# Embedding Unicritical Connectedness Loci

Malavika Mukundan  University of Michigan, Ann Arbor<EMAIL_ADDRESS>http://www-personal.umich.edu/$\sim$ malavim/index.html

###### Abstract.

In this article, for degree $d\geq 1$, we construct an embedding $\Phi_{d}$ of
the connectedness locus $\mathcal{M}_{d+1}$ of the polynomials $z^{d+1}+c$
into the connectedness locus of degree $2d+1$ bicritical odd polynomials.

##  1\. Introduction

Relationships between different families of rational maps have been studied in
various contexts in complex dynamics. In rational dynamics, quadratic
polynomials of the form $z^{2}+c$ are the fundamental objects of study, and
much of the field involves the study of the Mandelbrot set pioneered by Douady
and Hubbard, in [DH82], [Dou83],[DH85a] and [Hub93], and developed by Milnor
([Mil00]), Lyubich and Dudko ([DL18]) and several others. In general,
polynomials with a single critical point, normalized as $z^{d}+c$, and their
connectedness loci $\mathcal{M}_{d}$ \- that is, the set of parameters $c$ for
which the filled Julia set is connected, commonly referred to as the Multibrot
sets, have also been studied in [Sch04], [EMS16], etc., and the properties of
these sets have been used to conjecture and prove several results in both
rational and transcendental dynamics.

For example, the theory of matings and the work of Tan Lei, Rees, Shishikura
and others (see [Ree90], [Shi00], [Lei92]) created a link between polynomials
and rational maps, by combining two polynomials to create a rational map. This
made it easier to study certain hyperbolic components in rational parameter
spaces, as well as the structure of Julia sets of rational maps that arise as
matings.

(a) $\mathcal{CBO}_{1}$

(b) $\mathcal{CBO}_{2}$

(c) $\mathcal{CBO}_{3}$

(d) $\mathcal{CBO}_{4}$

(e) $\mathcal{CBO}_{5}$

Figure 1. The families $\mathcal{CBO}_{d}$ for $d=1,2,3,4,5$

Renormalization is a powerful tool in dynamics. It restricts certain
holomorphic functions to smaller domains in which they “look” like some
$z^{d}+c$ to make them easier to study. In his study of renormalizable maps in
[McM00], McMullen proved that unicriticals are universal in the sense that
there are small copies of the Multibrot sets found in any holomorphic family
of maps.

Branner and Douady constructed a continuous map from the basilica limb of the
Mandelbrot set $\mathcal{M}_{2}$ to the rabbit limb (see [BD88]). This was
later extended by Branner and Fagella in [BF99] into homeomorphisms between
various limbs of the Mandelbrot set, and in a different spirit, by Dudko and
Schleicher in [DS12]. In [RS98], Riedl and Schleicher also construct a
homeomorphism from a subset of any $\frac{p}{nq}$-limb of the Mandelbrot set
to the $\frac{p}{q}$-limb.

We establish another such relationship between two holomorphic families -
unicritical polynomials and the family of symmetric polynomials which we
introduce below and describe in detail in Section 2. We take symmetric
polynomials to mean polynomials that commute with some affine map $M$
satisfying $M^{\circ 2}=\text{Id}$. Symmetric cubic polynomials are
encountered, for example, in the study of core entropy (see [GT21]). We focus
on the more specific family of symmetric polynomials with exactly two critical
points. As we will show in Section 2, such a polynomial is affine conjugate to

$p_{a,d}(z)=a\int_{0}^{z}\Big{(}1-\frac{w^{2}}{d}\Big{)}^{d}dw$

for some $a\in\mathbb{C}^{*}$ and $d\geq 2$. For each $a$, $p_{a,d}$ is an odd
polynomial- that is, it commutes with $z\mapsto-z$. For a fixed $d$, we let
$p_{a}=p_{a,d}$. We let $\mathcal{CBO}_{d}$ denote the set of
$a\in\mathbb{C}^{*}$ such that $p_{a}$ has connected filled Julia set. This is
a closed, compact connected subset of $\mathbb{C}$. In this article we shall
prove the following.

######  Theorem 1.1.

For $d\geq 1$, there exists a continuous map
$\Phi_{d}:\mathcal{M}_{d+1}\longrightarrow\mathcal{CBO}_{d}$ that is a
homeomorphism onto its image.

Our proof is along the lines of Douady and Branner’s use of quasiconformal
surgery in [BD88]. We shall perform a quasiconformal surgery along a $\beta-$
fixed point and its pre-images. The map $\Phi_{d}$ is natural in the sense
that its inverse can be described by a renormalization operator on a subset of
$\mathcal{CBO}_{d}$. We shall also give a complete description of the image
under $\Phi_{d}$ (see Section 2.3).

Although we do not provide details here, our construction holds in the
following generality:

######  Theorem 1.2.

For any integer $k\geq 2$, there exists a continuous map from
$\mathcal{M}_{d+1}$ to the collection of $a\in\mathbb{C}^{*}$ such that
$a{\displaystyle\int_{0}^{z}}\Big{(}1-\frac{w^{k}}{d}\Big{)}^{d}dw$ has
connected Julia set, that is a homeomorphism onto its image.

The family $p_{a,d}$ is interesting in its own right: as
$d\longrightarrow\infty$, $p_{a,d}(z)\longrightarrow
a{\displaystyle\int_{0}^{z}}e^{-w^{2}}dw$ locally uniformly on $\mathbb{C}$.
The limit function is entire, odd, has two asymptotic values
$\pm\frac{a\sqrt{\pi}}{2}$ and no critical points. It is called an “error”
function (see [Nev70] for an introduction). Error functions belong to the
larger Speiser class- the family of entire functions with finitely many
critical and asymptotic values. This family is studied in [GK86], [EL92] and
several others.

