborhood of any ideal vertex of Π, while it does admit such extensions around all other boundary points of Π. It follows that S does not admit an anti-conformal extension in a neighborhood of any point of ψ(ω^j) (where ω=exp(2π i/d+1), and j∈{0,⋯,d}), but it does extend anti-conformally around other boundary points of ∂. The only obstructions for an anti-conformal extension of S across a boundary point of  is a cusp or a double point. By part 3) of this theorem, the pairs of end-points of the leaves of  are in one-to-one correspondence with the double points on ∂. Hence, the other ideal vertices of Π correspond bijectively to the cusps on ∂. The result is now a consequence of the above analysis and part (1) of this theorem.

5) The map _d carries each arc of 𝕊^1∖Fix(_d) to the complement of its closure. Hence, _d(∂ G∩𝕊^1) covers ∂ G∩𝕊^1 exactly ((G)-1) times. It follows that Λ(S)∩Ω_G covers itself ((G)-1) times under the map S|_Ω_G. Since the limit set is completely invariant, we conclude that S|_Ω_G:(S|_Ω_G)^-1(Ω_G)→Ω_G is a degree ((G)-1) branched covering. This observation and Proposition <ref> together imply that the degree of the uniformizing rational map of Ω_G is equal to (G).









 §.§ Coarse partitions in low degree

There are four fixed ray laminations in degree 2: the trivial lamination _0, and the three symmetric laminations _1:={{1/3,2/3}}, _2:={{0,1/3}}, _3:={{0,2/3}}. The corresponding space 𝒮__2, ℒ_1 (respectively, 𝒮__2, ℒ_0) is given by Schwarz reflections in a fixed cardioid and its circumcircles (respectively, by Schwarz reflections in deltoid-like curves). These families were studied in <cit.>.









 §.§ Compactness of connectedness locus 𝒮__d


Recall that 𝒮__d stands for the space of degree d normalized Schwarz reflection maps whose tiling set dynamics are conformally conjugate to the Nielsen map _d of the regular ideal polygon reflection group G_d. The domain of definition of such a Schwarz reflection is the closure of a disjoint collection of simply connected quadrature domains Ω_1,⋯, Ω_k such that
∪_i=1^k Ω_i is connected and simply connected.


Since there are only finitely many fixed ray laminations, we can assume (after possibly passing to a subsequence) that any sequence in 𝒮__d entirely lies in 𝒮__d,, where  is a particular fixed ray lamination. We enumerate the gaps of  as 𝒢_1,⋯,𝒢_k.
For maps S:→ in 𝒮__d,, we denote the components of  as Ω_1,⋯,Ω_k such that Ω_i corresponds to 𝒢_i. 

We topologize 𝒮__d as follows.

We say that a sequence {S_n:^n=⋃_r=1^kΩ_r^n→}⊂𝒮__d, converges to S:=⋃_r=1^k⋃_j=1^l_rΩ_r,j→ if 


  * {Ω_r,j: j∈{1,⋯,l_r}} is the collection of all Carathéodory limits of the sequence of domains {Ω_r^n}_n, r∈{1,⋯,k}, and 


  * for n large enough, the antiholomorphic maps S_n converge uniformly to S on compact subsets of .




For each Schwarz reflection S_n:^n=⋃_r=1^kΩ_r^n→, there are k Riemann uniformization maps ϕ_n,r: ⟶Ω_r^n, r = 1,..., k which extend as rational maps of  (see <ref>).
For S:=⋃_r=1^k⋃_j=1^l_rΩ_r,j→, there are ∑_r=1^k l_r Riemann uniformization maps ϕ_r, j: ⟶Ω_r, j which extend as rationals maps of .
It follows from the definition of Carathéodory limits that S_n → S if and only if there exist M_n, r, j∈() so that

    ϕ_n,r∘ M_n, r, j→ϕ_r, j

compactly away from finitely many points in  (where each map is viewed as maps on ).




The connectedness locus 𝒮__d is compact.



Let {S_n:^n=∪_r=1^kΩ_r^n→}⊂𝒮__d,. By definition, there exist conformal maps

    ψ_n:→ T^∞(S_n)

that conjugate _d to S (wherever defined). We can normalize ψ_n, possibly after conjugating S_n by Möbius maps, such that ψ_n(0)=∞, and ψ_n(z)=1/z+O(z) near the origin. 

Note that ψ_n(Π)=T^0(S_n). Since Π contains a round disk around the origin, it follows from the Koebe one-quarter theorem that each ψ_n(Π) contains a definite round disk centered at ∞. In other words, _n⊂ B(0,R), for some fixed R>0. 
Also note that the maps ψ_n|_ form a normal family. After possibly passing to a subsequence, we may assume that ψ_n converges normally to a conformal map ψ_∞ on . Let B_s be an open ball compactly contained in ρ_s(Π) centered at ρ_s(0), for s∈{1,…,d+1}. Normal convergence of ψ_n to ψ_∞ implies that for n large, the conformal disks Ω_r^n contain some fixed open set ψ_∞(B_s) (see Figure <ref>). It follows that for each r∈{1,⋯,k}, the sequence of conformal disks {Ω_r^n}_n has at least one non-trivial Carathéodory limit. We will now argue that there are finitely many such Carathéodory limits.


By conformality of ψ_∞, the set ψ_∞(∂Π∩) is a union of d+1 disjoint bi-infinite geodesics in the hyperbolic metric of ψ_∞(). Since each ψ_n(∂Π)=∂_n is a finite union of real-algebraic curves of uniformly bounded degree contained in the fixed compact set B(0,R), one concludes that the ends of the above bi-infinite geodesics have unique limit points on ∂ψ_∞(). Thus, ψ_∞ extends continuously to ∂Π and ψ_∞(Π) has only finitely many identifications on its boundary (i.e., it is a topological polygon possibly with some vertices identified). Specifically, only the ideal vertices of Π may be identified under ψ_∞.

It follows from the above discussion that ∖ψ_∞(Π) has finitely many (at least one) components, and these are precisely the Carathéodory limits of the sequences of domains {Ω_r^n}_n, r∈{1,⋯,k} (with all possible choices of base points). We denote these components as Ω_1,⋯,Ω_l.
As the Riemann uniformizations of Ω_r^n are rational maps, it follows from the Carathéodory kernel convergence theorem that the Riemann uniformization of each Ω_j, j∈{1,⋯,l}, is also a rational map. Thus, each such limit is a simply connected quadrature domain.  


The description of Schwarz reflection maps given in Proposition <ref> now implies that for n large enough, S_n converges to the Schwarz reflection map S associated with the quadrature domains Ω_1,⋯,Ω_l. We also note that 
T^0(S) is equal to ψ_∞(Π). The equivariance property of ψ_n implies that ψ_∞ conjugates _d to S, wherever defined. Hence, the escaping set of S is ψ_∞(), and the dynamics of S on its escaping set is conformally conjugate to _d via ψ_∞^-1. Hence, S:→ lies in 𝒮__d and is a limit point of the sequence {S_n:_n→}.





Since fixed ray laminations are equivalence relations on 𝕊^1, there is a natural partial order ≥ on them. 


If {S_n}⊂𝒮__d, _1 converges to some S∈𝒮__d, _2, then _2≥_1.


If ψ_n(a)=ψ_n(b) for some a,b∈∂Π∩𝕊^1 and for all n∈, then ψ_∞(a)=ψ_∞(b).




By using quasi-PCF degeneration studied in <cit.>, one can show that the converse of Corollary <ref> is also true.
Suppose _2≥_1, then there exists a sequence {S_n}⊂𝒮__d, _1 that converges to some S∈𝒮__d, _2.
Some explicit examples are constructed in section <ref>, which are the key ingredients for proving discontinuity.








 §.§ Discontinuity and change of signature


Recall from Section <ref> that there exists a natural bijective map Φ from geometrically finite polynomials 𝒞^-_d,gf onto geometrically finite Schwarz reflections 𝒮__d,gf.

Let H_0 and H_1 stand for the main hyperbolic component and the period two hyperbolic component contained in the (0,1/d+1)-limb of the Multicorn ℳ_d, which is the connectedness locus of unicritical anti-polynomials f_c(z)=z^d+c. Then, H_1 bifurcates from H_0 along an arc, and barring a unique double parabolic parameter, all points on this arc are simple parabolic parameters (see <cit.>, <cit.>, and <cit.>).

We denote the trivial fixed ray lamination by _0, and the fixed ray lamination {{0,1/d+1}} by _1.


Let c_∞ be a simple parabolic parameter on ∂ H_0∩∂ H_1. Then Φ is discontinuous at c_∞.


Suppose that {c_n}⊂ H_1 converges to c_∞∈∂ H_0∩∂ H_1.
By our assumption, the dynamical rays at angles 0 and 1/d+1 land at a common point of the Julia set of f_c_n. Hence, Φ(f_c_n) lies in 𝒮__d, _1. By Corollary <ref> and unicriticality of the maps, any subsequential limit of {Φ(f_c_n)} also lies in 𝒮__d, _1.

On the other hand, as f_c_∞ has a simple parabolic fixed point and hence a Jordan curve Julia set, it follows that Φ(f_c_∞) lies in 𝒮__d, _0. Hence, the sequence {Φ(f_c_n)} does not converge to Φ(f_c_∞).





Let c_0 be the unique double parabolic parameter on ∂ H_0∩∂ H_1. Then Φ^-1 is discontinuous at Φ(f_c_0).


Let c_∞ be a simple parabolic parameter on ∂ H_0∩∂ H_1, and {c_n} be a sequence in H_1 converging to c_∞. Then, the multiplier of the unique attracting 2-cycle of f_c_n converges to 1. By definition of Φ, the multiplier of the unique attracting 2-cycle of Φ(f_c_n) also converges to 1. This implies that the sequence {Φ(f_c_n)} converges to Φ(f_c_0)∈𝒮__d, while {c_n} converges to c_∞. Hence, Φ^-1 is discontinuous at Φ(f_c_0).


We are now ready to complete the proof of Theorem <ref>.

The discontinuity of Φ and Φ^-1 follow from Lemmas <ref> and <ref>, respectively. 





§ COMBINATORIAL CLASSES FOR POLYNOMIALS AND ANTI-POLYNOMIALS

Let 𝒞^±_d, r be the space of monic and centered polynomials (or anti-polynomials) with connected Julia set and all cycles repelling (also called `periodically repelling').
In this section, we discuss a natural way to organize the data for renormalizations of polynomials, or anti-polynomials in 𝒞^±_d, r.

We will use puzzle pieces to construct a decomposition of the dynamical plane.
To better organize these combinatorial data, we will inductively construct puzzles depending on their level of renormalizations and the depth of the puzzle pieces.



 §.§ Level zero renormalization puzzles

To start our construction, we consider rays landing at the fixed points.
If the polynomial or the anti-polynomial is periodically repelling, the following lemma guarantees that these rays cut through the Julia set.

Let f ∈𝒞^±_d, r.
Then the union of rays landing at all fixed points together with their landing points cut the Julia set of f.


If f is a polynomial, then there are d fixed points in .
Since f has connected Julia set and is periodically repelling, these fixed points are distinct and all lie on the Julia set.
Since there are d-1 fixed rays, at least one fixed point p is the landing point of some periodic rays of period ≥ 2.
Then these rays landing at p together with p cut the Julia set of f.

If f is an anti-polynomial, then there are d fixed points in  by Lefschetz fixed point theorem, as all fixed points are repelling (see <cit.>).
Since there are d+1 fixed rays, at least two fixed rays must land together at a fixed point p.
Then these two rays landing at p together with p cut the Julia set of f.


To define level zero renormalization puzzles, we let ϕ be the Riemann mapping from - to - K(f) (where K(f) is the filled Julia set of f), normalized so ϕ'(∞) = 1.
Consider the disk D bounded by the image of the circle {| z| =2} under ϕ.
This disk encloses K(f) and is bounded by an equipotential of the complement.

By Lemma <ref>, this disk is cut into q>1 pieces by the external rays landing at the fixed points of f that are also cut points.
These pieces form the puzzle pieces of depth zero for level zero renormalization.
Each piece is a closed disk, whose boundary consists of fixed points, segments of external rays landing at those fixed points, and part of ∂ D.

The puzzle pieces at depth k+1 are defined inductively as the closures of the components of f^-1((P)), where P ranges over all puzzle pieces at depth k.

The puzzle pieces at depth k have disjoint interiors and cover the Julia set.
As the depth increases, the puzzles become successively finer.
More precisely, a piece at depth k+1 is contained in a unique piece at depth k.

We will use the notation P_k, f^0(z) to denote the puzzle piece at depth k for level zero renormalization, that contains z in its interior.
Sometimes we simply write it as P_k^0(z) or P_k(z) if the map f and the level of renormalizations are not ambiguous.

We remark that the puzzle piece P_k, f^0(z) is well-defined for z∈ J(f) if f^k(z) is not a fixed point.
If f^k(z) is a fixed point, then z might be on the common boundary of multiple puzzle pieces at depth k.


Let c be a critical point for f.
We say that c is level zero renormalizable if c is not on the boundary of any puzzle piece and there exists an integer p ≥ 1 so that 

    P_k(c) = P_k(f^p(c))  for all  k≥ 0.

The smallest such integer p is called the renormalization period.

We say that f is level zero renormalizable if some critical point of f is level zero renormalizable, and level zero non-renormalizable otherwise.




If c is level zero renormalizable for a map f ∈𝒞^±_d, r, then there exist open sets U∋ c, V∋ c with U compactly contained in V so that 

    f^p: U ⟶ V

is a polynomial-like map with connected small Julia set J(c) := ⋂_k P_k(c).


Since P_k(c) is a nested decreasing sequence of closed disks, 
there exists some integer N so that for any critical point c', either


  * c' ∈ P_k(c) for all k ≥ N; or

  * c' ∉ P_k(c) for all k≥ N.

Let k ≥ N.
Consider the map

    f^p: P_k+p(c) ⟶  P_k(f^p(c)) = P_k(c).

Note that P_k+p(c) ⊆ P_k(c).
Since all cycles of f are repelling, we can find a small neighborhood U, V of P_k+p(c) and P_k(c) so that


  * a critical point c' ∈ U if and only if c' ∈ P_k+p(c);

  * U is compactly contained in V;

  * f^p: U ⟶ V is a branched covering map.

Then any point in J(c) ⊆ U is non-escaping for the map f^p: U ⟶ V.
Let c' ∈ U be a critical point.
Then c' ∈ P_k(c) for all k ≥ N, so c' is non-escaping.
Therefore, f^p: U ⟶ V is a polynomial-like map with connected small Julia set J(c).



If c is level zero renormalizable, then its renormalization period p is at least 2.


Suppose for contradiction that p = 1.
Then we have a polynomial-like map f:U ⟶ V.
By Lemma <ref>, the external rays landing at the fixed points of f: U ⟶ V would cut through the small Julia set J(c) = ⋂_k P_k(c), which is a contradiction.



If c is level zero renormalizable, then f^p: U ⟶ V is a simple renormalization, i.e., the small Julia sets J(c), J(f(c)), ..., J(f^p-1(c)) intersect possibly only at their tip points.


For sufficiently large k, we may assume that the puzzle pieces 

    P_k(c), P_k(f(c)), ..., P_k(f^p-1(c))
 
have disjoint interior.
Therefore, J(f^i(c)) = ⋂_k P_k(f^i(c)) cannot intersect at a cut point.





























 §.§ Level m renormalization puzzles

Let f ∈𝒞^±_d, r be a polynomial or anti-polynomial with connected Julia set and all cycles repelling.
Suppose f is level zero renormalizable.
Let c be a critical point that is level zero renormalizable with renormalization period p.
By Lemma <ref>, we have a polynomial-like map f^p: U ⟶ V with small Julia set J(c). 
The fixed points of f^p in J(c) are periodic points of f with period p.
Note that the polynomial-like map f^p: U ⟶ V has connected Julia set and is periodically repelling.
Thus, the external rays landing at the fixed points of f^p in J(c) cut the small Julia set.

To define the level one renormalization puzzle, we consider 


  * the external rays landing at all fixed points of f; as well as 

  * the external rays landing at the periodic points of f associated with the fixed points of the polynomial-like maps f^p: U ⟶ V for all level zero renormalizable critical points.

Then the closed disk D bounded by the image of the circle {| z| =2} under the Riemann mapping ϕ:  - ⟶ - K(f) is cut by these external rays into finitely many pieces.
These pieces form the puzzle pieces of depth zero for level one renormalization.

Similar as before, the puzzle pieces at depth k+1 are defined inductively by pulling back the puzzle pieces at depth k.
We use the notation P^1_k(z) to denote the puzzle piece at depth k for level one renormalization that contains z in its interior.