The simplest of the Speiser class is the family of exponential functions. A
lot of the analysis of exponential functions is a direct application of the
tools used in the analysis of unicritical polynomials, normalized as
$\lambda(1+\frac{z}{d})^{d}$ and using the fact that they converge to
$\lambda\exp{z}$ as $d\longrightarrow\infty$. This is a theme that is explored
in [BDH+00]. Our work in progress aims to carry out a similar analysis for the
error functions $a{\displaystyle\int_{0}^{z}}e^{-w^{2}}dw$. This paper
presents a structural similarity between the polynomials approximating
exponential functions, and the polynomials $p_{a,d}$ that approximate error
functions, and prompts us to make the following conjecture:

######  Conjecture 1.3.

Let $E_{c}(z)=\exp z+c$, and
$\mathscr{E}_{a}(z)=a{\displaystyle\int_{0}^{z}}e^{-w^{2}}dw$. There exists a
continuous map from $\big{\\{}c\in\mathbb{C}|\\{E_{c}^{\circ n}(c)\\}_{n\geq
0}\text{ is bounded}\big{\\}}$ to the set
$\big{\\{}a\in\mathbb{C}^{*}|\\{\mathscr{E}^{\circ n}_{a}(a)\\}_{n\geq
0}\text{ is bounded}\big{\\}}$ that is a homeomorphism onto its image.

There is some evidence to show that this is reasonable; work in progress
indicates that it may be possible to embed postsingularly finite exponential
functions into the collection of postsingularly finite error functions in a
combinatorially meaningful manner. We do not, however, address error functions
in this article.

The paper is organized as follows. In Section 2, we introduce symmetric
polynomials and establish some of their basic properties, provide motivation
for Theorem 1.1 while laying out our proof strategy, and describe the image of
$\Phi_{d}$. In Sections 3 and 4 respectively, we define $\Phi_{d}$ and prove
that it is continuous. We end in Section 5 by constructing a continuous
inverse for $\Phi_{d}$ on its image.

### Acknowledgements.

The author is indebted to John Hubbard and Sarah Koch for their continued
guidance throughout this study, and to Dierk Schleicher for helping widen the
horizons of the author’s research perspective. Special thanks to Jack Burkart,
Alex Kapiamba, Leticia Pardo-Simón and Vasiliki Evdoridou for helpful
conversations and resources, and to Lukas Geyer, Núria Fagella, Lasse Rempe,
Laurent Bartholdi, Joanna Furno, Giulio Tiozzo, Eriko Hironaka, David Martí-
Pete, Mikhail Hlushchanka, Nikolai Prochorov and others for their time and
insightful comments. The author also thanks Daniel Stoll for an introduction
to the Mandel software package.

This material is based upon work supported by the National Science Foundation
under Grant No. DMS-1928930 while the author was in residence at the
Mathematical Sciences Research Institute in Berkeley, California, during the
Spring 2022 semester.

(a) $\mathcal{M}_{3}$

(b) $\mathcal{CBO}_{2}$

(c) A portion of $\mathcal{CBO}_{2}$. The cut point on the mid-left is $a=1$.
Note the resemblance to $\mathcal{M}_{3}$

Figure 2. The figure on the top left is the unicritical locus
$\mathcal{M}_{3}$. The figure on the top right is the locus
$\mathcal{CBO}_{2}$ of odd bicritical polynomials of degree $5$. The figure at
the bottom zooms in on the right of the figure on the top right- we will show
that this region contains a copy of $\mathcal{M}_{3}$

##  2\. Preliminaries

For an introduction to polynomial dynamics and the Multibrot sets, see
[Mil00], [Mil06], [Hub16] and [EMS16].

###  2.1. Introduction

We call a polynomial $f$ of degree $>1$ symmetric if it commutes with an
affine map $M$ that satisfies $M^{\circ 2}(z)=z$. $M$ is of the form
$M(z)=-z+b$ for some $b\in\mathbb{C}$, and we may conjugate $M$ by a
translation $\tau$ so that $\tau^{-1}\circ M\circ\tau(z)=-z$. We have

$\displaystyle\tau^{-1}\circ f\circ\tau(z)$
$\displaystyle=\tau^{-1}\circ(M\circ f\circ M^{-1})\circ\tau(z)$
$\displaystyle=(\tau^{-1}\circ M\circ\tau)\circ(\tau^{-1}\circ
f\circ\tau)\circ(\tau^{-1}\circ M^{-1}\circ\tau)(z)$
$\displaystyle=-(\tau^{-1}\circ f\circ\tau)(-z)$

That is, $\tau^{-1}\circ f\circ\tau$ is an odd polynomial. Therefore, every
symmetric polynomial contains an odd polynomial in its affine conjugacy class.
We recall that an odd polynomial has only odd degree terms.