Just as in Definition <ref>, we define a critical point c to be level one renormalizable
if c is not on the boundary of any puzzle piece and there exists a number p ≥ 1 so that 

    P^1_k(c) = P^1_k(f^p(c))  for all  k.

If c is level one renormalizable with renormalization period p_1, then the same proof of Lemma <ref> shows that there exists a polynomial-like map

    f^p_1: U^1 ⟶ V^1

with small Julia set J^1(c)= ⋂_k P^1_k(c).

Inductively, if f is level m renormalizable, then we can define the level m+1 renormalization puzzle for f.
We remark that the choice of depth zero puzzle pieces for level n renormalization depends on the polynomial or the anti-polynomial f.

To make our notations uniform in the future, if f is not level m renormalizable, then we define level m+1 renormalization puzzle for f equal to the level m renormalization puzzle.

In this way, for a periodically repelling polynomial or an anti-polynomial f with connected Julia set, we can associate an infinite sequence of puzzles

    𝔓^m(f) = {P^m_k}, m=0, 1, ⋯.

We say that f is finitely renormalizable if this sequence of puzzle 𝔓^m(f)  is eventually constant, and infinitely renormalizable otherwise.

It is clear that we have the following basic nesting property.

If c is level m renormalizable, then c is level j renormalizable for all j ≤ m.
The corresponding renormalization periods satisfy

    p_0 | p_1 | ... | p_m.

Further, the small Julia sets are nested; i.e.,

    J^m(c) ⊆ J^m-1(c) ⊆ ... ⊆ J^0(c).






 §.§ Combinatorial distance


    Let f, g ∈𝒞^±_d, r. 


    We say that g is aligned with f at depth k of level m if for each point p ∈∂ P^m_k,f∩ J(f), there exists a point q ∈ J(g) so that the angles of the external rays landing at p are the same as the angles of those landing at q.

From the definition, it is clear that if g is aligned with f at depth k of level m, then g is aligned with f at depth k' of level m' for all k' ≤ k and m' ≤ m.

Given a puzzle piece P^m_k,f for f, its boundary consists of segments of external rays, their landing points and segments of an equipotential.
If g is aligned with f at depth k of level m, we can construct a puzzle piece for g bounded by the corresponding segments of external rays (with the same angle), their landing points and segments of the equipotential.
We call this the induced puzzle piece for g.
We use the notation P^m_k, g|f(z) to represent the induced puzzle piece for g at depth k of level m that contains z.




We remark that these induced puzzle pieces for g may be different from the level m renormalization puzzle 𝔓^m(g) for g.
Indeed, it is possible for a non-renormalizable polynomial g to align with a renormalizable polynomial f at some depth k and level m with m ≥ 1.
Then there are more induced puzzle pieces at depth zero of level 1 than those in 𝔓^1(g).  As an example, one can take f to be the quadratic Feigenbaum polynomial and pick any non-renormalizable g in the limb of the Mandelbrot set cut out by the parameter rays at angles 2/5 and 3/5.


We define level m renormalization distance from g to f as

    d_m(f, g) = e^-(k_m(f, g)+1)· (m+1),

where 

    k_m(f,g) = inf{j: g  is not aligned with f at depth j of level m}.

We define

    d(f, g) = ∑_m≥ 0 d_m(f, g).

We remark that d is not a distance function as d(f, g) may be different from d(g, f).



































Recall that two periodically repelling polynomials or anti-polynomials are in the same combinatorial class if they have the same rational lamination, and 𝒞^±_d, r consists of combinatorial classes of periodically repelling polynomials or anti-polynomials.

Let f, g ∈𝒞^±_d, r. Then d(f, g) = 0 if and only if f, g have the same rational lamination; i.e., [f] = [g] ∈𝒞^±_d, r.


It is clear that if f, g have the same rational lamination, then the induced puzzle for g agrees with the renormalization puzzle for f at all levels and depths. Thus, d(f, g) = 0.

Conversely, suppose that s, t ∈ form a leaf of the rational lamination for f; i.e., they coland at some pre-periodic point x for f.
If x is on the boundary of some puzzle piece, then s, t also coland for g as d(f, g) = 0.
So s, t form a leaf for the rational lamination for g as well.

Thus, we assume that x is not on the boundary of any puzzle piece.
Assume that x has pre-period l and period p, and y = f^l(x).
Then f^p: (P^m_k+p, f(y)) ⟶(P^m_k, f(y)) is a proper map for all m and k.
We can choose m large enough so that the level m renormalization period for any critical point is larger than p.
(Here we use the convention that the renormalization period is ∞ if the critical point is not renormalizable.)
Thus, we can then choose N large enough so that (P^m_k+p, f(y)) contains no critical points for k ≥ N.
So f^p: (P^m_k+p, f(y)) ⟶(P^m_k, f(y)) has degree 1 for all k ≥ N.
Therefore, ⋂_k P^m_k, f(y) = { y}, so ⋂_k P^m_k, f(x) = { x}.
Hence, we have a sequence of angles s_k, t_k colanding at boundary points of some puzzle pieces with s_k → s and t_k → t.
Since d(f, g) = 0, the angles s_k, t_k define a leaf for the rational lamination of g, for all k.
As rational laminations are closed, the angles s, t also define a leaf of the rational lamination for g.
Therefore, the rational lamination of f is contained in the rational lamination of g.
The other inclusion can be proved similarly.






 §.§ Combinatorial continuity

In this section, we will prove a combinatorial continuity statement for maps in 𝒞^±_d, r.
The proof is divided into two cases depending on whether there are critical points on boundaries of puzzle pieces or not.




  §.§.§ No boundary critical points

We first present the proof where no critical points are on the boundaries of puzzle pieces.

Let f ∈𝒞^±_d, r. Suppose that no critical point is on the boundary of any puzzle piece. 


  * If f_n → f in 𝒞^±_d, r, then d(f, f_n) → 0.

  * If d(f, f_n) → 0, then for any accumulation point g∈𝒞^±_d, r of f_n, we have that d(f, g) = 0.




Suppose that for some fixed level m, the distance d_m(f, f_n) → 0 as n→∞. We remark that it is not true that any accumulation point g∈𝒞^±_d, r of f_n satisfies d_m(f, g) = 0.

For example, let f be the quadratic Feigenbaum polynomial, and f_n be the tuning of the basilica polynomial with a sequence of real polynomials g_n that goes to the Chebyshev polynomial.
The limit g = lim_n f_n is the basilica polynomial tuned with the Chebyshev polynomial.
Then d_0(f, f_n) = 0 for all n, but d_0(f, g) ≠ 0 since two rays (lying on the boundary of level zero puzzles) that land at distinct Julia points for f coland at the critical point of g.





Let f ∈𝒞^±_d, r. Suppose that no critical point is on the boundary of any puzzle piece.
Let c be a critical point for f, and P^m_k(c) be the level m puzzle piece at depth k that contains c.
Then there exist l, j so that P^l_j(c) is compactly contained in (P^m_k(c)).










Suppose for contradiction that there exists a Julia point x on the boundary of P^m_k(c) so that x ∈ P^l_j(c) for all j, l.
Let q be the pre-period and p be the period of the pre-periodic point x. The assumption x ∈ P^l_j(c) for all j, l implies that f^q+np(x)=f^q(x)∈ P^l_j(f^q+np(c)) for all j, l, and n. It follows that P^l_j(f^q(c)) = P^l_j(f^q+rp(c)), where r is the valence of f^q(x) as a branch point of J(f).


If for some l, P^l_j(f^q(c)) contains no critical points for all large j, then ⋂_j P^l_j(f^q(c)) = f^q(x); i.e., f^q(c)=f^q(x).
This is a contradiction to the fact that c is not on the boundary of any puzzle piece.
Otherwise, there exists some critical point c' (potentially equal to c) which is infinitely renormalizable, and f^q(x) is in the small Julia set J^l(c') for all l. But this is not possible as the period of f^q(x) is a fixed integer.


We are now ready to prove Theorem <ref>.

To prove the first statement, it suffices to show d_m(f, f_n) → 0 for all m.
Let P be a puzzle piece of level m and depth k.
Since there are no critical points of f on the boundaries of any puzzle pieces, any points in ∂ P is eventually repelling and its forward iterate avoids any critical points.
Thus, by stability of landing external rays (see <cit.>), there exists a small neighborhood U_m,k of f so that any map g ∈ U_m,k is aligned with f at depth k of level m.
Therefore d_m(f, f_n) → 0 as n →∞.

For the second statement, by passing to a subsequence, we may assume that f_n → g ∈𝒞^±_d, r. 
It suffices to show that d_m(f, g) = 0 for all m.
Let m be the minimal level where d_m(f, g) ≠ 0, and let k:= k_m(f, g).

We claim that k > 0.
By way of contradiction, suppose that k=0.
Let {t_i} be the set of angles of external rays landing at boundaries of the puzzle pieces at depth 0 of 𝔓^m(f).
Note that the Julia points on boundaries of puzzle pieces at depth 0 for f are periodic points, so t_i are periodic under the map m_d(t) = dt  or m_-d(t)=-dt.
Since d(f, f_n) → 0, the external rays with angles in {t_i} have the same landing patterns in the dynamical planes of f_n and f for all sufficiently large n.
By our assumption, the landing patterns of the external rays with angles in {t_i} in the dynamical plane of g are different from the common landing patterns of these rays in the dynamical planes of f and f_n (with sufficiently large n).
Thus, by stability of landing external rays, some angle must land at critical points.
But this is not possible as each t_i is periodic.

Let P_k, f^m be a puzzle piece for f which does not correspond to an induced puzzle piece for g.
Since k > 0, P_k-1, f^m:=f(P_k, f^m) is a puzzle piece at depth k-1.
It corresponds to some induced puzzle piece P_k-1, g|f^m for g.
We claim that some critical value v of g must lie on ∂P_k-1, g|f^m.
Indeed, otherwise, the Julia points on the boundary of g^-1(P_k-1, g|f^m) are pre-repelling and avoid critical points. Since P_k, f^m does not correspond to a puzzle piece for g, this is a contradiction by stability of landing rays.

Let c_g be the critical point of g that is mapped to v.
Since d(f, f_n) → 0, the puzzle pieces of f_n at depth k and level m align with f for all large n.
Since no critical point of f is on the boundary of puzzle pieces, for all large n, no critical point of f_n is on the boundary of puzzle pieces of f_n at depth k and level m.
Since f_n → g, there exists a sequence of critical points c_n for f_n with c_n → c_g.
Let P_k, f_n^m(c_n) be the puzzle piece that contains c_n, which corresponds to some puzzle piece P_k, f^m(n) for f.
Since there are only finitely many puzzle pieces at depth k and level m, we may assume that P_k, f^m(n) is eventually constant.
Denote this puzzle piece by P_k, f^m. Note that it contains some critical point c of f.
One can verify that P_k, f^m does not correspond to an induced puzzle piece for g.
Since f(P_k, f^m) corresponds to an induced puzzle piece for g, there exist two angles α, β landing at distinct points a, b ∈∂ P_k, f^m for f, but they land at a critical point for g.
By Lemma <ref>, there exists some puzzle piece P^m_1_k_1, f(c) that is compactly contained in (P_k, f^m) = (P_k, f^m(c)).
Inductively, we obtain a sequence 

    ⋯⋐(P^m_2_k_2, f(c)) ⋐(P^m_1_k_1, f(c)) ⋐(P^m_k, f(c)).

Therefore, we can find a sequence of pairs of angles t_i, s_i so that 


  * t_i, s_i coland at a point on ∂ P^m_i_k_i, f(c);

  * the union of external rays ℛ_f(t_i)∪ℛ_f(s_i) separates the two points a, b ∈∂ P_k, f^m(c).

Therefore, ℛ_f_n(t_i) and ℛ_f_n(s_i) land at the same point for all sufficiently large n. 
On the other hand, since α, β coland for g, the union ℛ_g(α)∪ℛ_g(β) separates the external rays ℛ_g(t_i) and ℛ_g(s_i).
It follows that both ℛ_g(t_i) and ℛ_g(s_i) land at c_g (see <cit.>).
This is a contradiction, as there can be at most finitely many external rays landing at a pre-periodic point.














  §.§.§ With boundary critical points

To deal with the situation where critical points lie on boundaries of puzzle pieces, we need to modify the definition of combinatorial distances.


Let f ∈𝒞^±_d, r with puzzles 𝔓^m.
Let c be a critical point on the boundary of some puzzle piece.
Then c is pre-periodic.
Denote by 𝒪 the periodic cycle associated to c.
We define a modification of 𝔓^m by removing all external rays landing on 𝒪 and their iterated preimages in the definition of puzzle pieces.

As the first step of the modification, we denote by 𝔓^m the modified puzzle after removing all external rays associated with all critical points on the boundaries of puzzle pieces.





We say that c is statically renormalizable with respect to the modified puzzle 𝔓^m_0 if c is renormalizable with respect to the modified puzzle 𝔓^m_0 and

    ⋂_k P^m_k(c) = ⋂_k P^m_0_k(c)
 
for all m ≥ m_0.
We denote by

    J^∞(c) = J^m_0(c) := ⋂_k P^m_0_k(c) = ⋂_m ⋂_k P^m_k(c)

the static small Julia set for c.
Similarly, we say that a critical point c is statically pre-renormalizable if c is eventually mapped to some static small Julia set.



    We remark that if c is statically renormalizable, then c is renormalizable with respect to 𝔓^m for all m.
    However, for all m ≥ m_0, the renormalization with respect to 𝔓^m stays the same, as the condition ⋂_k P^m_k(c) = ⋂_k P^m_0_k(c) implies that all higher level puzzle pieces do not cut through the small Julia set J^m_0(c).
    This is a consequence of the fact we generate the puzzle by rays landing at the fixed points of renormalizations.





Let f ∈𝒞^±_d, r.
Let c be a statically renormalizable critical point with respect to 𝔓^m_0.
Then the corresponding polynomial-like map

    f^p_m_0: U^m_0⟶ V^m_0

is finitely renormalizable.


If f^p_m_0: U^m_0⟶ V^m_0 is infinitely renormalizable, then as m →∞, there are infinitely many puzzle pieces of 𝔓^m at depth 0 that cut through the small Julia set.
The modified puzzles can only omit finitely many of them.
Therefore, as m →∞, there exist an unbounded number of modified puzzle pieces of 𝔓^m at depth 0 that cut through the small Julia set J^∞(c). 
Since c is a statically renormalizable, there are no modified puzzle pieces of 𝔓^m at depth 0 that cut through J^∞(c).
This is a contradiction.



























The second step of the modification is not canonical.
Let c be a statically renormalizable critical point for 𝔓^m_0, with static small Julia set J^∞(c).
We choose periodic cut cycles in J^∞(c) disjoint from the critical orbit of f.
We add puzzle pieces induced by the external rays landing at these cut points as depth zero puzzles in 𝔓^m for all m ≥ m_0.
Pulling back by dynamics, we add depth k puzzles in 𝔓^m.
By Lemma <ref>, we can choose these periodic cut cycles so that no critical point in J^∞(c) is renormalizable with respect to the new modified puzzle.
By making such modifications for each static renormalization critical point, we obtain a new puzzle system.
We denote it by 𝔓.
We denote by d(f, g) the distance from g to f with respect to the modified puzzle 𝔓.

By our construction, any static critical point is only finitely renormalizable with respect to the modified puzzle 𝔓. 
A similar argument as Lemma <ref> gives

Let f, g∈𝒞^±_d, r. 
Then d(f, g) = 0 if and only if f, g have the same rational lamination.



By construction, no critical point is on the boundary of puzzle pieces in 𝔓. Thus, the proof of Theorem <ref> applies to this setting as well, and yields:

Let f, f_n ∈𝒞^±_d, r.


  * If f_n → f in 𝒞^±_d, r, then d(f, f_n) → 0.


  * If d(f, f_n) → 0, 
then for any accumulation point g∈𝒞^±_d, r of f_n, we have that d(f, g) = 0.





















































 §.§ Approximation by post-critically finite maps

In this subsection, we will prove the following theorem about approximating any combinatorial class using post-critically finite maps.

Let f∈𝒞^±_d, r. Then there exists a sequence of post-critically finite polynomials f_n ∈𝒞^±_d, r so that d(f, f_n) → 0.




    Let λ(f) be the rational lamination of f.
    By <cit.>, there exists a sequence of post-critically finite f_n converging to a periodically repelling map g so that λ(g) = λ(f).
    Since f is periodically repelling, f_n can be chosen to be periodically repelling as well.
    Then by Theorem <ref>, we have  d(g, f_n) → 0. Since λ(g) = λ(f) (and the construction of the puzzles can be uniquely recovered from the lamination), it follows that d(f, f_n) → 0.




































§ COMBINATORIAL CLASSES FOR POLYGONAL SCHWARZ REFLECTIONS


Recall that for a Schwarz reflection map S:→, the set 𝔖 stands for the singular (fixed) points on the boundary ∂. The local dynamics near a singular point is parabolic in nature. In fact, for S∈𝒮__d, these singular fixed points have attracting directions in the tiling set. Indeed, the points in 𝔖 correspond, under the conformal map ψ:→ T^∞(S) that conjugates _d to S, to the (parabolic) fixed points of _d. The set of (parabolic) fixed points of _d, denoted by Fix(_d), consists of the (d+1)-st roots of unity. It is easy to see that points in Fix(_d) attract nearby points in  under iterations of _d.

We define the space 𝒮__d,r⊂𝒮__d as the collection of those maps S for which 


  * no point in 𝔖 has an attracting direction in K(S), and 

  * each periodic point of S not in 𝔖 is repelling.





 §.§ Renormalization puzzles

Recall that T^0(S) is the fundamental tile for the Schwarz reflection map S, and connected components of S^-n(T^0(S)) are called tiles of rank n. As in Proposition <ref>, we denote the union of the tiles of rank 0 through k by E^k. Clearly, E^0=T^0(S).
We define the depth zero equipotential for S as ∂ E^0.
Similarly, the depth k equipotential is defined as ∂ E^k.

Note that these equipotentials enclose K(S); however, unlike the polynomial or anti-polynomial setting, an equipotential for a Schwarz reflection always intersects the limit set Λ(S).
We also remark that any intersection point of an equipotential of depth zero with the limit set Λ(S) is a fixed point of S.

To define level zero renormalization puzzles for a Schwarz reflection S ∈𝒮__d,r, we consider the union ⋃ of closures of all dynamical rays landing at all fixed points of S (see Section <ref> for the definition of dynamical rays for Schwarz reflections in 𝒮__d).
A similar counting argument as in Lemma <ref> shows that this closure cuts the limit set Λ(S).
We define puzzle pieces at depth zero for level zero renormalization as closures of the connected components of 

    - E^0 - ⋃

intersecting Λ(S).
Similarly as in the polynomial case, the puzzle pieces at depth k+1 are defined inductively as closures of the components of S^-1((P)), where P ranges over all puzzle pieces at depth k.
We will use a similar notation as in the polynomial case, where P_k(z) represents the depth k puzzle piece that contains z in its interior.

Renormalization can be defined similarly.

Let c be a critical point for S.
We say that c is level zero renormalizable if there exists an integer p ≥ 1 so that 

    P_k(c) = P_k(S^p(c))  for all  k≥ 0.

The smallest such integer p is called the renormalization period.

We say that S is level zero renormalizable if any some critical point of S is level zero renormalizable, and level zero non-renormalizable otherwise.


Similar to the polynomial setting, we have the following lemma (c.f. Lemma <ref>).

If c is level zero renormalizable with renormalization period p>2, then there exist open sets U∋ c, V∋ c with U compactly contained in V so that 

    S^p: U ⟶ V

is a polynomial/anti-polynomial-like map with small Julia set J(c) := ⋂_k P_k(c).


We comment on the nuance of the assumption that the renormalization period p is larger than two.
Indeed, the same argument as in Lemma <ref> gives a branched covering between puzzles P_k+p(c) ⊆ P_k(S^p(c)) = P_k(c),

    S^p: P_k+p(c) ⟶  P_k(c).

The subtlety here is to thicken the puzzle pieces slightly to ensure compact containment between them.
Since S is assumed to be periodically repelling, this can be done if there are no singular points on the boundaries of the puzzle pieces.

Now let x be a singular point. Then there are at most two different nested sequences of puzzle pieces P_n with x ∈∂ P_n (recall that at most two quadrature domains can touch at a point).
Thus, we have S^2: P_n+2⟶ P_n.
Therefore, if the renormalization period p is larger than two, no singular point can lie on the boundary, and the lemma follows.


A modification of the proof of Corollary <ref> also gives the following lemma.

If c is level zero renormalizable, then its renormalization period p is at least 2.


By way of contradiction, let us assume that the renormalization period p is equal to 1.
The proof is identical to Corollary <ref> if there are no singular points on J(c) := ⋂_k P_k(c).
Thus, we may assume that there is a singular point x ∈ J(c).
Then for all large n, we have a map S: P_n+1(c) ⟶ P_n(c) of degree e>1.
Since we assume that the map is periodically repelling, we can extend the map S: P_n+1(c) ⟶ P_n(c) to a topological degree e branched covering F of the sphere so that x has (Lefschetz fixed point) index -1 and there is a unique fixed point in - P_n(c) with index +1.
Thus, by Lefschetz–Hopf fixed point theorem, there exists another fixed point z ≠ x in P_n(c).
The arguments used in the proof of Corollary <ref> now show that the dynamical rays landing at the fixed
points of S: P_n+1(c) ⟶ P_n(c) cut through the small Julia set J(c).


We can define level m renormalization in this setting as in the polynomial or anti-polynomial case.
The following lemma says that except possibly level zero renormalizations, all subsequent renormalizations are actually polynomial/anti-polynomial-like maps.


If c is level m renormalizable with m ≥ 1, then there exist open sets U∋ c, V∋ c with U compactly contained in V so that 

    S^p_0p_1...p_m: U ⟶ V

is a polynomial/anti-polynomial-like map with small Julia set J^m(c):= ⋂_k P^m_k(c).


Note that the arguments of Lemma <ref> imply that p_0, p_1 ≥ 2, so p_0p_1...p_m>2.
Thus the lemma follows from arguments used in the proof of Lemma <ref>.


Recall that 𝒮__d, r consists of combinatorial classes of periodically repelling Schwarz reflections in 𝒮__d.
The same proof as Lemma <ref> also gives the following lemma.

Let S, R ∈𝒮__d,r. Then d(S, R) = 0 if and only if S, R have the same rational lamination; i.e., [S] = [R] ∈𝒮__d, r.






Let S ∈𝒮__d,r. Suppose that no critical point is on the boundary of any puzzle piece. 


  * If S_n → S in 𝒮__d,r, then d(S, S_n) → 0.

  * If d(S, S_n) → 0, then for any accumulation point R∈𝒮__d,r of S_n, we have that d(S, R) = 0.



    The first statement follows from the same argument as in Theorem <ref>, as an external ray landing at a repelling periodic point, or a repelling pre-periodic point whose orbit does not contain a critical point is stable under perturbation (see Proposition <ref>).

    By Lemma <ref>, all renormalization of level m ≥ 1 are (anti-)polynomial-like maps.
    Thus, the argument of the Separation Lemma <ref> applies.
    Now the same argument in Theorem <ref> applies to show the second statement.


If there are critical points on the boundary of a puzzle piece, then we can define modified puzzles 𝔓^m for S exactly as in the (anti-)polynomial setting.
Then, no critical points lie on the boundary of puzzle pieces for 𝔓. This gives the following combinatorial continuity result, where d(S_1, S_2) stands for the distance from S_2 to S_1 with respect to the modified puzzle 𝔓.




Let S, S_n ∈𝒮__d,r.


  * If S_n → S in 𝒮__d,r, then d(S, S_n) → 0.


  * If d(S, S_n) → 0 
then for any accumulation point R∈𝒮__d,r of S_n, we have that d(S, R) = 0.








§ COMBINATORIAL STRAIGHTENING

In this section, we shall assemble all the ingredients to prove Theorem <ref>.

To construct the map Φ, we first note that if f is post-critically finite, then there exists unique conformal mating S ∈𝒮__d,r of f with _d (by Theorem <ref>).
Let [f] and [S] be the combinatorial class of f and S respectively.
We define Φ([f]) = [S].

Let f ∈𝒞^-_d, r be a periodically repelling anti-polynomial.
By Theorem <ref>, there exists a sequence of post-critically finite anti-polynomials f_n with d(f, f_n) → 0. Let  S_n ∈𝒮__d,r be the unique conformal mating of f_n with _d.
By the compactness result Theorem <ref>, after passing to a subsequence, we may assume that S_n converges.
Denote this limit by S.
Using a similar argument in <cit.>, one can show that S cannot have parabolic cycles or irrationally neutral cycles.
Thus S ∈𝒮__d,r.
We define Φ([f]) = [S].

Since S_n → S, by Theorem <ref>, d(S, S_n) → 0.

We claim that the rational lamination (ℰ_d)_*(λ(S)) = λ(f).
Indeed, since (ℰ_d)_*(λ(S)) is m_-d-invariant, there exists a periodically repelling anti-polynomial g with λ(g) = (ℰ_d)_*(λ(S)) (see <cit.>).
Since (ℰ_d)_*(λ(S_n)) = λ(f_n), and the construction of the puzzles can be uniquely recovered from the lamination, we conclude that d(g, f_n) → 0.
Therefore, by Theorem <ref>, d(f, g) = 0, and hence λ(f) = λ(g) by Lemma <ref>.
This also shows that the map Φ is well-defined.












We now show that Φ is continuous.
Suppose f_n → f. 
Then d(f, f_n) → 0.
Let S_n ∈Φ([f_n]) and let S∈𝒮__d,r be an accumulation point of S_n.
By Theorem <ref>, d(S, S_n) → 0.
The same argument as above shows that (ℰ_d)_*(λ(S)) = λ(f).
Therefore S ∈Φ([f]), proving continuity of Φ.

A similar argument can be applied to construct a well-defined continuous inverse Φ^-1, and the theorem follows.




§ RIGIDITY OF AT MOST FINITELY RENORMALIZABLE SCHWARZ REFLECTIONS

In this section, we will show that a periodically repelling, finitely renormalizable Schwarz reflection is rigid, and deduce Theorem <ref>.
More precisely, we have

Let S ∈𝒮__d,r be finitely renormalizable.
Then


  * the limit set Λ(S) is locally connected;

  * S has no invariant line field on Λ(S);

  * if S∈𝒮__d,r has the same rational lamination as S, then S = S. In particular, the combinatorial class of S is a singleton.



The proof uses rigidity of complex box mappings, which is a refinement of the notion of generalized polynomial-like maps (cf. <cit.>, <cit.>). Although the proof is almost identical to the polynomial setting (see <cit.>, cf. <cit.>), there is a subtle difference from the polynomial setting, as the complement - 𝒟 of the domain of a Schwarz reflection S: 𝒟⟶ may be pinched (i.e., not Jordan), which requires some additional treatment.
(<cit.>)
A holomorphic (or anti-holomorphic) map F: U ⟶ V between open sets U ⊆ V in  is a complex box mapping if


  * F has finitely many critical points;

  * V is a union of finitely many pairwise disjoint Jordan disks;

  * let V' be a component of V, then either V' is a component of U or V' ∩ U is a union of Jordan disks with pairwise disjoint closures which are compactly contained in V';

  * let U' be a component of U, then F(U') is a component of V.





Since S is finitely renormalizable, we can choose the level m puzzle 𝔓 = 𝔓^m so that S is non-renormalizable with respect to 𝔓.
The same argument as in <cit.> allows us to construct a finite collection of open puzzle pieces V that contains all critical points, such that the first return map into V defines a complex box mapping F:U ⟶ V.

Let {a_1,..., a_r} be the collection of pinching points for - 𝒟 where 𝒟 is the domain for S.
Note that each a_i is a fixed point for S.
We may assume that {a_1,..., a_r}∩V = ∅.

Note that for each depth k, there are at most two puzzle pieces attached at a_i.
Since S is non-renormalizable with respect to 𝔓, the puzzle pieces attached at a_i contain no critical points at all sufficiently large depth k.
Since the local dynamics on Λ(S) near these fixed points are parabolic repelling, these puzzle pieces shrink to points as the depth k →∞.

Now choose k_0 large enough so that these puzzle pieces have pairwise disjoint closure except at a_i and their closure are also disjoint from V.
Let W be the union of all depth k_0 puzzle pieces attached to a_i for some i. 

Then rigidity of complex box mappings <cit.> and the above definition of W imply that all puzzle pieces in V∪ W shrink to points, and it carries no measurable invariant line field on its Julia set.
Note that the set of points which are never mapped into V ∪ W is hyperbolic, so all puzzle pieces shrink to points and its area is 0.
This proves (1) and (2) in Theorem <ref>.

Let S∈𝒮__d,r be a normalized polygonal Schwarz reflection that has the same lamination as S.
Then S is also finitely renormalizable.
Since the limit sets Λ(S) and Λ(S) are locally connected, and all cycles of S, S are repelling, the maps S|_K(S) and S|_K(S) are topologically conjugate.
By the same argument as in <cit.>, the corresponding complex box mappings F:U ⟶ V and F: U⟶V are quasiconformally conjugate.
By spreading around the conjugacy (see the Spreading Principle <cit.>), we can promote this local quasiconformal conjugacy to a quasiconformal map Ψ: ⟶ that conjugates the dynamics of S and S.
Since the external dynamics of S, S are conformally conjugate, and the Julia set Λ(S) is a dendrite that does not support any invariant line field, Ψ is a conformal map.
This proves (3) in Theorem <ref>.



The theorem now follows from Theorem <ref> and Theorem <ref>.









§ CORRESPONDENCES ASSOCIATED WITH POLYGONAL SCHWARZ REFLECTIONS


Let S:𝒟→ be a conformal mating of a degree d anti-polynomial f with connected Julia set and the Nielsen map _d. Recall that 

    = ⋃_i=1^k Ω_i,

where Ω_i are Jordan quadrature domains, and S|_Ω_i is the Schwarz reflection map of Ω_i.
In this section, we will associate anti-holomorphic correspondences with such matings, and show that they can also be interpreted as matings of anti-polynomials and ideal (d+1)-gon reflection groups. 

Let R_i:→Ω_i be uniformizing rational maps. To make book-keeping easier, we denote the domain of R_i (which is a copy of the Riemann sphere) by _i and the unit disk in _i by _i. We also set ℐ:={1,⋯,k}.

We recall that ∂ meets the limit set Λ(S) in the finite set 𝔖, which is the set of singular points on ∂.
We define 

    𝔖_i := (R_i|_∂_i)^-1(𝔖∩∂Ω_i).
 





 §.§ Construction of the correspondence




Let us now consider the disjoint union 

    𝔘 := _i∈ℐ_i ≅ ×ℐ,

and define the maps

    R: 𝔘⟶,   (z,i)↦ R_i(z),

and

    η^- : 𝔘⟶𝔘,   (z,i)↦ (η^-(z),i).

Note that R is a branched covering of degree d+1, and η^- is a homeomorphism.

The conformal mating S gives rise to an anti-holomorphic correspondence on 𝔘 as follows. For i,j∈ℐ, define

    ℭ_i,j:= {(z,w)∈_i×_j: R_j(w)-R_i(η^-(z))/w-η^-(z)=0},    if i=j,

and

    ℭ_i,j:= {(z,w)∈_i×_j: R_j(w)-R_i(η^-(z))=0},    if i≠ j.

The union of the real-algebraic curves ℭ_i,j can be written succinctly as

    {(𝔲_1,𝔲_2)∈𝔘×𝔘: R(𝔲_2)-R(η^-(𝔲_1))/𝔲_2-η^-(𝔲_1)=0}.

(The division in Equation (<ref>) makes sense since the numerator and the denominator can be viewed as points of .)
The first and second coordinate projection maps π_1^i and π_2^j from ℭ_i,j onto _i and _j define an anti-holomorphic correspondence (i.e., a multi-valued map with anti-holomorphic local branches) from _i onto _j (cf. <cit.>):

    ℭ_i,jdl[swap]π_1^idrπ_2^j
    _i       _j.

Combining all these anti-holomorphic correspondences for various i,j∈ℐ, we obtain an anti-holomorphic correspondence on 𝔘 defined by the (in general reducible) curve ∑_i,jℭ_i,j. We denote this correspondence by ℭ^⊛.

The following observations show that ℭ^⊛ is obtained by lifting S and its appropriate backward branches via the rational maps R_i. 


∙ Fix z∈_i. Then, S(R_i(z))=R_i(η^-(z)), and hence,

    ((z,,i),(w,j))∈ℭ^⊛ R_j(w)=R_i(η^-(z))=S(R_i(z)), (w,j)≠η^-(z,i).



∙ Now fix z∈_i∖_i. Then, S(R_i(η^-(z)))=R_i(z); i.e.,

    ((z,,i),(w,j))∈ℭ^⊛ R_j(w)=R_i(η^-(z))=S^-1(R_i(z)), (w,j)≠η^-(z,i),

where S^-1 is a suitable backward branch of S. 