###  2.2. Bicritical odd polynomials

$f$ is bicritical if it has, upto multiplicity, exactly two critical points on
the plane. Let $f$ be a bicritical odd polynomial of degree $2d+1$, with
$d\geq 1$. The Riemann Hurwitz formula shows that $f$ has local degree $d+1$
at both critical points. Furthermore, the critical points are of the form $\pm
x$ for some $x\in\mathbb{C}^{*}$. Let $\phi(z)=kz$ be such that
$\phi(\\{x,-x\\})=\\{\sqrt{d},-\sqrt{d}\\}$. Then there exists a constant
$a\in\mathbb{C}^{*}$ such that

$\displaystyle(\phi\circ f\circ\phi^{-1})^{\prime}(z)$
$\displaystyle=a\Big{(}1-\frac{z^{2}}{d}\Big{)}^{d}$

Therefore,

$\displaystyle\phi\circ f\circ\phi^{-1}(z)$
$\displaystyle=a\int_{0}^{z}\Big{(}1-\frac{w^{2}}{d}\Big{)}^{d}dw$

For $a\in\mathbb{C}^{*}$, let

$\displaystyle p_{a}(z)$
$\displaystyle=a\int_{0}^{z}\Big{(}1-\frac{w^{2}}{d}\Big{)}^{d}dw$

It is evident that $p_{a}$ is affine conjugate to $p_{a^{\prime}}$ if and only
if $a=a^{\prime}$. Therefore, the space of bicritical odd polynomials modulo
conjugation by scaling (or, the space of symmetric bicritical polynomials
modulo affine conjugation) is the family $a\mapsto p_{a}$ over
$\mathbb{C}^{*}$. We shall denote this family $\mathcal{BO}_{d}$, and let

$\displaystyle\mathcal{CBO}_{d}=\\{a:K_{p_{a}}\text{ is connected}\\}$

Figure 1 illustrates $\mathcal{CBO}_{d}$ for $d=1,2,3,4,5,19$.

We consider the part of $\mathcal{CBO}_{2}$ illustrated in Figure 2(c), and
present some of the Hubbard trees of postcritically finite polynomials in this
region, in Figure 3. The pictures indicate a relationship between
$\mathcal{M}_{3}$ and $\mathcal{CBO}_{2}$, and in general, between
$\mathcal{M}_{d+1}$ and $\mathcal{CBO}_{d}$.

(a) A section of $\mathcal{CBO}_{1}$

(b) The $(+,-)$ type $z\mapsto z^{2}$

(c) The $(+,-)$ type basilica

Figure 3. Postcritically finite polynomials along with their Hubbard trees in
the family $p_{a}(z)=a{\displaystyle\int_{0}^{z}}(1-w^{2})dw$. $x_{0}^{-}=-1$,
$x_{0}^{+}=1$ are the two critical points, with $x_{i}^{\pm}=p_{a}^{\circ
i}(x_{0}^{\pm})$. Terminology:‘$(+,-)$ type polynomial $p$’ refers to the
polynomial in $\mathcal{CBO}_{1}$ that looks like a pair of copies of the
polynomial $p=f_{c}$, where $c\in\mathcal{M}_{2}$

(a) The $(+,-)$ type airplane

(b) The $(+,-)$ type rabbit

(c) The $(+,-)$ type $z\mapsto z^{2}-0.10003+0.95227i$

Figure 4. More examples of Hubbard trees in $\mathcal{CBO}_{1}$

###  2.3. Monic representatives of polynomials in $\mathcal{BO}_{d}$

Any polynomial $p_{a}$ for $a\in\mathcal{BO}_{d}$ has leading coefficient
$T(a):=\frac{(-1)^{d}a}{d^{d}(2d+1)}$ attached to $z^{2d+1}$. Let
$w(z)=\frac{z}{s}$. We note that $P_{s}(z)=w^{-1}\circ p_{a}\circ w$ is monic
if and only if $s^{2d}=T(a)$.

The polynomial $P_{s}$ admits a unique Böttcher chart $\varphi_{s}$ in a
neighborhood of $\infty$ that satisfies
$\lim_{z\rightarrow\infty}\frac{\varphi_{s}(z)}{z}=1$, and if
$s_{1}^{2d}=s_{2}^{2d}$, then $\varphi_{s_{1}}(z)=\omega\varphi_{s_{2}}(z)$
where $\omega=\frac{s_{2}}{s_{1}}$ is a $2d-$th root of unity. Let
$\mathcal{R}_{\theta}(s)$ denote the ray at angle $\theta$ in the dynamical
plane of $P_{s}$. Then it is easy to see that if
$\frac{s_{2}}{s_{1}}=e^{\frac{2\pi ij}{2d}}$ for some integer $j$, then for
all $\theta\in\mathbb{R}/\mathbb{Z}$,

$\displaystyle\mathcal{R}_{\theta}(s_{2})$
$\displaystyle=\mathcal{R}_{\theta+\frac{j}{2d}}(s_{1})$

Additionally, since $P_{s}$ is odd,
$\varphi_{s}(z)=\lim_{n\rightarrow\infty}P_{s}^{\circ
n}(z)^{\frac{1}{(2d+1)^{n}}}$ satisfies

$\displaystyle\varphi_{s}(-z)$ $\displaystyle=-\varphi_{s}(z)$
$\displaystyle\implies\mathcal{R}_{\theta+\frac{1}{2}}(s)$
$\displaystyle=-\mathcal{R}_{\theta}(s)$

For any $s$ such that $0\in J_{P_{s}}$, there exists a subset $\Theta$ of
$\big{\\{}0,1,...,\frac{2d-1}{2d}\big{\\}}$ satisfying
$\Theta+\frac{1}{2}=\Theta$ such that the dynamical rays landing at $0$ are
exactly those with angles in $\Theta$. Moreover, if $s^{\prime}=e^{\frac{2\pi
ij}{2d}}s$, then the set of angles that land at $0$ in the dynamical plane of
$P_{s^{\prime}}$ is $\frac{j}{2d}+\Theta$.

This shows that for any $a\in\mathcal{CBO}_{d}$ such that $0\in J_{p_{a}}$,
there exists a monic representative $P_{s}$ of $p_{a}$ so that $0$ is the
landing point of $\mathcal{R}_{0}(s)$ and $\mathcal{R}_{\frac{1}{2}}(s)$. The
union of the rays $\mathcal{R}_{0}(s)$ and $\mathcal{R}_{\frac{1}{2}}(s)$
separates the plane into two connected components $F^{L}_{s}$ and $F^{R}_{s}$,
named so that $F^{L}_{s}$ is the component that contains the critical point
$-s\sqrt{d}$, and $F^{R}_{s}$ contains $s\sqrt{d}$ ($L$ and $R$ stand for left
and right).

####  2.3.1. The Image under $\Phi_{d}$

The set of $a\in\mathcal{CBO}_{d}$ such that $0\in J_{p_{a}}$ is exactly
$\mathcal{CBO}_{d}\setminus\mathbb{D}$. Let $H$ be the component of
$\mathcal{CBO}_{d}\setminus\mathbb{D}$ that intersects the right half plane.
By the previous paragraph, on $H$, there exists a branch of
$a\mapsto(T(a))^{\frac{1}{2d}}$, which we shall denote $s(a)$, so that $0$ is
the landing point of $\mathcal{R}_{0}(s(a))$ and
$\mathcal{R}_{\frac{1}{2}}(s(a))$.

We note that the critical points of $P_{s}$ are $\pm s\sqrt{d}$. Let
$\mathscr{O}^{R}_{s}$ and $\mathscr{O}^{L}_{s}$ be the orbits under $P_{s}$ of
$s\sqrt{d}$ and $-s\sqrt{d}$ respectively.

$\displaystyle\Phi_{d}(\mathcal{M}_{d+1})$
$\displaystyle=\big{\\{}a\big{|}\mathscr{O}^{L}_{s(a)}\subset
F^{L}_{s(a)}\cup\\{0\\}\text{ and }\mathscr{O}_{s(a)}^{R}\subset
F^{R}_{s(a)}\cup\\{0\\}\big{\\}}$

That is, the image is the set of polynomials where the dynamical rays at
angles $0,\frac{1}{2}$ separate the orbits of the two distinct critical
points. It is easy to see that the latter is a closed set in
$\mathcal{CBO}_{d}$. We shall henceforth denote this image as
$\mathcal{CBO}^{(+,-)}_{d}$. This is also a proper subset of
$\mathcal{CBO}_{d}$: for example, $a=-1$ is not in this set. We have described
in detail the dynamics of polynomials in $\mathcal{CBO}_{d}^{(+,-)}$ in
Section 5.1.

Figure 3(a) illustrates the way $\Phi_{d}$ maps $\mathcal{M}_{2}$ by pointing
out the position of the images of well-known polynomials like the rabbit, co-
rabbit, airplane, etc.

###  2.4. Quotienting by $z^{2}$

Given $p_{a}\in\mathcal{BO}_{d}$, there exists a unique polynomial
$\mathcal{P}_{a}:\hat{\mathbb{C}}\longrightarrow\hat{\mathbb{C}}$ so that the
following diagram commutes.

${\hat{\mathbb{C}}}$${\hat{\mathbb{C}}}$${\hat{\mathbb{C}}}$${\hat{\mathbb{C}}}$$\scriptstyle{z\mapsto
z^{2}}$$\scriptstyle{p_{a}}$$\scriptstyle{z\mapsto
z^{2}}$$\scriptstyle{\mathcal{P}_{a}}$

The critical points of $\mathcal{P}_{a}$ are $d$ and
$\big{\\{}x_{\ell}^{2}\big{\\}}_{\ell=1}^{d}$, where $\pm x_{\ell}$,
$\ell=1,2,...,d$, are the pre-images of $0$ that are not equal to $0$. When
$d=1$, the family $\mathcal{P}_{a}$ corresponds to the collection of cubic
polynomials where one critical point is a pre-image of a $\beta-$fixed point
(that is, the landing point of a dynamical ray at angle $0$ or $\frac{1}{2}$)
and the other is free. This is isomorphic to the collection
$\mathcal{F}=\big{\\{}(a,b)|Q_{a,b}(a)=-2a\big{\\}}$, where
$Q_{a,b}(z)=z^{3}-3a^{2}z+b$ discussed in [BD88, Chapters I,II] in the
following way: letting $\widetilde{a}=9a^{2}$, we have

$\displaystyle\mathcal{P}_{-a}$ $\displaystyle=\mathcal{P}_{a}\simeq
Q_{\widetilde{a},2\widetilde{a}^{3}-2\widetilde{a}}$

where $\simeq$ refers to affine conjugacy.

Let $F_{+}\subset\mathcal{F}$ be the collection of polynomials $Q_{a,b}$ for
which the critical point $-a$ maps to the landing point of the dynamical ray
at angle $0$, and the other critical point $a$ is in the filled Julia set.
Douady and Branner show that there exists a homeomorphism $\Phi_{B}$ from the
basilica limb of the Mandelbrot set to $F_{+}$. The relationship between
$\mathcal{CBO}_{1}^{(+,-)}$ and $F_{+}$ is as follows:

$\displaystyle\big{\\{}\big{(}\widetilde{a},2{\widetilde{a}}^{3}-2\widetilde{a}\big{)}\big{|}a\in\mathcal{CBO}_{1}^{(+,-)},\widetilde{a}=9a^{2}\big{\\}}\subsetneq
F_{+}$

For $d=1$, the map $\Phi_{d}$ we construct in this paper exhibits different
behaviour from the $\Phi_{B}$ that the authors construct in [BD88, Chapter
II]. Firstly, it is defined on the whole of the Mandelbrot set, and not just
the basilica limb. Secondly, generally, given $c$ in the basilica limb, if
$Q_{\widetilde{a},2\widetilde{a}^{3}-2\widetilde{a}}$ is the polynomial
corresponding to $\Phi_{B}(c)$, $\widetilde{a}$ does not equal
$9\Phi_{d}(c)^{2}$. Thirdly, it is evident that our map does not change the
combinatorics of critical portraits, whereas $\Phi_{B}$ does.