To capture our desired mating phenomenon, we need to pass to a correspondence induced by ℭ^⊛ on a quotient of 𝔘. To this end, we endow 𝔘 with the following finite equivalence relation:


For z∈𝔖_i⊂_i and w∈𝔖_j⊂_j,


(z,i)∼_w (w,j) R_i(z)=R_j(w).




The fact that  is the quotient of  by a finite lamination and that R_i|_∂_i is injective (for all i∈ℐ) imply that 

    𝔚 := 𝔘∼_w

has a trivial fundamental group (it can be thought of as a compact, simply connected, noded Riemann surface). 
By definition, the map R descends to a map from 𝔚⟶. This induced map, which we denote by R, is also a branched covering. Abusing notation, we denote the image of a set X⊂𝔘 (respectively, a point p∈𝔘) under the quotient map 𝔘⟶𝔚 by X (respectively, p). We further set 

    𝔇:=⋃_i∈ℐ_i,  𝔖:=⋃_i∈ℐ𝔖_i.
 



The homeomorphism η^-:𝔘→𝔘 descends to a homeomorphism η:𝔚⟶𝔚.


Let us suppose that (z,i)∼_w (w,j); i.e., z∈𝔖_i, w∈𝔖_j, and R_i(z)=R_j(w). We need to show that η^-(z,i)∼_wη^-(w,j). However, this is obvious since η^-(z,i)=(η^-(z),i)=(z,i)∈𝔖_i and η^-(w,j)=(η^-(w),j)=(w,j) ∈ 𝔖_j. 
 

Thus, the correspondence ℭ^⊛ on 𝔘 also descends to a correspondence on 𝔚. We denote this correspondence by ℭ.



 §.§ Dynamical partition for ℭ



The invariant partition of the dynamical plane of S, given by K(S) and T^∞(S), can be pulled back by R to produce an invariant partition of the dynamical plane of the correspondence ℭ.
We define

    𝒦:=R^-1(K(S)),  𝒯:= R^-1(T^∞(S)).




  * η(𝒯)=𝒯, and η(𝒦)=𝒦.


  * Let (𝔲_1,𝔲_2)∈ℭ. Then 𝔲_1∈𝒯 (respectively, 𝔲_1∈𝒦) if and only if 𝔲_2∈𝒯 (respectively, 𝔲_2∈𝒦).



1) It suffices to show that η(𝒦)=𝒦. 

Let us first assume that (z,i)∈_i∩𝒦. Then, R_i(z)∈ K(S) and R_i(η^-(z))=S(R_i(z)). As K(S) is invariant under S, it follows that R_i(η^-(z))∈ K(S). We conclude that η(z,i)=(η^-(z),i)∈𝒦.

Next let (z,i)∈𝒦∖_i. Then, R_i(z)∈ K(S) and S(R_i(η^-(z)))=R_i(z). As K(S) is also backward invariant under S, it follows that R_i(η^-(z))∈ K(S), and hence η(z,i)=(η^-(z),i)∈𝒦.

2) Let (𝔲_1,𝔲_2)∈ℭ, where 𝔲_1=(z,i) and 𝔲_2=(w,j). It suffices to show that if (z,i)∈𝒯 (respectively, if (z,i)∈𝒦), then (w,j)∈𝒯 (respectively, (w,j)∈𝒦). 

To this end, first suppose that z∈_i, which implies that R_j(w)=R_i(η^-(z))=S(R_i(z)). Now let (z,i)∈𝒯 (respectively, (z,i)∈𝒦). The S-invariance of T^∞(S) (respectively, of K(S)) implies that R(w,j)=R_j(w)∈ T^∞(S) (respectively, R(w,j)=R_j(w)∈ K(S)). Hence, (w,j)∈𝒯 (respectively, (w,j)∈𝒦). 

Next suppose that z∈_i∖_i, which implies that S(R_j(w))=R_i(z). Now let (z,i)∈𝒯 (respectively, (z,i)∈𝒦). The backward invariance of T^∞(S) (respectively, of K(S)) under S implies that R(w,j)=R_j(w)∈ T^∞(S) (respectively, R(w,j)=R_j(w)∈ K(S)). Hence, (w,j)∈𝒯 (respectively, (w,j)∈𝒦). 





 §.§ Polynomial structure of ℭ


Note that η maps 𝔇 onto 𝔚∖𝔇, and preserves 𝔖. 
We set 𝒦_1:=𝒦∩𝔇 and 𝒦_2:=𝒦∖𝔇.





We have that 𝒦_2=η(𝒦_1), and 𝒦_1∩𝒦_2=𝔖.


The first statement follows from the fact that η(𝒯)=𝒯. 

For the second statement, first observe that

    𝒦_1∩𝒦_2={𝔲∈∂𝔇: R(𝔲)∈ K(S)}.

The result is now a consequence of the fact that R(𝔖)=𝔖⊂ K(S) and R(∂𝔇∖ 𝔖)=∂∖𝔖⊂ T^∞(S).






  * 𝒦_2 is forward invariant, and hence, 𝒦_1 is backward invariant under ℭ.


  * ℭ has a forward branch carrying 𝒦_1 onto itself with degree d, and this branch is conformally conjugate to S:K(S)→ K(S). 


  * ℭ has a backward branch carrying 𝒦_2 onto itself with degree d, and this branch is topologically conjugate to S:K(S)→ K(S).



1) This follows immediately from the definition of ℭ and Lemma <ref>. 
 
2) The forward branch of ℭ carrying 𝒦_1 onto itself (with degree d) acts as 

    𝔟: (z,i)↦(R|_𝔇)^-1(R(η(z,i)))= ((R_j|__j)^-1(R_i(η^-(z))),j),

where (z,i)∈𝒦_1, and R_i(η^-(z))∈Ω_j.
It is easy to see from the construction that R:𝒦_1⟶ K(S) is a homeomorphism. We claim that R|_𝒦_1 is the desired conjugating map. To this end, note that 

    S(R(z,i))=S(R_i(z))=R_i(η^-(z)),

and hence

    R(𝔟(z,i))=R((R|_𝔇)^-1(R(η(z,i))))=R_i(η^-(z))=S(R(z,i)).


It follows that R|_𝒦_1∘𝔟=S∘R|_𝒦_1.

3) This follows from the previous part and the fact that η conjugates a forward branch of the correspondence on 𝒦_1 to a backward branch of the correspondence on 𝒦_2.






 §.§ Group structure of ℭ


The fact that S: T^∞(S)∩⟶ T^∞(S) is conformally conjugate to the Nielsen map _d implies that S has no critical value in T^∞(S). Hence, in light of the description of S in terms of η^- and R_i, we conclude that no R_i has a critical value in 𝒯.
Thus, R:𝒯→ T^∞(S) is a covering map. Simple connectedness of T^∞(S) now implies that 𝒯 is the union of d+1 many disjoint open topological disks U_0,⋯,U_d.

We define a conformal automorphism

    τ:𝒯→𝒯
 
satisfying the conditions


  * τ(U_j)=U_j+1, j∈/(d+1),


  * R∘τ=R, and


  * τ^d+1=id,


  * R^-1(R(z))={z,τ(z),⋯,τ^d(z)} ∀ z∈𝒯.

By construction, the forward branches of ℭ on 𝒯 are given by the anti-conformal automorphisms τ∘η,⋯,τ^d∘η. 

Recall the notation T^0≡ T^0(S)=∖(∪𝔖). We call T^0 the rank zero tile of T^∞(S), and connected components of S^-n(T^0) tiles of rank n. This yields a natural tessellation of the lifted tiling set 𝒯.  A connected component of the pre-image of a rank n tile of T^∞(S) under R is called a rank n tile of 𝒯.




  * The grand orbits of the correspondence ℭ on 𝒯 are equal to the orbits of ⟨η⟩∗⟨τ⟩. 

  * The group ⟨η⟩∗⟨τ⟩ acts properly discontinuously on , and /ℭ is biholomorphic to /G_d (where G_d is the regular ideal (d+1)-gon reflection group).



1) We have already observed that the forward branches of ℭ on 𝒯 are given by τ∘η,⋯,τ^d∘η. Furthermore, τ=(τ^2∘η)∘(τ∘η)^-1, and hence ⟨τ∘η,⋯,τ^d∘η⟩=⟨η,τ⟩. It now remains to justify that ⟨η,τ⟩ is the free product of the cyclic groups ⟨η⟩ and ⟨τ⟩.

To this end, first note that any relation in ⟨η,τ⟩ other than η^2=id and τ^d+1=id can be reduced to one of the form 

    (τ^k_1∘η)∘⋯∘(τ^k_r∘η)=id
 
or 

    (τ^k_1∘η)∘⋯∘(τ^k_r∘η)=η,

where k_1,⋯,k_r∈{1,⋯,n}.


Case 1: Let us first assume that there exists a relation of the form (<ref>) in ⟨η,τ⟩. It is readily seen using Relation (<ref>) that (τ^k_p∘η) maps the interior of a tile of rank s in 𝒯∖𝔇 to the interior of a tile of rank (s+1) in 𝒯∖𝔇. Hence, the group element on the left of Relation (<ref>) maps the tile of rank 0 to a tile of rank r. Clearly, such an element cannot be the identity map.


Case 2: Now we consider a relation of the form (<ref>) in ⟨τ,η⟩. Each (τ^k_p∘η) maps 𝒯∖𝔇 to itself. Hence, the group element on the left of Relation (<ref>) maps 𝒯∖𝔇 to itself, while η maps 𝒯∖𝔇 to 𝒯∩𝔇. This shows that there cannot exist a relation of the form (<ref>) in ⟨τ,η⟩.

We conclude that η^2=id and τ^d+1=id are the only relations in ⟨η,τ⟩, and hence ⟨η,τ⟩=⟨η⟩∗⟨τ⟩≅/2∗/(d+1). 


2) Recall that ⟨τ⟩⩽⟨η⟩∗⟨τ⟩ acts transitively on the components U_0,⋯,U_d of . Hence it suffices to study the action of the stabilizer subgroup

    Stab(U_0):={γ∈⟨η⟩∗⟨τ⟩: γ(U_0)=U_0}

of U_0 in ⟨η⟩∗⟨τ⟩. It is easily seen that Stab(U_0) is generated by {τ^j∘η∘τ^-j, j∈{0,⋯,d}}. 

We denote the rank zero tile R^-1(T^0)∩ U_0 of  in the component U_0 by T^0_0. As R:U_0→ T^∞(S) is a conformal isomorphism, it follows that T^0_0 is a regular ideal (d+1)-gon in the conformal disk U_0. Further, each of the above generators of Stab(U_0) is an anti-conformal involution of U_0 which fixes a side of T^0_0 pointwise. This implies that the Stab(U_0)-action on U_0 is conformally conjugate to the G_d-action on . The result now follows.


We are now ready to prove a part of Theorem <ref>.


Let f be a degree d geometrically finite or finitely renormalizable, periodically repelling anti-polynomial with connected Julia set. By Theorem <ref>, there exists a polygonal Schwarz reflection S:→ that is a conformal mating of f with the Nielsen map _d. 

We claim that the desired correspondence ℭ on 𝔚 is the one constructed in Section <ref>. To this end, note that the required ℭ-invariant partition of 𝔚 is given in Proposition <ref>. Finally, the desired dynamical properties of ℭ on  follow from Proposition <ref>, and those on  follow from Lemma <ref>, Proposition <ref>, and the fact that S|_K(S) is topologically conjugate (conformally on the interior) to f|_(f).






§ ANTI-FAREY SCHWARZ REFLECTIONS


We now look at degenerate anti-polynomial-like maps having the anti-Farey map _d (see Section <ref>) as their external map. Since _d:∖ℋ→ fixes ∂ℋ pointwise and since ∂ℋ is a Jordan curve touching the unit circle at exactly one point, it follows that any such degenerate anti-polynomial-like map is the restriction of a Schwarz reflection map S:→ of a simply connected quadrature domain . Moreover, the tiling set dynamics of this Schwarz reflection map is conformally conjugate to _d. 

Following <cit.>, we will denote by 𝒮__d the space of Schwarz reflection maps S:→ such that  is a Jordan quadrature domain and the tiling set dynamics of S is conformally conjugate to _d. We denote by 𝔘_d+1 the space of degree d+1 polynomials p such that p|_ is injective and p has a unique (non-degenerate) critical point on 𝕊^1.

<cit.>
Let R be a rational map of degree d+1 that is injective on , :=R(), and S the Schwarz reflection map associated with . Then the following are equivalent.

  * There exists a conformal conjugacy ψ between 

    _d:∖ℋ⟶  and   S:T^∞(S)∖T^0(S)⟶ T^∞(S).

In particular, T^∞(S) is simply connected.


  * After possibly conjugating S by a Möbius map and pre-composing R with an element of Aut(), the uniformizing map R can be chosen to be a polynomial with a unique critical point on 𝕊^1; i.e., R can be chosen to be a member of 𝔘_d+1. Moreover, K(S) is connected. 


  *  is a Jordan domain with a unique conformal cusp on its boundary. Moreover, S has a unique critical point in its tiling set T^∞(S), and this critical point maps to T^0(S) with local degree d+1.




That fact that rational uniformizing map R of the quadrature domain of a Schwarz reflection S∈𝒮__d can be chosen to be a polynomial follows from the existence of a multiplicity d critical point of S in its tiling set (recall that _d has such a critical point). 


<cit.>

The moduli space [𝒮__d]:= 𝒮__dPSL_2() is compact.


We normalize S∈𝒮__d so that the conformal map ψ:→ T^∞(S) that conjugates _d to S sends 0 to ∞, and has asymptotics z↦ 1/z+O(z) as z→ 0.

We say two maps S_1, S_2∈𝒮__d are combinatorially equivalent, denoted by S_1∼ S_2, if they have the same rational lamination (cf. Definition <ref>). As in the case for the space 𝒮__d, we denote by 𝒮__d, r⊂𝒮__d the subset of periodically repelling Schwarz reflections, and by 𝒮__d, r := 𝒮__d, r/∼ the space of combinatorial classes.
Let 𝒮__d, fr⊂𝒮__d (respectively, 𝒮__d, gf⊂𝒮__d) be the subset of at most finitely renormalizable periodically repelling Schwarz reflections (respectively, geometrically finite Schwarz reflections).



The proofs of Theorems <ref>, <ref>, and <ref> work mutatis mutandis for the space 𝒮__d of Schwarz reflections and yield the following result.




  * There is a dynamically natural homeomorphism 

    Φ: 𝒞^-_d, r⟶𝒮__d, r,

where `dynamically natural' means that the circle homeomorphism conjugating _d to z^d carries the rational lamination of an anti-polynomial to the rational lamination of a Schwarz reflection.


  * There is a dynamically natural homeomorphism 

    Φ: 𝒞^-_d, fr⟶𝒮__d, fr.

Here, dynamically natural means that for any f∈𝒞^-_d, fr, Φ(f) is the unique conformal mating of f with the anti-Farey map _d.


  * There is a dynamically natural bijection 

    Φ: 𝒞^-_d, gf⟶𝒮__d, gf.
 
Here, dynamically natural has the same meaning as in item (2) of this theorem.





We believe that the bijection Φ: 𝒞^-_d, gf⟶𝒮__d, gf is not continuous in general. Although the proof of discontinuity given in Subsection <ref> does not apply to the family 𝒮__d (essentially because all Schwarz reflections in 𝒮__d are defined by a single quadrature domain), one possible way of proving discontinuity in the current setting is to use the arguments of <cit.>.



We end this section with the proof of Theorem <ref> in the case of the anti-Hecke group.


Let f a degree d anti-polynomial as in the statement of the theorem. By Theorem <ref>, there exists a Schwarz reflection S=Φ(f):→ in 𝒮__d such that the non-escaping set dynamics of S is topologically conjugate (conformally on the interior) to f|_(f) and the tiling set dynamics of S is conformally conjugate to _d.

By Propositions <ref> and <ref>, after possibly Möbius conjugating S (which amounts to replacing  by a Möbius image of it), we can assume that p()=, for some polynomial p∈𝔘_d+1. We note that S≡ p∘η^-∘ (p|_)^-1.

We now define the antiholomorphic correspondence ℭ on  by the relation

    (z,w)∈ℭp(w)-p(η^-(z))/w-η^-(z)=0,
 
and set :=p^-1(K(S)), :=p^-1(T^∞(S)). By <cit.>, the sets  and  are completely invariant under the action of the correspondence. 

Note that by construction of S, the tiling set T^∞(S) is simply connected, and it contains exactly one critical point of S. More precisely, this critical point lies at p(0), and it maps to ∞ with local degree d+1 under S. Thus, by <cit.>, the dynamics of ℭ on  is conformally orbit-equivalent to the action of the anti-Hecke group G_d on .

Next, we set _1:=∩ and _2:=∖. Since p carries 𝕊^1∖{1} onto ∂∖{s}⊂ T^∞(S) (where s is the unique cusp of ∂), it follows that 𝕊^1∖{1} is disjoint from . As s∈ K(S), it follows that (p|_𝕊^1)^-1(s) is the unique point of intersection of _1 and _2. 

By <cit.>, ℭ has a forward branch carrying _1 onto itself with degree d, and this branch is topologically conjugate to S|_K(S) such that the conjugacy is conformal on the interior. The result now follows from the fact that S|_K(S) is topologically conjugate (conformally on the interior) to f|_(f).











PART: 
Holomorphic world




§ FACTOR BOWEN-SERIES MAPS FOR GENUS ZERO ORBIFOLDS


We will now describe a collection of circle coverings that were introduced in <cit.>.
Let us first define a class of orbifolds that give rise to these external maps:

 𝔉:= hyperbolic orbifolds Σ of genus zero with
	
	
  * at least one puncture,
	
  * at most one order two orbifold point,
	
  * at most one order ν≥ 3 orbifold point.
	
We set

    n=
    ν  if Σ∈𝔉  has an order ν≥ 3  orbifold point,
    
    1   otherwise.

Each Σ∈𝔉 has an n-fold cyclic cover Σ. Specifically, if Σ has δ_1≥1 punctures and δ_2∈{0,1} order two orbifold points, then Σ is a genus zero orbifold with n(δ_1-1)+1 punctures and nδ_2 order two orbifold points. Topologically, Σ is obtained by skewering the surface Σ along an infinite geodesic (in Σ) connecting the order ν orbifold point and a cusp, and gluing n copies of it cyclically.

The surface Σ is uniformized by a Fuchsian group Γ which admits a (closed) ideal m-gon Π as a fundamental domain. This fundamental domain Π can be chosen so that it has ideal vertices at 1 and exp(2π i/n) which are identified, and such that Π is symmetric under rotation by 2π/n around the origin. Here, 

    m=
    
    2n(δ_1-1)    when  δ_2=0,
     
    n(2δ_1-1)    when  δ_2=1.







The Bowen-Series map A^BS_Σ≡ A^BS_Γ:∖Π→ of Γ equipped with the fundamental domain Π has jump discontinuities at the n-th roots of unity. However, the 2π/n rotation symmetry of Π and of the associated side-pairing transformations imply that A^BS_Σ commutes with rotation by 2π/n. This allows one to pass to a factor of the above Bowen-Series map on the quotient cone /⟨ M_ω⟩ (where M_ω(z)=ω z, and ω=e^2π i/n). The resulting map is denoted by 

    A^BS_Σ: (∖Π)/⟨ M_ω⟩→/⟨ M_ω⟩.
 
We uniformize the cone /⟨ M_ω⟩ by the closed disk  and call this uniformizing map ξ:/⟨ M_ω⟩→ (note that ξ is induced by z↦ z^n). Finally, we define the factor Bowen-Series map associated with Σ as

    A^f_Σ:=ξ∘A^BS_Σ∘ξ^-1: ∖ℋ→,
 
where ℋ:=ξ(Π/⟨ M_ω⟩). The set ℋ has p:=m/n ideal boundary points on 𝕊^1, and the map A_Σ^f restricts to a self-homeomorphism of order two on ∂ℋ.


1) In the anti-holomorphic world, a Nielsen map or an anti-Farey map has a unique domain of definition which is bounded by the unit circle and curves comprising fixed points of the map. However, this is not the case for factor Bowen-Series maps since there is no unique collection of curves in the disk on which a factor Bowen-series map acts as an involution. As a convention, we will fix ∖ℋ as the canonical domain of definition of the factor Bowen-Series map A_Σ^f.

2) The superscript `f' in A^f_Σ stands for `factor'.



The factor Bowen-Series map A^f_Σ turns out to be an expansive covering map of 𝕊^1 of degree  

    d≡ d(Σ)= m-1 = 
    2n(δ_1-1)-1   when  δ_2=0,
    
    n(2δ_1-1)-1   when  δ_2=1.
 
Moreover, when n≥ 3, the map A_Σ^f has p=m/n critical points, each of multiplicity n-1.
All these critical points are mapped to the same critical value. 


Two special classes of examples of factor Bowen-Series maps are given by the cases 

    n=1, δ_1≥ 2, δ_2∈{0,1},
 and 

    n≥ 3, δ_1=1, δ_2=1.
 
In the former case, Σ is a punctured sphere with possibly an order two orbifold point, and Σ=Σ. In particular, the factor Bowen-Series map of Σ is simply the Bowen-Series map of Σ (see Figure <ref>). In the latter case, Σ is a Hecke surface, and Σ is a sphere with one puncture and ν≥ 3 order two orbifold points (see Figure <ref>).



 §.§ Conformal mating of factor Bowen-Series maps with polynomials


We continue to use the notation used above.
Let f be a monic, centered complex polynomial of degree d with a connected and locally connected Julia set. We denote the Böttcher coordinate of f by

    ψ_f:∖→ℬ_∞(f):=∖𝒦(f).
 
This map, which conjugates z^d to f, is normalized to be tangent to the identity map near infinity.
As 𝒥(f) is locally connected, ψ_f extends continuously to 𝕊^1 to yield a semi-conjugacy between z^d|_𝕊^1 and f|_𝒥(f). 
By <cit.>, there exists a homeomorphism 𝔥:𝕊^1→𝕊^1 that conjugates z^d to A_Σ^f. We normalize 𝔥 so that it sends the fixed point 1 of z^d to the fixed point 1 of A_Σ^f.

We define an equivalence relation ∼ on 𝒦(f) generated by 

    ψ_f(ζ)∼𝔥(ζ), for all z∈𝕊^1.
 




The maps f and A_Σ^f are said to be conformally mateable if there exist a continuous map S: Dom(S)⊊→ (called a conformal mating of A_Σ^f and f) that is complex-analytic in the interior of Dom(S) and continuous maps 
	
    𝔛_f:𝒦(f)→ and 𝔛_Σ: →,

	conformal on 𝒦(f) and  (respectively), satisfying
	
		
  *  𝔛_f(𝒦(f))∪𝔛_Σ() =,
		
		
  *  Dom(S)= 𝔛_f(𝒦(f))∪𝔛_Σ(∖ℋ),
		
		
  * 𝔛_f∘ f(z) = S∘𝔛_f(z),  for z∈𝒦(f),
		
		
  * 𝔛_Σ∘ A_Σ^f(w) = F∘𝔛_Σ(w),  for w∈∖ℋ,  and 
		
		
		
  *  𝔛_f(z)=𝔛_Σ(w) if and only if z∼ w where ∼ is the equivalence relation on 𝒦(f)⊔ defined by Relation (<ref>).
	



<cit.>
Let f be a hyperbolic complex polynomial of degree d with connected Julia set.
Then, there exists a conformal mating S of f:𝒦(f)→𝒦(f) and A_Σ^f:∖ℋ→. Moreover, S is a unique up to Möbius conjugacy.








§ B-INVOLUTIONS


We now introduce a holomorphic analogue of Schwarz reflections.




Let {Ω_1,⋯,Ω_k} be a disjoint collection of proper simply connected sub-domains of  such that Ω_j=Ω_j, j∈{1,⋯, k}, and let :=_j=1^kΩ_j. Further, let 𝔖⊂∂ be a (possibly empty) finite set such that ∂^0:=∂∖𝔖 is a finite union of disjoint non-singular real-analytic curves.

The set  is called an inversive multi-domain if it admits a continuous map S:→ satisfying the properties:


  *  S is meromorphic on ,

  *  S(∂Ω_j)=∂Ω_j', for some j'∈{1,⋯,k}, and

  *  S:∂→∂ is an orientation-reversing involution preserving 𝔖; i.e., S(𝔖)=𝔖.

The map S is called a B-involution of the inversive multi-domain .

When k=1, the domain  is called an inversive domain.


For a B-involution S:→, we set T:=∖, and T^0:= T∖𝔖.


1) If each Ω_j is a Jordan domain, then Condition <ref> of Definiion <ref> follows from Condition <ref>.

2) The nomenclature `B-involution' is chosen to emphasize the property that such a map induces an involution on the boundary of an inversive multi-domain .





The main result of this section is an explicit algebraic description of B-involutions, which can be thought of as a holomorphic counterpart of Proposition <ref>. We first prove this characterization for an inversive domain.
Recall that η^+(z)=1/z.


Let  be an inversive domain with associated B-involution S:→. 
Then, there exists a rational map R of degree d+1 and a Jordan disk 𝔇 such that the following hold.

  * 1, -1∈∂𝔇.

  * η^+:𝔇→∖𝔇 is a homeomorphism.

  * ∂𝔇 is a piecewise non-singular real-analytic curve.

  * R:𝔇→ is a conformal isomorphism.

  * S|_:= R∘η^+∘ (R|_𝔇)^-1.

  * S:S^-1()→ is a proper map of degree d.

  * S:S^-1(T)→T is a proper map of degree d+1.




Step 1: Extending S to a conformal involution around ∂^0.
By assumption, S:∂→∂ is an order two homeomorphism preserving 𝔖. Hence, it carries each component of ∂^0 (which is a non-singular real-analytic curve) onto some component of ∂^0. Depending on the parity of the number of components of ∂^0, no component or exactly one component of ∂^0 is mapped onto itself by S. Since S is meromorphic on  and continuous up to ∂, it now follows by the Schwarz Reflection Principle that S extends to a holomorphic map (which we call S) in a neighborhood of ∂^0. Since S carries ∂^0 onto itself, the map S^∘ 2 is also well-defined in a neighborhood of ∂^0. Moreover, as S^∘ 2=id on ∂^0, the Identity Principle for holomorphic maps implies that S is a conformal involution in a neighborhood of ∂^0.











Step 2: Constructing a welded Riemann surface Σ with S as the welding map.
We now weld the bordered Riemann surface ∪∂^0 with itself using the conformal involution S:∂^0→∂^0 as the boundary identification map. The resulting (non-compact) Riemann surface is denoted by Σ. Note that Σ is a genus zero surface with finitely many boundary components.

By definition of Σ, there exist homeomorphisms ψ_1,ψ_2:∪∂^0→Σ such that


  * ψ_i, i∈{1,2}, is conformal on ,

  * ψ_1()∩ψ_2()=∅,

  * ψ_1(∂^0)=ψ_2(∂^0) is a union of non-singular real-analytic curves (which we denote by 𝔍),

  * ψ_1(w)=ψ_2(S(w)), for w∈∂^0, and

  * Σ=ψ_1(∪∂^0)∪ψ_2().



Step 3: Defining a conformal involution M on Σ.
Let us define a map 

    M:Σ→Σ,  
    M=
    ψ_2∘ψ_1^-1  on  ψ_1(∪∂^0),
    ψ_1∘ψ_2^-1  on  ψ_2().
 
The relations ψ_1=ψ_2∘ S (on ∂^0) and S^∘ 2=id (on ∂^0) imply that the piecewise definitions of M match continuously along 𝔍. Moreover, M is easily seen to be holomorphic, and hence a conformal involution on Σ. We also note that ψ_1 conjugates the involution S|_∂^0 to M|_𝔍. 


Step 4: A finite degree branched cover R and the type of Σ. We define a holomorphic map R:Σ∖ψ_2(S^-1(𝔖))⟶∖𝔖 as follows:

    R=
    ψ_1^-1  on  ψ_1(∪∂^0),
    
    S∘ψ_1^-1∘ M = S∘ψ_2^-1  on  ψ_2(∖ S^-1(𝔖)).
 
The assumption that S extends continuously to an involution of ∂ fixing 𝔖 implies that R is a proper holomorphic map. Hence, R is a finite degree branched covering map. Suppose that the degree of R is d+1, for some non-negative integer d.

As the punctured sphere ∖𝔖 is a finite type Riemann surface, it follows that Σ∖ψ_2(S^-1(𝔖)) must also be a finite type Riemann surface. Hence, Σ∖ψ_2(S^-1(𝔖)) is also a punctured sphere. This means that S^-1(𝔖) is a finite set and the boundary components of Σ arising from the welding procedure are points (i.e., not holes). 


Step 5: Extending R to a rational map and M to a Möbius involution. Since R is of finite degree, it extends holomorphically to the punctures of Σ∖ψ_2(S^-1(𝔖)). This produces a holomorphic map R of the Riemann sphere, and hence R is a rational map of degree d+1.

The fact that Σ is a punctured sphere also implies that the conformal involution M extends to the entire Riemann sphere, and yields a Möbius involution. Note that M preserves the set of punctures of Σ.
After possibly a change of coordinates, we can assume that M=η^+.


Step 6: Putting everything together.
Since Σ is a punctured sphere, the curve 𝔍 can also extended to a Jordan curve 𝔍⊂ by throwing in the punctures. Moreover, the assumption that S:∂→∂ is orientation-reversing implies that η^+ is an orientation-reversing self-homeomorphism of 𝔍. Hence, η^+ must have exactly two fixed points (namely, ± 1) on 𝔍.

We set 𝔇:=ψ_1() and note that 𝔇 is one of components of ∖𝔍. Finally, the definition of R and the above discussion imply that 

    R|_∖𝔇=S|_∘ R|_𝔇∘η^+|_∖𝔇  S|_= R∘η^+∘ (R|_𝔇)^-1.

Finally, the degrees of the branched coverings S:S^-1()→ and S:S^-1(T)→T follow directly from Expression (<ref>).




The next result gives a similar description for inversive multi-domains with two components such that the boundary components are swapped by the B-involution.


Let =Ω_0⊔Ω_1 be an inversive multi-domain with associated B-involution S:→. 
Assume further that S(∂Ω_0)=∂Ω_1.
Then, there exist rational maps R_0, R_1 and Jordan disks 𝔇_0, 𝔇_1 such that the following hold for j∈{0,1}.

  * η^+:𝔇_j→∖𝔇_1-j is a homeomorphism.

  * ∂𝔇_j is a piecewise non-singular real-analytic curve.

  * R_j:𝔇_j→Ω_j is a conformal isomorphism.

  * S|_Ω_j:= R_1-j∘η^+∘ (R_j|_𝔇_j)^-1.

  * S:S^-1(Ω_1-j)∩Ω_j⟶Ω_1-j is a branched covering of degree (R_1-j)-1.

  * S:S^-1(Ω_1-j^c)∩Ω_j⟶Ω_1-j^c is a branched covering of degree (R_1-j).



The proof essentially follows the ideas used in the proof of Lemma <ref>.

As in the case of inversive domains, the map S extends to a conformal involution in a neighborhood of ∂^0. 
We set 𝔖_j:=𝔖∩∂Ω_j and ∂^0Ω_j:=∂Ω_j∖𝔖_j=∂^0∩∂Ω_j, j∈{0,1}. The map S:∂Ω_j→∂Ω_1-j carries 𝔖_j (respectively, ∂^0Ω_j) onto 𝔖_1-j (respectively, onto ∂Ω^0_1-j). We now weld the bordered Riemann surfaces Ω_0∪∂^0Ω_0 and Ω_1∪∂^0Ω_1 using the real-analytic homeomorphism S:∂^0Ω_0→∂^0Ω_1, and call the resulting Riemann surface Σ. 

By construction of Σ, there exist homeomorphisms ψ_0:Ω_0∪∂^0Ω_0→Σ and ψ_1:Ω_1∪∂^0Ω_1→Σ such that


  * ψ_j is conformal on Ω_j, j∈{0,1}, 

  * ψ_0(Ω_0)∩ψ_1(Ω_1)=∅,

  * 𝔍:=ψ_0(∂^0Ω_0)=ψ_1(∂^0Ω_1) is a union of non-singular real-analytic curves,

  *  ψ_0=ψ_1∘ S on ∂^0Ω_0, and

  * Σ=ψ_0(Ω_0)⊔𝔍⊔ψ_1(Ω_1).

We now define a holomorphic map 

    R_0:Σ∖ψ_1((S|_Ω_1)^-1(𝔖_0))⟶∖𝔖_0


    R_0=
    ψ_0^-1  on  ψ_0(Ω_0∪∂^0Ω_0),
    
    S|_Ω_1∘ψ_1^-1  on  ψ_1(Ω_1 ∖(S|_Ω_1)^-1(𝔖_0)).
 
Condition (<ref>) and the relation S|_∂Ω_1=(S|_∂Ω_0)^-1 together guarantee that the piecewise definitions of R_0 match continuously along the piecewise real-analytic curve 𝔍.
Moreover, the assumption that the continuous extension of S to ∂ fixes 𝔖 implies that R_0 is a proper holomorphic map, and hence a finite degree branched covering map. As in the proof of Lemma <ref>, this implies that Σ∖ψ_1((S|_Ω_1)^-1(𝔖_0)) is biholomorphic to a punctured sphere. This allows us to extend R_0 to a rational map of the Riemann sphere. Moreover, the closure of 𝔍 in the Riemann sphere (obtained by adding finitely many points to 𝔍) is a Jordan curve.