###  2.5. Properties of $\mathcal{CBO}_{d}$

Let $\mathcal{MBO}_{d}$ denote the set of $s$ such that $P_{s}$ has connected
filled Julia set. Using the methods used by Douady and Hubbard in their proof
that the Mandelbrot set is connected (see, for example, [Hub16, Chapter 10]),
we find that the map

$\displaystyle\hat{\mathbb{C}}\setminus\mathcal{MBO}_{d}$
$\displaystyle\longrightarrow\hat{\mathbb{C}}\setminus\overline{\mathbb{D}}$
$\displaystyle c$ $\displaystyle\mapsto\varphi_{c}(P_{s}(-s\sqrt{d}))$
$\displaystyle\infty$ $\displaystyle\mapsto\infty$

is a proper map of degree $2d$ ramified at $\infty$. This implies
$\mathcal{MBO}_{d}$, and consequently $\mathcal{CBO}_{d}$, is connected and
compact.

###  2.6. Proof Strategy and Tools

We will use all the theorems listed in this section. Their statements are
borrowed from [DH85b].

######  Theorem 2.1.

(The Measurable Riemann Mapping Theorem) Let $\mu_{0}$ be the standard complex
structure on $\mathbb{C}$. If $\mu$ is a complex structure on a simply
connected domain $U\subset\mathbb{C}$ that has bounded dilitation with respect
to $\mu_{0}$, then there exists a quasiconformal map $f:U\longrightarrow
V\subset\hat{\mathbb{C}}$ satisfying

$\displaystyle f^{*}\mu_{0}=\mu$

unique up to post composition by a Möbius transformation.

  1. (1)

Let $\mu_{n}$ be a sequence of Beltrami forms on a bounded domain
$U\subset\mathbb{C}$ such that $||\mu_{n}||_{\infty}\leq m<1$ and
$\mu_{n}\longrightarrow\mu$ in the $L^{1}$ norm, where $\mu$ is a Beltrami
form on $U$ with $||\mu||_{\infty}\leq m$. There exists a sequence of
integrating maps $\phi_{n}$ for $\mu_{n}$ and an integrating map $\phi$ for
$\mu$ such that $\phi_{n}\longrightarrow\phi$ uniformly on $U$.

  2. (2)

Let $\Lambda$ be an open set in $\mathbb{C}^{n}$ and
$(\mu_{\lambda})_{\lambda\in\Lambda}$ be a family of Beltrami forms on $U$.
Suppose $\lambda\mapsto\mu_{\lambda}(z)$ is holomorphic for almost every $z\in
U$ , and that there exists a constant $m<1$ such that
$||\mu_{\lambda}||_{\infty}<m$ for each $\lambda$. For each $\lambda$, extend
$\mu_{\lambda}$ to $\mathbb{C}$ by $\mu_{\lambda}=0$ on $\mathbb{C}\setminus
U$, and let $f_{\lambda}:\mathbb{C}\ \longrightarrow\mathbb{C}$ be the unique
quasi-conformal homeomorphism such that
$f_{\lambda}^{*}\mu_{0}=\mu_{\lambda}$, and
$\frac{f_{\lambda}(z)}{z}\longrightarrow 1$ when $|z|\longrightarrow\infty$.
Then $(\lambda,z)\mapsto(\lambda,f_{\lambda}(z))$ is a homeomorphism of
$\Lambda\times\mathbb{C}$ onto itself, and for each $z\in\mathbb{C}$ the map
$\lambda\mapsto f_{\lambda}(z)$ is holomorphic.

######  Definition 2.2 (Polynomial-like maps).

Given Jordan domains $U,V\subset\mathbb{C}$ with $\overline{U}\subset V$, a
polynomial-like map $f:U\longrightarrow V$ is an analytic proper map of finite
degree $d$.

The filled Julia set of $f$ is the set

$\displaystyle K_{f}$ $\displaystyle=\bigcap_{n\geq 0}f^{\circ n}(U)$

Given a polynomial $p:\hat{\mathbb{C}}\longrightarrow\hat{\mathbb{C}}$ of
degree $d$, we can always find suitable domains $U,V$ such that
$\overline{U}\subset V$ and $p\big{|}_{U}:U\longrightarrow V$ is polynomial-
like of degree $d$.

######  Definition 2.3 (Hybrid Equivalence).

Given two polynomial like maps $f:U\longrightarrow V$ and
$g:U^{\prime}\longrightarrow V^{\prime}$, we say that $f,g$ are hybrid
equivalent if there exists a quasiconformal homeomorphism
$\psi:(V,U)\longrightarrow(V^{\prime},U^{\prime})$ satisfying
$g\circ\psi=\psi\circ f$, with zero dilitation on $K_{f}$.

The following theorem is due to Douady and Hubbard, and we shall be using it
several times.

######  Theorem 2.4 (The Straightening Theorem for polynomial-like maps).

Every polynomial-like map of degree $d$ is hybrid equivalent to a polynomial
of degree $d$.