Let us now set 𝔇_0:=ψ_0(Ω_0) and 𝔇_1:=η^+(∖𝔇_0). We further define a rational map R_1 as

    R_1=
    ψ_1^-1∘η^+   on  𝔇_1,
    
    S|_Ω_0∘ψ_0^-1∘η^+   on  ∖𝔇_1.
 
Once again, Condition (<ref>) shows that the piecewise definitions of R_1 match continuously along the Jordan curve 𝔍 (where the closure is taken in ).

By construction, each R_j:𝔇_j→Ω_j is a conformal isomorphism, j ∈ {0,1}. To justify that R_0, R_1 are the desired rational maps, it only remains to check that S|_Ω_j:= R_1-j∘η^+∘ (R_j|_𝔇_j)^-1, j∈{0,1}. To this end, first note that

    S|_Ω_1∘ψ_1^-1 = R_0    on   ∖𝔇_0
        S|_Ω_1 = R_0∘ψ_1 = R_0∘η^+∘η^+∘ψ_1
        S|_Ω_1 = R_0∘η^+∘(ψ_1^-1∘η^+|_𝔇_1)^-1 = R_0∘η^+∘(R_1|_𝔇_1)^-1.

On the other hand,

    S|_Ω_0∘ψ_0^-1∘η^+ = R_1    on   ∖𝔇_1
        S|_Ω_0 = R_1∘η^+∘ψ_0
        S|_Ω_0 = R_1∘η^+∘(ψ_0^-1|_𝔇_0)^-1 = R_1∘η^+∘(R_0|_𝔇_0)^-1.

Finally, the statements about the degrees of S follow from the relations S|_Ω_j≡ R_1-j∘η^+∘(R_j|_𝔇_j)^-1, j∈{0,1}.
The proof is now complete.






We are now ready to establish the promised algebraic description of a general B-involution.


Let S:→ be a B-involution of the inversive multi-domain =_j=1^kΩ_j.
Then, there exist Jordan domains 𝔇_j and rational maps R_j, j∈{1,⋯,k}, such that the following hold.


  * η^+:𝔇_j→∖𝔇_κ(j) is a homeomorphism.

  * ∂𝔇_j is a piecewise non-singular real-analytic curve.

  * R_j:𝔇_j→Ω_j is a conformal isomorphism.

  * S|_Ω_j≡ R_κ(j)∘η^+∘(R_j|_𝔇_j)^-1.

Here, κ is a permutation of the set {1,⋯,k} of order two.


The assumption that S:∂→∂ is an involution which maps each ∂Ω_j homeomorphically onto some ∂Ω_j' implies that S induces an order two permutation on the set {1,⋯,k}. We denote this permutation by κ.
Since κ is of order two, it can be factored as a product of disjoint 1-cycles and 2-cycles; i.e., S either preserves ∂Ω_j, or swaps ∂Ω_j and ∂Ω_κ(j).

If κ(j)=j, then Ω_j is an inversive domain with associated B-involution S|_Ω_j. If κ(j)≠ j, then Ω_j⊔Ω_κ(j) is an inversive multi-domain with associated B-involution given by the restriction of S. The existence of the desired Jordan domains 𝔇_j and rational maps R_j now follow from Lemmas <ref> and <ref>.




§ B-INVOLUTIONS WITH FACTOR BOWEN-SERIES EXTERNAL MAPS


Throughout this section, we will work with a fixed Σ∈𝔉. Let m,n∈ be as in Section <ref>.


We define the space ℬ_Σ to be the collection of all degenerate polynomial-like maps with connected non-escaping set that admit the factor Bowen-Series map A_Σ^f as their external map.




We equip every degenerate polynomial-like map (with connected filled Julia set) with an invariant external access to its filled Julia set (or equivalently, by a fixed point on the ideal boundary of its escaping set). Two marked degenerate polynomial-like maps are identified if they are conjugate via a Möbius map that preserves the marking.      



By definition, for a map (S,P_1,P_2)∈ℬ_Σ, there exists a conformal isomorphism 

    ψ:→ T^∞(S):=∖ K(S)
 
that conjugates A_Σ^f to S, wherever the maps are defined. We set 

    :=∖ψ(ℋ)
 
and remark that  is a pinched polygon (note that the image of ∂ℋ under the conformal isomorphism ψ consists of finitely many non-singular real-analytic curves). The map S extends to a continuous map on  which is meromorphic on . We call  the canonical domain of definition of S (see Remark <ref>). 


We normalize S∈ℬ_Σ so that the conformal map ψ:→ T^∞(S) that conjugates A_Σ^f to S sends 0 to ∞, and has asymptotics z↦ 1/z+O(z) as z→ 0.

We also remark that since S:∂ S^-1()→∂ has degree d, the map S:S^-1()→ can be extended to a degree d branched cover of the sphere with a fixed critical point of multiplicity d-1 at ∞. In fact, this can be done in such a way that there is no other critical point of the branched cover outside K(S). The fact that a degree d branched cover of the sphere has 2d-2 critical points (counted with multiplicities), it follows that S has d-1 critical points (counted with multiplicities) in K(S).



 §.§ Coarse lamination

We now proceed to give an explicit description of the pinched polygon . 
Let us denote the set of p=m/n ideal boundary points of ℋ on 𝕊^1 by S_Σ. These points are periodic of period at most two under A_Σ^f; in fact, at most two points of S_Σ are fixed by A_Σ^f.
Under the circle homeomorphism 𝔥 that conjugates z^d to A_Σ^f, the set S_Σ is pulled back to the set 

    𝒜:={i/p:i∈{0,⋯,p-1}}.

(Here we identify 𝕊^1 with =/.) If p is even, A_Σ^f has two fixed points in S_Σ and these correspond to 0,1/2∈𝒜. For odd p, the map A_Σ^f has a unique fixed point in S_Σ and this fixed point corresponds to 0∈𝒜. On the other hand, the 2-cycles of A_Σ^f in S_Σ correspond to the 2-cycles ±r/p of m_d:/→/, θ↦ dθ.

Following Section <ref>, we can associate a coarse lamination ℒ with S∈ℬ_Σ that records which points of 𝒜 are identified on the boundary of K(S). Then,  is homoemorphic to the pinched disk /ℒ (cf. <cit.>.

As a slight abuse of terminology, we will call the components of ∖ of infinite (respectively, finite) hyperbolic area gaps (respectively, polygons) of .
For a gap 𝒢, the `boundary at infinity' ∂𝒢∩ has length equal to k/p for some integer k≥ 1.
We shall call this number k the degree of the gap, denoted by (𝒢). 
We also denote the number of points of 𝒜 in (∂𝒢∩) by (𝒢), and call it the cusp number of the gap.

The next result is immediate from the definition of coarse laminations.


Let S∈ℬ_Σ with canonical domain of definition .
The components of  are in bijective correspondence with the gaps of ℒ. Two such components share at most one boundary point; moreover, they share a boundary point if and only if the corresponding gaps are cobordant on a leaf or a polygon of .


We enumerate the gaps of as 𝒢_1,⋯,𝒢_k such that 𝒢_j corresponds to Ω_j.

Note that no leaf of ℒ connects the fixed points 0 or 1/2 to the rest of (2-peridoic) points of 𝒜 (cf. <cit.>).
If r/p, s/p∈𝒜 lie in the same -equivalence class, then so do -r/p=m_d(r/p) and -s/p=m_d(s/p). Hence, the gaps of the lamination ℒ either intersect the real line, or come in complex conjugate pairs. It is easily seen that


∙ if 𝒢_j intersects the real line, then S(∂Ω_j)=∂Ω_j; otherwise,


∙ 𝒢_j and 𝒢_j' (for some j'∈{1,⋯,k}) form a complex conjugate pair of gaps, and S(∂Ω_j)=∂Ω_j'.


The next result demonstrates that degenerate polynomial-like maps having factor Bowen-Series maps as their external maps are B-involutions.


Let S∈ℬ_Σ with canonical domain of definition .
The map S:→ is a B-involution of an inversive multi-domain. In particular, the conformal mating S described in Theorem <ref> is a B-involution of an inversive multi-domain. 


We have already observed that  is a pinched polygon with. Let us set 𝔖=ψ(S_Σ), where ψ:→ T^∞(S) is the conformal conjugacy between A_Σ^f to S. The first statement is now an immediate consequence of the facts that a factor Bowen-Series map A_Σ^f:∖ℋ→ restricts to a self-homeomorphism of order two on ∂ℋ and the discussion above (or from the fact that each component of  is a Jordan domain, see Remark <ref>). The second statement is now immediate as the conformal matings of Theorem <ref> lie in the class ℬ_Σ.





 §.§ Degrees and critical points of uniformizing rational maps

Throughout this subsection, we fix an S∈ℬ_Σ with canonical domain of definition .
By Theorem <ref> and Proposition <ref>, there exist Jordan domains 𝔇_j and rational maps R_j, j∈{1,⋯,k}, such that 

  * η^+:𝔇_j→∖𝔇_κ(j) is a homeomorphism,

  * R_j:𝔇_j→Ω_j is a conformal isomorphism, and

  * S|_Ω_j≡ R_κ(j)∘η^+∘(R_j|_𝔇_j)^-1,

where κ is a permutation of the set {1,⋯,k} of order two. We denote the domain of R_j by _j.

Let d_j:=R_j. The relation S|_Ω_j≡ R_κ(j)∘η^+∘(R_j|_𝔇_j)^-1 implies that S:S^-1(Ω_κ(j))∩Ω_j⟶Ω_κ(j) is a branched covering of degree d_κ(j)-1, and
S:S^-1(Ω_κ(j)^c)∩Ω_j⟶Ω_κ(j)^c is a branched covering of degree d_κ(j). As S:S^-1()→ has degree d, we conclude that

    ∑_j=1^k d_j = d+1.



For each j∈{1,⋯,k},

  * (𝒢_j)=(𝒢_κ(j)), and

  * d_j=n·(𝒢_j), and hence d_j=d_κ(j).



1) This follows from the structure of the gaps of  described in Section <ref>.

2) Recall that S:S^-1(Ω_κ(j))∩Ω_j⟶Ω_κ(j) is a branched covering of degree d_j-1.

Note that under the map m_d, each arc of ∖𝒜 is wrapped onto the whole circle (n-1) times and onto the complement of the closure of its complex conjugate arc one more time. Hence, m_d(∂𝒢_j∩) covers ∂𝒢_κ(j)∩ exactly 

    (n-1)·(𝒢_j)+((𝒢_j)-1)=n·(𝒢_j)-1
 
times. It follows that ∂ K(S)∩Ω_j covers ∂ K(S)∩Ω_κ(j) exactly n·(𝒢_j)-1 times under the map S. Since ∂ K(S) is completely invariant under S, we conclude that S:S^-1(Ω_κ(j))∩Ω_j⟶Ω_κ(j) is a degree n·(𝒢_j)-1 branched covering. Therefore, d_j=n·(𝒢_j).



To describe the structure of critical points of the rational maps R_j concisely, we employ the following notation. 
Consider the disjoint union 

    𝔘 := _j=1^k _j ≅ ×{1,⋯,k},

and define the maps

    R: 𝔘⟶,   (z,j)↦ R_j(z),

and

    η^+ : 𝔘⟶𝔘,   (z,j)↦ (η^+(z),κ(j)).

Note that R is a branched covering of degree d+1, and η^+ is a homeomorphism. We also set 

    𝔇:= _j=1^k 𝔇_j⊂𝔘.

Recall the notation 𝔖=ψ(S_Σ). We denote the set of points in 𝔖 that do not disconnect ∂ by 𝔖^cusp. Note that |𝔖^∩∂Ω_j| = (𝒢_j).

The following result was proved in <cit.> in certain special cases. We include the proof for completeness (and since the current setting is more general).




  * R has no critical points in 𝔇.


  * R has d-1=np-2 critical points in R^-1()∖𝔇. On the other hand, for n≥ 3, the map R has p distinct critical points, each of multiplicity n-1, in R^-1(T^∞(S))∖𝔇, and all these critical points are mapped by R to the same point in T^∞(S). 


  * crit(R)∩∂𝔇=(R|_∂𝔇)^-1(𝔖^cusp).



1) This follows from injectivity of R_j on 𝔇_j.

2) Recall that S has d-1=np-2 critical points in K(S), and p critical points, each of multiplicity n-1 (when n≥ 3), in T^∞(S) (coming from the p critical points of A_Σ^f). Moreover, S maps all the p(n-1) critical points in T^∞(S) to the same critical value since A_Σ^f sends all of its p(n-1) critical points to the origin.

The result now follows from the relation S|_Ω_κ(j)≡ R_j∘η^+∘(R_κ(j)|_𝔇_κ(j))^-1. 




3) Since A_Σ^f does not admit an analytic continuation in a neighborhood of any point of S_Σ, the map S does not admit an analytic continuation in a neighborhood of any point of 𝔖. Suppose that x∈𝔖^cusp lies on the boundary of Ω_j.
Then, the relation S|_Ω_j=R_κ(j)∘η^+∘ (R_j|_𝔇_j)^-1 implies that (R_j|_𝔇_j)^-1 does not extend complex-analytically to a neighborhood of x. This forces x to be a critical value of R_j with a corresponding critical point on ∂𝔇_j. Hence, (R|_∂𝔇)^-1(𝔖^cusp)⊂crit(R)∩∂𝔇.

It remains to show that R has no other critical point on ∂𝔇. This will be derived from the following counting argument.

    |crit(R)| =∑_j=1^k (2d_j-2)≥ (np-2)+p(n-1)+∑_j=1^k (𝒢_j)
        2np-2k ≥ 2np-p-2+ ∑_j=1^k (𝒢_j)
          ( recall that  d_j=n·(𝒢_j) and ∑_j=1^k(𝒢_j)=p ) 
        p+2≥ 2k+ ∑_j=1^k (𝒢_j). ()


Claim: The lamination  contains no polygon.
 If the lamination  is empty; i.e., if k=1, then (𝒢_1) consists of p points, and the two sides of Inequality () coincide. Every time an r-gon is introduced in , the number k (i.e., the number of gaps) increases by r-1 and the number ∑_j=1^k (𝒢_j) drops by r. Thus, if r>2, the introduction of an r-gon in  would increase the right side of Inequality () by 2(r-1)-r=r-2>0. But this would violate the inequality, proving that  contains no polygon.


Thanks to the above claim, the leaves of  have disjoint closures in . It is now easily seen that Inequality () is actually an equality, and hence |crit(R)∩∂𝔇|=|𝔖^cusp|. We conclude that (R|_∂𝔇)^-1(𝔖^cusp)= crit(R) ∩ ∂𝔇.



The coarse lamination  associated with S∈ℬ_Σ has no polygon.




 §.§ Coarse partition and topology of ℬ_Σ

As in Section <ref>, the above coarse laminations lead to a coarse partition of ℬ_Σ:

    ℬ_Σ = ⋃_ℒℬ_Σ,ℒ,

where ℬ_Σ,ℒ consists of all maps in ℬ_Σ with coarse lamination ℒ, and the union is taken over all possible coarse laminations. This coarse partition can be used to topologize ℬ_Σ as in Definition <ref>. Further the proof of Theorem <ref> also applies mutatis mutandis to show compactness of ℬ_Σ.




 §.§ Parameter space homeomorphisms


The local dynamics of the B-involution S near the singular points of ∂ is similar to that of a Schwarz reflection near the singular points of the boundary of its domain of definition, thanks to the algebraic description of B-involutions given in Theorem <ref>. This fact allows one to give a classification of Fatou components of S (i.e., connected components of K(S)) and to establish the classical relations between Fatou components/non-repelling cycles and critical points. Moreover, the tessellation of  given by the Fuchsian group Γ (uniformizing the cyclic cover Σ of Σ) and the preferred fundamental domain Π can be used to construct external dynamical rays for S (following Definition <ref>). The proof of Proposiion <ref> can also be applied to current setting to prove landing of pre-periodic dynamical rays, and this gives rise to the notion of the rational lamination of a B-involution S (see Definition <ref>).

One can now define the spaces ℬ_Σ, r, ℬ_Σ, r, ℬ_Σ, fr, and ℬ_Σ, gf as in the case of polygonal (or anti-Farey) Schwarz reflections.

Let d≥ 2 be the degree of A_Σ^f:𝕊^1→𝕊^1.