Our strategy for constructing $\Phi_{d}$ follows the general layout in [BD88].
Given $c\in\mathcal{M}_{d}$, we will perform a topological surgery in the
dynamical plane using the dynamics around one of the $\beta-$ fixed points. At
the end of this surgery, we will construct a quasiregular map $g_{c}$ from a
simply-connected Riemann surface $X_{1}$ to a simply connected Riemann surface
$X$ with $\overline{X}_{1}\subset X$.

Next, we will show that $g_{c}$ has an invariant complex structure, and is
therefore equivalent to a polynomial-like map of degree $2d+1$. We will
finally show that this map is hybrid equivalent to an odd polynomial $p_{a}$
with $a\in\mathcal{CBO}_{d}$. The strategy for constructing an inverse for
$\Phi_{d}$ is similar.

All the figures in this paper are illustrations of the case $d=2$.

Figure 5. The dynamical plane of $z^{3}+c$ Figure 6. Cutting the dynamical
plane of $z^{3}+c$.

##  3\. Construction of $\Phi_{d}$

As mentioned before, we proceed along the lines of holomorphic surgery as
outlined in [BD88]. Let $f_{c}(z)=z^{d+1}+c$, and $\varphi_{c}$ be a Böttcher
chart at $\infty$ that satisfies
$\lim_{z\rightarrow\infty}\frac{\varphi_{c}(z)}{z}=1$. In the absence of the
latter condition, $\varphi_{c}$ is unique only upto multiplication by a $d$th
root of unity. By including the condition, we fix a choice of $\varphi_{c}$
for every $c$ that makes it continuous in the following sense: given
$c\in\mathbb{C}$ and $z\in\mathbb{C}$ such that $\varphi_{\tilde{c}}(z)$ is
well-defined for $\tilde{c}$ in a neighborhood of $c$,
$\tilde{c}\mapsto\varphi_{c}(z)$ is continuous in $\tilde{c}$.

Now fix $c\in\mathcal{M}_{d+1}$. The dynamical ray $\mathcal{R}_{0}$ at angle
$0$ lands at a fixed point $\beta$ on the dynamical plane of $c$. For a fixed
$r>0$, choose $q,\eta$ such that $q\eta<r$. We will explain how to choose $r$
in the following passages. Let $G_{c}$ denote the Green’s escape rate
function.
Let

$\displaystyle W_{0}$ $\displaystyle=\\{z:G_{c}(z)<\eta\\}$

and $W_{i}=f_{c}^{-1}(W_{0})$. We note that

$\displaystyle W_{i}$
$\displaystyle=\Biggl{\\{}z:G_{c}(z)<\frac{\eta}{(d+1)^{i}}\Biggl{\\}}$

Also define

$\displaystyle\widetilde{S}(0)$ $\displaystyle=\\{\varphi_{c}^{-1}(e^{s+2\pi
it}):s\in(0,\eta),|t|<qs\\}$

We call $\widetilde{S}(0)$ a “sector” based at $\beta$. It is invariant under
$f_{c}$, and its inverse image under $f_{c}$ is a union of similar sectors,
each based at a pre-image of $\beta$. More precisely, for
$\ell\in\\{1,2,...,d\\}$, let

$\displaystyle\widetilde{S}\Big{(}\frac{\ell}{d+1}\Big{)}$
$\displaystyle=\Big{\\{}\varphi_{c}^{-1}(e^{s+2\pi
it}):s\in(0,\eta),\Big{|}t-\frac{\ell}{d+1}\Big{|}<qs\Big{\\}}\subset W_{0}$

Then

$\displaystyle f_{c}^{-1}(\widetilde{S}(0))$
$\displaystyle=\bigcup_{\ell=0}^{d}\Big{(}W_{1}\cap\widetilde{S}\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}$

Figure 7. The Riemann Surface $\widetilde{X}$

When imposing the condition $q\eta<r$, we choose $r$ small enough so that the
sectors $\widetilde{S}(\frac{\ell}{d+1})$, $\ell=0,1,...,d$, are pairwise
disjoint (see Figure 5).

Additonally, form open subsets
$\widetilde{S_{i}}(0)=W_{i}\cap\widetilde{S}(0)$ of $\widetilde{S}(0)$. All
points in $\widetilde{S}_{i}(0)$ have escape rates in the interval
$\Big{(}0,\frac{\eta}{(d+1)^{i}}\Big{)}$, and $f_{c}$ maps
$\widetilde{S}_{i}(0)$ conformally onto $\widetilde{S}_{i-1}(0)$. By
definition, there is a branch of $\log$ that satisfies

$\log\circ\varphi_{c}(\widetilde{S}(0))=\\{z:Re(z)>0\text{ and }|Im(z)|<2\pi
qRe(z)\\}$

###  3.1. Steps in the definition of $\Phi_{d}(c)$

As in [BD88, Chapter II], we shall follow this sequence of steps:

  1. (1)

First, we cut along $\mathcal{R}_{0}$ and glue together two copies of $W_{0}$,
one rotated by $180^{\circ}$, to get a quotient Riemann surface
$\widetilde{X}$

  2. (2)

We then construct $f$ on an open subset of $\widetilde{X}$ that is

     * •

analytic and acts like $f_{c}$ away from the sectors
$\widetilde{S}\big{(}\frac{\ell}{d+1}\big{)}$, $\ell\in\\{0,1,2,...,d\\}$ on
both copies of $W_{0}$

     * •

has lines of discontinuities at the two copies of
$\mathcal{R}_{\frac{\ell}{d+1}}$ for $\ell\in\\{1,2,...,d\\}$

  3. (3)

We show that by changing $f$ in sectors around these rays, and by modifying
the boundary of these sectors, we may construct a quasiregular map
$g:X_{1}\longrightarrow X$ between simply connected Riemann surfaces with
$\overline{X_{1}}\subset X$, and an almost complex structure $\sigma$ on $X$
that is $g$ invariant. Under the measurable Riemann mapping theorem, there
exists a quasi-conformal map $\psi$ such that $\psi\circ g\circ\psi^{-1}$ is
analytic.