  * There is a dynamically natural homeomorphism 

    Φ: 𝒞^+_d, r⟶ℬ_Σ, r,

where `dynamically natural' means that the circle homeomorphism conjugating A_Σ^f to z^d carries the rational lamination of a polynomial to the rational lamination of a B-involution.


  * There is a dynamically natural homeomorphism 

    Φ: 𝒞^+_d, fr⟶ℬ_Σ, fr.

Here, dynamically natural means that for any f∈𝒞^+_d, fr, the B-involution Φ(f) is the unique conformal mating of f with the factor Bowen-Series map A_Σ^f.


  * There is a dynamically natural bijection 

    Φ: 𝒞^+_d, gf⟶ℬ_Σ, gf.
 
Here, dynamically natural has the same meaning as in item (2) of this theorem.





As in the antiholomorphic setting, the bijection Φ: 𝒞^+_d, gf⟶ℬ_Σ, gf should be discontinuous at infinitely many parameters; however, we refrain from justifying this statement here (see <cit.>).



The David surgery technique used in Theorem <ref> works mutatis mutandis in the current setting, and yields a dynamically natural bijection between geometrically finite parameters as asserted in (3).
To construct puzzles for B-involutions, we need a slight modification at level 0; namely, we consider all external rays landing at fixed and period 2 cycles.
This is to accommodate the coarse lamination coming from the degenerate polynomial-like structure of the B-involution (see <ref>).
With this modification, the proofs of (1) and (2) are the same as the proofs for Theorem <ref> and Theorem <ref>.






 §.§ Correspondences associated with conformal matings


As in Section <ref>, the map S∈ℬ_Σ can be lifted by the degree d+1 branched cover R:𝔘→ to obtain a bi-degree d:d correspondence ℭ^⊛ on 𝔘. Specifically,  ℭ^⊛ is defined the algebraic equation:

    {(𝔲_1,𝔲_2)∈𝔘×𝔘: R(𝔲_2)-R(η^+(𝔲_1))/𝔲_2-η^+(𝔲_1)=0}.

We then pass to the quotient 

    𝔚 := 𝔘∼_w,

where ∼_w is the finite equivalence relation defined as


For z∈∂𝔇_i⊂_i and w∈∂𝔇_j⊂_j,


(z,i)∼_w (w,j) R_i(z)=R_j(w).


The space 𝔚 can be viewed as a compact, simply connected, (possibly) noded Riemann surface.

As in Section <ref>, the maps R, η^+ descend to 𝔚, defining a bi-degree d:d correspondence ℭ on 𝔚.

 


Let f be a degree d=d(Σ)


  * geometrically finite; or 

  * finitely renormalizable and periodically repelling

polynomial with connected Julia set. By Theorem <ref>, there exists S∈𝒮_Σ that is a conformal mating of f with the factor Bowen-Series map A_Σ^f. The above discussion allows us to associate with S a bi-degree d:d correspondence ℭ on a compact, simply connected, (possibly) noded Riemann surface 𝔚. The proof of <cit.> applies verbatim to the current setting to show that ℭ is a mating of f with the Fuchsian group G uniformizing Σ.






§ TEICHMÜLLER SPACES AND CONNECTEDNESS LOCI IN THE SPACE OF DEGENERATE POLYNOMIAL-LIKE MAPS




 §.§ Hybrid classes and vertical fibers


Recall that the triple (g, P_1, P_2) is a degenerate (anti-)polynomial-like map of degree d if g: P_1 ⟶ P_2 is a degree d holomorphic/anti-holomorphic map between two pinched polygons P_1⊂ P_2 satisfying conditions in Definition <ref>.


Let (g_1,P_1, P_2) and (g_2,Q_1,Q_2) be two degenerate (anti-)polynomial-like maps. We say that g_1 and g_2 are hybrid conjugate if there exists a quasiconformal map Φ→ such that:


	
  * Φ sends the pinched points (respectively, corners) of P_2 to the pinched points (respectively, corners) of Q_2,
	
  * Φ conjugates g_1 to g_2, restricted to the closure of a neighborhood of K(g_i) pinched at the singular points of ∂ P_1, ∂ Q_1,
	
  * ∂Φ≡ 0 almost everywhere on K(g_1).



Hybrid conjugacy defines an an equivalence relation on the space of degenerate (anti-)polynomial-like maps, and we call the corresponding equivalence classes hybrid classes of degenerate (anti-)polynomial-like maps.

Recall that for a degenerate (anti-)polynomial-like map, the external map can be defined in a similar way as external maps of polynomial-like maps, which is a piecewise real-analytic, degree d, covering map of 𝕊^1.
As in the classical setting of polynomial-like maps, equivalence classes of piecewise real-analytic degree d coverings of the circle under the equivalence relation induced by piecewise real-analytic circle conjugacy are called external classes (cf. <cit.>). Two (piecewise) conformally conjugate degenerate (anti-)polynomial-like maps have the same external class. For our purposes, however, it is more convenient to follow the approach of <cit.> and consider the space of degenerate (anti-)polynomial-like maps up to Möbius equivalence. Analogously, we will identify two external maps when they are conjugate by a Möbius automorphism of the disk.
Finally, we define a vertical fiber in the space of degree d degenerate (anti-)polynomial-like maps with connected Julia set as the collection of all such maps with a fixed external map.




 §.§ Total space as union of vertical fibers


The space 𝒮_G_d≡𝒮__d introduced in Section <ref> consists of degree d degenerate anti-polynomial-like maps with connected Julia set having the Nielsen map _d of the regular ideal (d+1)-gon reflection group G_d as their external map. The Teichmüller space Teich(G_d) of the group G_d is real (d-2)-dimensional. Specifically,
the space Teich(G_d) consists of Möbius conjugacy classes of discrete, faithful, strongly type-preserving representations of G_d into Aut^±(). Any such representation ρ with image G is induced by a quasiconformal homeomorphism ψ_ρ of  that preserves the disk  and carries the preferred fundamental domain Π of G_d to a preferred fundamental domain Π(G) of G:

    ρ(g)=ψ_ρ∘ g∘ψ_ρ^-1, g∈G_d

(cf. <cit.>).
Thus, the Nielsen map _G of G associated with the fundamental domain Π(G) is simply the conjugate of _d via ψ_ρ. We denote by 𝒮_G≡𝒮__G the space of degree d degenerate anti-polynomial-like maps with connected Julia set having the Nielsen map _G as their external map. Note that each 𝒮_G is a vertical fiber in the space of degree d degenerate anti-polynomial-like maps with connected Julia set (see Subsection <ref>).
Putting all these vertical fibers together, we get the total space

    𝒯:=_G∈Teich(G_d) 𝒮_G.



On the holomorphic side, for a marked surface Σ_0∈𝔉 with associated Fuchsian model G_0 (see Section <ref>), the slice ℬ_G_0≡ℬ_Σ_0 consists of all degree d≡ d(Σ_0) degenerate polynomial-like maps with connected Julia set that have the factor Bowen-Series map A^f_Σ_0 as their external map. It follows from the construction of factor Bowen-Series maps that for Σ∈Teich(Σ_0) (or equivalently, for G∈Teich(G_0)), the map A_Σ^f is quasiconformally conjugate to A_Σ_0^f. Specifically, the quasiconformal conjugacy between A_Σ^f, A_Σ_0^f is induced by the quasiconformal conjugacy between the Fuchsian models G, G_0 of Σ, Σ_0 (cf. <cit.>). We will use the notation ℬ_Σ and ℬ_G interchangeably.
As in the antiholomorphic case, we consider the total space

    𝒯:=_G∈Teich(G_0) ℬ_G,

which is the disjoint union of all the vertical fibers ℬ_G where G runs over the Teichmüller space of G_0. 



    To make things interesting, we will assume that Teich(G_d) (respectively, Teich(G_0)) is not a singleton set. This is equivalent to saying that the ideal polygon defining G_d has at least four sides (in the antiholomorphic case) or the surface Σ_0 has at least four marked points (in the holomorphic case). In both cases, this implies that d>2.




 §.§ Cartesian product structure of the total space


In what follows, the notation _G will stand for the vertical fiber 𝒮_G or ℬ_G.
It is easy to see that maps in _G are completely determined by their hybrid class. This can be proved using the arguments of <cit.> (cf. <cit.>).




Let G_0 stand for the regular ideal polygon reflection group G_d or the Fuchsian model of a surface Σ_0∈𝔉.


  * For each S∈_G_0, there exists an injective continuous map 

    Ψ_S:(Teich(G_0),G_0)↪(𝒯,S)

such that Ψ_S(G) is hybrid conjugate to S and has A_G^f (respectively, _G) as its external map.

  * For G∈Teich(G_0), the map

    Ψ_G:_G_0→_G 
     
    S↦Ψ_S(G)

is a bijection.

  * The space 𝒯 has a Cartesian product structure; i.e., 
the map

    Ψ:Teich(G_0)×_G_0⟶𝒯
    
    (G,S)↦Ψ_S(G)=Ψ_G(S)

is a bijection.




1) The factor Bowen-Series maps A_G^f (respectively, the Nielsen maps _G), G∈Teich(G_0), are quasiconformally conjugate to A_G_0^f (respectively, to _G_0), and these conjugacies can be chosen to depend continuously on Teich(G_0). With these quasiconformal conjugacies at hand, one can employ a standard quasiconformal surgery argument to replace the external map of S with A_G^f (respectively, _G), for any given G∈Teich(G_0). By construction, this quasiconformal deformation of S is supported on its escaping set (i.e., it is conformal on K(S)), and hence, any such deformed map is hybrid conjugate to S and lies in the vertical fiber _G. It follows that the deformed map is unique up to affine conjugacy (preserving marking).
This produces a well-defined map Ψ_S:Teich(G_0)↪𝒯.

Finally, injectivity of Ψ_S is a consequence of the fact that for G≠ G'∈Teich(G_0), the maps Ψ_S(G), Ψ_S(G') lie in distinct vertical fibers.


2) Injectivity of Ψ_G follows from the definition of Ψ_G and the fact that maps in the vertical fiber _G_0 are completely determined by their hybrid class. That the map Ψ_G is surjective can be seen from the observation that the quasiconformal deformation of the previous part can be reversed. More precisely, the
external map of any map S_1∈_G can be replaced with the external map A_G_0^f (respectively, _G_0) producing a degenerate (anti-)polynomial-like map S∈_G_0 with Ψ_G(S)=S_1. 


3) By definition of Ψ and part (2) of this theorem, the map Ψ induces a bijection between {G}×_G_0 and the vertical fiber _G. Since 𝒯 is the disjoint union of all vertical fibers _G as G runs over Teich(G_0), the result follows.



The upshot of Theorem <ref> can be summarized as follows. The image Ψ_S(Teich(G_0))⊂𝒯 passes through any map S∈_G_0, and it is a single hybrid class [S]. We call this, which is defined by quasiconformal deformation of the external map, the horizontal leaf through S in 𝒯. Such a horizontal leaf intersects each vertical fiber _G at a unique point. This defines a bijective holonomy map Ψ_G between the vertical fibers _G_0 and _G. These holonomy maps piece together to yield a product structure 𝒯≅Teich(G_0)×_G_0.


It is desirable to replace _G_0 with 𝒞_d^± in the product structure of 𝒯 given in Theorem <ref>. However, our techniques do not allow us to deduce this result due to analytic limitations of David surgery and combinatorial rigidity issues. However, thanks to Theorems <ref>, <ref>, and <ref>, we have the following weaker substitute. Here, 𝒯_gf (respectively, 𝒯_fr) denotes the collection of geometrically finite (respectively, periodically repelling and finitely renormalizable) maps in 𝒯.


There exists a natural bijection between the restricted total space 𝒯_gf∪𝒯_fr and the product space Teich(G_0)×(𝒞_d,gf^±∪𝒞_d,fr^±).





 §.§ A topological product structure for a restricted total space

The Cartesian product structure of Theorem <ref> may not be upgradable to a topological product structure due to quasiconformally non-rigid parameters. 
On the other hand, if we restrict to the smaller space of periodically repelling, finitely renormalizable maps, we have transverse continuity and hence the desired product structure.

There exists a dynamically natural homeomorphism

    𝒯_fr≅Teich(G_0) ×𝒞_d,fr^±.



For simplicity of the presentation, we will present the sketch of the proof for the holomorphic case.
Let Φ:𝒞_d,fr^+→ℬ_G_0,fr be the homeomorphism of Theorem <ref>. We will show that the map

    Ψ: Teich(G_0) ×𝒞_d,fr^+⟶𝒯_fr
    
    (G,f)↦Ψ(G,Φ(f))

is the desired homeomorphism, where Ψ:Teich(G_0)×ℬ_G_0⟶𝒯 is the bijection of Theorem <ref>.
For this, we only need to justify that Ψ is a homeomorphism.

We proceed to demonstrate continuity of the map Ψ. Let (G_n,S_n)→ (G_∞,S_∞) in Teich(G_0) ×ℬ_G_0. We need to verify that the sequence of maps Ψ(G_n,S_n)=Ψ_S_n(G_n) converges to the map Ψ(G_∞,S_∞)=Ψ_S_∞(G_∞).
Since {G_n} is contained in a compact set of Teich(G_0), it is easy to see that Ψ(G_n,S_n) is pre-compact (cf. Theorem <ref>.
Let 𝒮 be a subsequential limit of Ψ(G_n,S_n).
By Theorem <ref>, 𝒮 = Ψ(G_∞',S_∞') for some G_∞' ∈Teich(G_0) and S_∞' ∈ℬ_G_0.
Consider the uniformization map ϕ_n of the tiling set U_n of Ψ(G_n,S_n).
Then ϕ_n conjugates the dynamics of the factor Bowen-Series maps A_G_n^f to Ψ(G_n,S_n)|_U_n.
It is easy to see that after appropriate normalization, ϕ_n is pre-compact.
Thus we can take a subsequential limit, and conclude that G_∞' = G_∞.
Since Φ(G_n,S_n) is hybrid conjugate to S_n = Φ(G_0,S_n), and the hybrid conjugacy is uniformly quasiconformal, after passing to a subsequence, we conclude that 𝒮 is quasiconformally conjugate to S_∞ = Φ(G_0,S_∞).
As S_∞ is periodically repelling and finitely renormalizable,the non-escaping set K(S_∞) does not support any non-trivial quasiconformal deformation. Hence, the quasiconformal deformation space of the map S_∞ agrees with the horizontal leaf (or hybrid class) Ψ_S_∞(Teich(G_0)).
It follows that 𝒮 = Ψ(G_∞,S_∞).

 Continuity of the inverse Ψ^-1 can be proved using similar arguments. This completes the proof of the result.



By the discussion preceding this theorem, the sequence of holomorphic maps 
{Ψ_S_n:Teich(G_0)→𝒯} is pre-compact, and any subsequential limit Ψ_∞:Teich(G_0)→𝒯 satisfies the properties

    
  * Ψ_∞(G)∈ℬ_G, for each G∈Teich(G_0), and
    
  * Ψ_∞(Teich(G_0)) is contained in the quasiconformal deformation space of the map S_∞.

The fact that S_∞ is periodically repelling and finitely renormalizable implies that K(S_∞) does not support any non-trivial quasiconformal deformation. Hence, the quasiconformal deformation space of the map S_∞ agrees with the horizontal leaf (or hybrid class) Ψ_∞(Teich(G_0)). It now follows that Ψ_∞=Ψ_S_∞; i.e., the maps Ψ_S_n converge to Ψ_S_∞ locally uniformly.

Let 𝒳⊂Teich(G_0) be a compact set containing the sequence {G_n} and its limit G_∞. Since Ψ_S_n→Ψ_S_∞ uniformly on 𝒳, it follows that

    Ψ(G_n,S_n)=Ψ_S_n(G_n)→Ψ_S_∞(G_∞)=Ψ(G_∞,S_∞),  as n→+∞.


 Continuity of the inverse Ψ^-1 can be proved using similar arguments. This completes the proof of the result.






 §.§ A holomorphic motion and source of discontinuity

We will end our discussion with a brief interpretation of the picture in terms of holomorphic motions. The account given below has roots in the analogous description of the space of quadratic-like maps given in <cit.>.




  §.§.§ The holomorphic case




We note that the domain of definition of a map in 𝒯 is a finite tree of closed Jordan disks (a tree-like inversive multi-domain), and these Jordan domains can be uniformized by rational maps. 
One can regard  each strata

    𝒯_ℒ:=_G∈Teich(G_0)ℬ_G,ℒ

of the total space as a subset of a certain quotient of the product space ∏_j=1^k Rat_d_j(), where d_j are the degrees of the uniformizing rational maps and the quotients come from normalizations of the rational maps (cf. <cit.>). Here, ℬ_G,ℒ consists of maps in ℬ_G with coarse lamination ℒ, see Subsection <ref>.