  4. (4)

Finally, we will apply the straightening theorem to obtain a unique polynomial
$p_{a}$ hybrid equivalent to $\psi\circ g\circ\psi^{-1}$.

We will now implement these steps one by one.

####  3.1.1. Cutting along $\mathcal{R}_{0}$

Let us cut along $\mathcal{R}_{0}$. In this slit disk, $\widetilde{S}(0)$ is
now split into two components $\widetilde{S}_{1}$ and $\widetilde{S}_{2}$; we
will call the copy of $\mathcal{R}_{0}$ bounding $\widetilde{S}_{1}$
$\mathcal{R}_{0}^{(A)}$ and the one bounding $\widetilde{S}_{2}$,
$\mathcal{R}_{0}^{(B)}$. Every $x\in\mathcal{R}_{0}$ now has two copies
$x^{(A)}$ and $x^{(B)}$.

Consider a second copy of this slit $W_{0}$, and rotate it by $\pi$. We will
accent all objects in this (slit) second copy with a - superscript), and all
objects in the original copy with a + superscript. Glue the slit copies
$W_{0}^{+},W_{0}^{-}$ together using the following rule:

$\displaystyle\forall x\in\mathcal{R}_{0},\hskip 10.0ptx^{(A^{+})}$
$\displaystyle\sim x^{(B^{-})}$ $\displaystyle x^{(B^{+})}$ $\displaystyle\sim
x^{(A^{-})}$

This gives a quotient map

$\displaystyle\pi:W^{+}_{0}\sqcup W_{0}^{-}\longrightarrow W^{+}_{0}\sqcup
W_{0}^{-}/\sim$

This quotient surface can be endowed with a Riemann surface structure that
makes $\pi$ analytic away from $\beta^{\pm}$. We can think of $\widetilde{X}$
as an open subset of the branched cover over $\mathbb{C}$ corresponding to
$w\mapsto w^{2}+\beta$, and $\pi$ as a branch of $\sqrt{z-\beta}$ on each of
the slit copies $W_{0}^{+}$ and $W_{0}^{-}$.

Figure 8. The set
$\Delta_{q}\subset\log\circ\varphi_{c}(\widetilde{S}_{1}(0))$. We will
eventually define a map that is conformal on the white and lightly shaded
regions, and quasiconformal on the darkly shaded region

We name this Riemann surface $\widetilde{X}$, and note that $\widetilde{X}$
has smooth boundary. Let $\widetilde{X_{1}}=\pi(W^{+}_{1}\sqcup W_{1}^{-})$.
Then $\widetilde{X_{1}}$ is an open subset of $X$. We also define the
“sectors” $\widetilde{A}$ and $\widetilde{B}$ as follows:

$\displaystyle\widetilde{A}$
$\displaystyle=\pi(\widetilde{S}_{1}^{-}\cup\widetilde{S}_{2}^{+})$
$\displaystyle\widetilde{B}$
$\displaystyle=\pi(\widetilde{S}_{2}^{-}\cup\widetilde{S}_{1}^{+})$

See Figure 7 for an illustration.

######  Remark 3.1.

We could have performed our cut and paste surgery by cutting along
$\mathcal{R}_{\frac{j}{d}}$ for any $i\in\\{0,1,...,d-1\\}$ (the landing
points of these rays are precisely the $\beta$ fixed points of $f_{c}$). To
get a continuous embedding $\Phi_{d}$ of $\mathcal{M}_{d+1}$, however, we will
use the same $j$ for all $c\in\mathcal{M}_{d+1}$.

####  3.1.2. Constructing a map $f$ on a subset of $\widetilde{X}_{1}$

For $z\in\pi(W_{1}^{\pm})$, define

$\displaystyle f(\pi(z))$
$\displaystyle=\begin{cases}\pi(f_{c}(z))&z\not\in\mathcal{R}_{0}^{A^{\pm}}\cup\mathcal{R}_{0}^{B^{\pm}}\\\
\pi(f_{c}(z^{(A^{+})}))=\pi(f_{c}(z^{(B^{-})}))&z\in(\mathcal{R}_{0}^{(A^{+})}\cap
W_{1}^{+})\cup(\mathcal{R}_{0}^{(B^{-})}\cap W_{1}^{-})\\\
\pi(f_{c}(z^{(B^{+})}))=\pi(f_{c}(z^{(A^{-})}))&z\in(\mathcal{R}_{0}^{(B^{+})}\cap
W_{1})\cup(\mathcal{R}_{0}^{(A^{-})}\cap W_{1}^{-})\\\
\pi(\beta^{+})=\pi(\beta^{-})&z\in\\{\beta,\beta^{-}\\}\end{cases}$

For $\ell\in\\{1,2,...,d\\}$, $f$ is not well defined on
$\pi(\mathcal{R}^{\pm}(\frac{\ell}{d+1}))$ and we cannot extend it over any of
these rays continuously since one component of the complement of such a ray in
$\pi(\widetilde{S}^{\pm}_{i-1}(\frac{\ell}{d+1}))$ is mapped to
$\widetilde{A}$, and the other is mapped to $\widetilde{B}$.