The map Ψ of Theorem <ref> defines a `holomorphic motion' of ℬ_G_0 over the Teichmüller space Teich(G_0) in the above ambient space; indeed, the maps Ψ_S can be shown to be holomorphic (cf. <cit.>). However, the dimension of ℬ_G_0 is larger than one as d> 2. Hence, there is no λ-lemma in this setting that guarantees transverse continuity of the holomorphic motion. Note that transverse continuity of the motion is equivalent to continuity of the holonomy maps Ψ_G:ℬ_G_0→ℬ_G, G∈Teich(G_0).

We will now analyze the source of possible discontinuity of the above holomorphic motion in the transverse direction. To this end, let us first fix G_1∈Teich(G_0). For G close to G_1 in Teich(G_0), the conjugacies between G and G_0 can be chosen to be K-quasiconformal, for some K>1. This implies that the hybrid conjugacies between S∈ℬ_G_0 and Ψ_S(G) are given by global K-quasiconformal homeomorphisms, for G close to G_1 (note that K does not depend on S∈ℬ_G_0). Consequently, the family of holomorphic maps 

    {Ψ_S:Teich(G_0)→𝒯_ℒ}_S∈ℬ_G_0,ℒ

is locally bounded and hence normal (recall that ℬ_G_0 is compact).
Therefore, for a sequence S_n→ S_∞ in ℬ_G_0,ℒ, the holomorphic maps Ψ_S_n have (injective) holomorphic subsequential limits that send each G∈Teich(G_0) into the vertical fiber ℬ_G. However, although each Ψ_S_n takes values in the hybrid class [S_n], a priori, the images of the holomorphic subsequential limits are contained in the quasiconformal deformation space of the map S_∞, which is possibly larger than the hybrid class [S_∞] (cf. <cit.>). In other words, if the filled Julia set K(S_∞) supports non-trivial quasiconformal deformations, then the map Ψ_S_∞ may not be the local uniform limit of {Ψ_S_n}. This is exactly the reason why the holonomy maps Ψ_G may fail to be continuous at quasiconformally non-rigid parameters. (This is also the same phenomenon that is responsible for discontinuity in Theorem <ref> and Theorem <ref>, and c.f. Theorem <ref>.)




    The situation discussed above is analogous to the holomorphic motion of the Mandelbrot set over  in the space of quadratic rational maps (where one varies the multiplier of the fixed point at ∞ throughout , cf. <cit.>). Since the Mandelbrot set lives in , this motion enjoys transverse continuity (in fact, quasiconformality).




  §.§.§ The antiholomorphic case


Much of the discussion of the previous subsection can be applied to Schwarz reflections after passing to the second iterate. Nevertheless, since the Teichmüller space of the reflection group G_d only admits a real-analytic structure (as opposed to complex structures on Teichmüller spaces of Fuchsian groups), the motion of vertical fibers is only real-analytic in this context. 


To set up the framework, let us consider the second iterates _G^∘ 2 of Nielsel maps of the reflection groups G∈Teich(G_d). Such a map is defined on the complement of the interior of an ideal d(d+1)-gon Π(G) in . In Figure <ref> (where d=3), the polygon Π(G) is bounded by the red geodesics, while the blue geodesics bound the original polygon Π(G) whose boundary reflections generate G. If r_1,⋯, r_d+1 are the reflections in the sides of Π(G), then for each j∈{1,⋯,d+1}, 
    Ξ_j:={r_j∘ r_k: k∈{1,⋯,d+1}, k≠ j}
 
is a minimal generating set of the index two Fuchsian subgroup G of G. In fact, the elements of Ξ_j are the side-pairing transformations for the fundamental domain Π(G)∪ r_j(Π(G)) of G. It is also easily seen that 

    Π(G)= Π(G)∪⋃_j=1^d+1r_j(Π(G)),

and /G is a (d+1)-times punctured sphere that admits an antiholomorphic involution (induced by r_j). It can now be readily verified that the maps _G^∘ 2, G∈Teich(G_d), are higher Bowen-Series maps associated with (d+1)-times punctured spheres in the sense of <cit.> (cf. <cit.>).

According to <cit.>, one can associate a higher Bowen-Series map to each member Γ∈Teich(G_d) (where G_d is the index two Fuchsian subgroup of G_d). Further, a quasiconformal conjugation between G_d and Γ induces a quasiconformal conjugation between their higher Bowen-Series maps. We denote the higher Bowen-Series map by A_Γ^h. We denote the vertical fiber of A_Γ^h in the space of degree d^2 degenerate polynomial-like maps with connected Julia set by _Γ^h. We define the total space

    𝒯^h:=_Γ∈Teich(G_d) _Γ^h.

The arguments of the previous subsection now implies that the `base' vertical fiber _G_d^h moves holomorphically over the Teichmüller space of G_d. Here, we embed the total space 𝒯^h into the product space of holomorphic maps on finitely many domains in .

The operation of passing to the second iterate embeds the vertical fiber 𝒮_G_d (in the space of degree d degenerate anti-polynomial-like maps with connected Julia set) into the vertical fiber _G_d^h (in the space of degree d^2 degenerate polynomial-like maps with connected Julia set). Thus, this copy of 𝒮_G_d in _G_d^h moves holomorphically over Teich(G_d). This holomorphic motion can be used to explain the topological product structure in the antiholomorphic case of Theorem <ref>.

Finally, note that Teich(G_d) is the fixed-point set of an antiholomorphic involution on Teich(G_d) (elements of Teich(G_d) correspond to `symmetric' (d+1)-times punctured spheres, cf. <cit.>), and hence Teich(G_d) can be regarded as a real slice of the complex manifold Teich(G_d). It follows that the above copy of 𝒮_G_d moves real-analytically over Teich(G_d).


LLMM23a


[AS76]AS76
D. Aharonov and H. S. Shapiro.
Domains on which analytic functions satisfy quadrature identities.
J. Analyse Math., 30:39–73, 1976.

[BB09]BB09
G. Bassanelli and F. Berteloot.
Lyapunov exponents, bifurcation currents and laminations in bifurcation loci.
Math. Ann., 345:1–23, 2009.

[BF03]BF03
S. Bullett and M. Freiberger.
Hecke groups, polynomial maps and matings.
Internat. J. Modern Phys. B, 17:3922–3931, 2003.

[BF05]BF05
S. Bullett and M. Freiberger.
Holomorphic correspondences mating Chebyshev-like maps with Hecke groups.
Ergodic Theory Dynam. Systems, 25:1057–1090, 2005.

[BH07]BH07
S. Bullett and P. Haïssinsky.
Pinching holomorphic correspondences. 
Conform. Geom. Dyn., 11:65–89, 2007.

[BL20]BL20
S. Bullett and L. Lomonaco.
Mating quadratic maps with the modular group II.
Invent. Math., 220:185–210, 2020.

[BL22]BL22
S. Bullett and L. Lomonaco.
Dynamics of Modular Matings.
Adv. Math., 410, Part B, 108758, 2022.

[BLLM24]BLLM
S. Bullett, L. Lomonaco, M. Lyubich, and S. Mukherjee.
Mating parabolic rational maps with Hecke groups.
<https://arxiv.org/abs/2407.14780>, 2024.

[BP94]BP
S. Bullett and C. Penrose.
Mating quadratic maps with the modular group.
Invent. Math., 115:483–511, 1994.

[CDKvS22]CDKvS22
T. Clark, K. Drach, O. Kozlovski, and S. van Strien.
The dynamics of complex box mappings. 
Arnold Math. J., 8:319–-410, 2022.

[DS06]DS06
T.-C. Dinh and N. Sibony.
Distribution des valeurs de transformations méromorphes et applications.
Comment. Math. Helv. 81:221–258, 2006.

[DH85a]DH1
A. Douady and J. H. Hubbard.
Étude dynamique des polynômes complexes I, II.
Publications Mathématiques d'Orsay. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 - 1985.
  
[DH85b]DH2
A. Douady and J. H. Hubbard.
On the dynamics of polynomial-like mappings.
Ann. Sci. Éc. Norm. Sup., 18:287–343, 1985.

[Dud11]Dud11
D. Dudko.
Matings with laminations.
<https://arxiv.org/abs/1112.4780>, 2011.

[Eps93]Eps93
A. L. Epstein.
Towers of finite type complex analytic maps.
PhD thesis, The City University of New York, 1993.

[Fat29]Fatou29
P. Fatou.
Notice sur les travaux scientifiques de M. P. Fatou.
Astronome titulaire de l'observatoire de Paris, 5–29, 1929, <https://www.math.purdue.edu/ eremenko/dvi/fatou-b.pdf>.

[Gey20]Gey20
L. Geyer.
Classification of critically fixed anti-rational maps.
<https://arxiv.org/abs/2006.10788v3>, 2020.

[GM93]GM93
L. Goldberg and J. Milnor.
Fixed points of polynomial maps, part II: fixed point portraits.
Ann. Sci. Éc. Norm. Supér. (4), 26(1):51–98, 1993.

[GT21]GT21
Y. Gao and G. Tiozzo.
The core entropy for polynomials of higher degree.
J. Eur. Math. Soc. (JEMS), 24:2555–2603, 2022.

[HS14]HS
J. H. Hubbard and D. Schleicher.
Multicorns are not path connected.
Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday, pages 73–102, 2014.

[Ino09]Ino09
H. Inou.
Combinatorics and topology of straightening maps II: Discontinuity.
<http://arxiv.org/abs/0903.4289>, 2009.

[IK12]IK
H. Inou and J. Kiwi.
Combinatorics and topology of straightening maps, I: Compactness and bijectivity.
Adv. Math., 231:2666–2733, 2012.

[IM21]IM21
H. Inou and S. Mukherjee.
Discontinuity of straightening in anti-holomorphic dynamics: I.
Trans. Amer. Math. Soc., 374:6445–6481, 2021.
  
[KvS09]KvS09
O. Kozlovski, S. van Strien. 
Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials. 
Proc. Lond. Math. Soc. (3), 99:275–296, 2009.

[Kiw97]Kiw97
J. Kiwi.
Rational rays and critical portraits of complex polynomials. 
Thesis, SUNY at Stony Brook, 1997.

[Kiw01]Kiw01
J. Kiwi.
Rational laminations of complex polynomials.
In Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, editors, Laminations and Foliations in Dynamics, Geometry and Topology, volume 269 of
  Contemporary Mathematics, pages 111–154. American Mathematical Society, 2001.
  
[Kiw04]Kiw04
J. Kiwi.
ℝeal laminations and the topological dynamics of complex polynomials.
Adv. Math., 184:207–267, 2004.
 
[LMM22]LMM2
K. Lazebnik, N. G. Makarov, and S. Mukherjee.
Bers slices in families of univalent maps.
Math. Z., 300:2771–2808, 2022.


[LLMM21]LLMM3
S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee.
Schwarz reflections and anti-holomorphic correspondences.
Adv. Math., 385:Paper No. 107766, 88, 2021.

[LLMM23a]LLMM1
S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee.
Dynamics of Schwarz reflections: the mating phenomena.
Ann. Sci. Éc. Norm. Supér. (4), 56:1825–1881, 2023.
  
  
[LLMM23b]LLMM2
S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee.
Schwarz reflections and the Tricorn.
To appear in Ann. Inst. Fourier (Grenoble), <https://arxiv.org/abs/1812.01573v2>, 2023.

[LLMM23]LLMM4
R. Lodge, M. Lyubich, S. Merenkov, and S. Mukherjee.
On dynamical gaskets generated by rational Maps, Kleinian groups, and Schwarz reflections.
Conform. Geom. Dyn., 27:1–54, 2023. 

[LLM22a]LLM1
R. Lodge, Y. Luo, and S. Mukherjee.
Circle packings, kissing reflection groups and critically fixed anti-rational maps.
Forum Math. Sigma, 10, paper no. e3, 38 pp., 2022.

[LM14]LM14
S. Y. Lee and N. G. Makarov.
Sharpness of connectivity bounds for quadrature domains.
<https://arxiv.org/abs/1411.3415v1>, 2016.

[LM16]LM16
S. Y. Lee and N. G. Makarov.
Topology of quadrature domains.
J. Amer. Math. Soc., 29(2):333–369, 2016. 

[Lom15]Lom15
L. Lomonaco.
Parabolic-like mappings.
Ergodic Theory Dynam. Systems, 35:2171–2197, 2015.

[Luo23]L23
Y. Luo.
On geometrically finite degenerations I: boundaries of main hyperbolic components.
to appear in J. Eur. Math. Soc., 2023.

[Lyu97]Lyu97
M. Lyubich.
Dynamics of quadratic polynomials. I, II.
Acta Math., 178:185–297, 1997.

[Lyu99]Lyu99
M. Lyubich.
Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture. 
Ann. of Math. (2), 149:319–420, 1999.

[Lyu24]Lyu23
M. Lyubich.
Conformal geometry and dynamics of quadratic polynomials, vol i-ii.
<http://www.math.stonybrook.edu/ mlyubich/book.pdf>, 2023.

[LMM23]LMM23
M. Lyubich, J. Mazor, and S. Mukherjee.
Antiholomorphic correspondences and mating I: realization theorems.
To appear in Comm. Amer. Math. Soc., <https://arxiv.org/abs/2303.02459>, 2023.

[LMMN20]LMMN
M. Lyubich, S. Merenkov, S. Mukherjee, and D. Ntalampekos.
David extension of circle homeomorphisms, welding, mating, and removability.
To appear in Mem. Amer. Math. Soc., <https://arxiv.org/abs/2010.11256v2>, 2020.

[LM93]LM93
M. Lyubich and J. Milnor.
The Fibonacci unimodal map.
J. Amer. Math. Soc., 6:425–457, 1993.

[Mak93]Mak93
P. Makienko.
Pinching and plumbing deformations of  quadratic rational maps.
IC/93/32, International Centre for Theoretical Physics, Trieste, Italy, 1993, <https://inis.iaea.org/collection/NCLCollectionStore/_Public/24/036/24036519.pdf>.

[Man93]Man93
R. Mañé.
On a theorem of Fatou.
Bol. Soc. Brasil. Mat. (N.S.), 24:1–11, 1993.

[McM88]McM
C. McMullen.
Automorphisms of rational maps.
In Holomorphic functions and moduli, vol. I (Berkeley, CA, 1986), 31–60, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988. 

[McM94]McM1
C. McMullen.
Complex dynamics and renormalization.
Annals of Mathematics Studies, 135. Princeton University Press, Princeton, NJ, 1994. x+214 pp. 
         
[Mil00]Mil00
J. Milnor.
Periodic orbits, external rays and the Mandelbrot set.
Astérisque, 261:277–333, 2000.


[Mil06]Mil06
J. Milnor.
Dynamics in one complex variable.
Princeton University Press, New Jersey, 3rd edition, 2006.         

[MM23a]MM1
M. Mj and S. Mukherjee.
Combining rational maps and Kleinian groups via orbit equivalence.
Proc. Lond. Math. Soc (3), 126:1740–1809, 2023.

[MM23b]MM2
M. Mj and S. Mukherjee.
Matings, holomorphic correspondences, and a Bers slice. 
<https://arxiv.org/abs/2304.12699>, 2023.

[MM23c]MM3
M. Mj and S. Mukherjee.
The Sullivan dictionary and Bowen-Series maps.
EMS Surv. Math. Sci., 10:179–221, 2023.

[MNS17]MNS
S. Mukherjee, S. Nakane, and D. Schleicher.
On Multicorns and Unicorns II: bifurcations in spaces of antiholomorphic polynomials.
Ergodic Theory Dynam. Systems, 37:859–899, 2017.

[PR21]PR21
C. L. Petersen and P. Roesch.
The Parabolic Mandelbrot Set.
<https://arxiv.org/abs/2107.09407>, 2021.

[Sak91]Sak91
M. Sakai.
Regularity of a boundary having a Schwarz function.
Acta Math., 166:263–297, 1991.

[Sul85]Sul
D. Sullivan.
Quasiconformal homeomorphisms and dynamics I. solution of the
  Fatou-Julia problem on wandering domains.
Ann. of Math. (2), 122(2):401–418, 1985.

[Sul92]Sul1
D. Sullivan.
Bounds, quadratic differentials, and renormalization conjectures. 
American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988), 417–466, Amer. Math. Soc., Providence, RI, 1992.

[SM98]SulMc 
D. Sullivan and C. T. McMullen.
Quasiconformal homeomorphisms and dynamics III: The Teichmüller space of a holomorphic dynamical system.
Adv. Math., 135:351–395, 1998.

[Urb03]Urb1
M. Urbanski.
Measures and dimensions in conformal dynamics.
Bull. Amer. Math. Soc. (N.S.), 40:281–321, 2003.