Figure 9. The Riemann map $\widetilde{h}$ maps the lightly shaded region
$T_{1}(0)$ to the region $V_{0}$ (the non-white region at the bottom). The
darkly shaded region in the bottom figure is $f_{c}(Y_{1}(0))$

However, on the complement in $\widetilde{X}_{1}$ of the rays above , $f$ is
analytic.

####  3.1.3. A new map on some sectors

By our definition of sectors, note that
$\widetilde{S}\Big{(}\frac{\ell}{d+1}\Big{)}=\omega^{\ell}\tilde{S}(0)$.

Our strategy will be to produce a quasiregular map $g$ that agrees with $f$ on
the complement of the sets
$\pi\Big{(}\widetilde{S}_{1}^{\pm}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ for
$\ell=1,2,...,d$. We let

$\displaystyle\widetilde{S}_{i}\Big{(}\frac{\ell}{d+1}\Big{)}$
$\displaystyle=\widetilde{S}\Big{(}\frac{\ell}{d+1}\Big{)}\cap W_{i}$

$f_{c}$ maps $\widetilde{S}_{i}(\frac{\ell}{d+1})$ conformally to
$\widetilde{S}_{i-1}(0)$.

Choose $p,q^{\prime}$ such that $0<p<q^{\prime}<q$, and consider the set
$\Delta_{q}$ in $\log$ Böttcher coordinates as illustrated in Figure 8. Its
boundary is defined so that it is smooth away from the points
$0,\log{(\frac{\eta}{d+1})}(1\pm 2\pi qi)$, and such that it coincides with an
arc of the circle $x^{2}+y^{2}=\frac{\eta^{2}}{(d+1)^{2}}$ on the two
connected regions region bounded by $y=\pm 2\pi px$ and $y=\pm 2\pi
q^{\prime}x$. We will also require the boundary of $\Delta_{q}$ to be
symmetric about the $x$ \- axis in Figure 8. Additionally, let

$\displaystyle\Delta_{q^{\prime}}$ $\displaystyle=\Delta_{q}\cap\\{|y|<2\pi
q^{\prime}x\\}$ $\displaystyle\Delta_{p}$
$\displaystyle=\Delta_{q}\cap\\{|y|<2\pi px\\}$

For $\ell\in\\{0,1,...,d\\}$, define

(1) $\displaystyle S_{1}\Big{(}\frac{\ell}{d+1}\Big{)}$
$\displaystyle=\omega^{\ell}\varphi_{c}^{-1}\circ\exp(\Delta_{q})$ (2)
$\displaystyle Q_{1}\Big{(}\frac{\ell}{d+1}\Big{)}$
$\displaystyle=\omega^{\ell}\varphi_{c}^{-1}\circ\exp(\Delta_{q^{\prime}})$
(3) $\displaystyle T_{1}\Big{(}\frac{\ell}{d+1}\Big{)}$
$\displaystyle=\omega^{\ell}\varphi_{c}^{-1}\circ\exp(\Delta_{p})$ (4)
$\displaystyle Y_{1}\Big{(}\frac{\ell}{d+1}\Big{)}$
$\displaystyle=\omega^{\ell}\varphi_{c}^{-1}\circ\exp(\Delta_{q}\setminus\overline{\Delta_{q^{\prime}}})=S_{1}\Big{(}\frac{\ell}{d+1}\Big{)}\setminus\overline{Q_{1}\Big{(}\frac{\ell}{d+1}\Big{)}}$

Clearly,

$\displaystyle T_{1}\big{(}\frac{\ell}{d+1}\big{)}\subset
Q_{1}\big{(}\frac{\ell}{d+1}\big{)}\subset
S_{1}\big{(}\frac{\ell}{d+1}\big{)}\subset\widetilde{S}_{1}\big{(}\frac{\ell}{d+1}\big{)}$

On the slit disk $W_{0}\setminus\mathcal{R}_{0}$, we define $V_{0}$ as
follows:

(5) $\displaystyle V_{0}$
$\displaystyle=f_{c}\Big{(}W_{1}\setminus\bigcup_{\ell=0}^{d}S_{1}\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}\cup
f_{c}(Y_{1}(0))$

See Figure 9 for details.

Figure 10. The darkly shaded region above indicates the set $\Delta$\- the
image under $\hat{h}$ of a connected component of
$\Delta_{q^{\prime}}\setminus\overline{\Delta_{p}}$

By the Riemann mapping theorem, there exist analytic maps
$\widetilde{k}:\mathbb{D}\longrightarrow V_{0}$ and
$\widetilde{m}:T_{1}(0)\longrightarrow\mathbb{D}$ such that

$\displaystyle\widetilde{k}(\widetilde{m}(\beta))$ $\displaystyle=\beta$
$\displaystyle\widetilde{k}(\widetilde{m}(z_{1}))$
$\displaystyle=\widetilde{z}_{1}$
$\displaystyle\widetilde{k}(\widetilde{m}(z_{2}))$
$\displaystyle=\widetilde{z}_{2}$

Let $\widetilde{h}=\widetilde{k}\circ\widetilde{m}$. The map
$\widetilde{h}:T_{1}(0)\longrightarrow V_{0}$ is the unique analytic function
that sends the triple $(\beta,z_{1},z_{2})$ to
$(\beta,\widetilde{z}_{1},\widetilde{z}_{2})$ (see Figure 9). $\partial
V_{0},\partial T_{1}(0)$ are quasi-circles. Therefore,
$\widetilde{k},\widetilde{m}$ extend to quasisymmetric maps on the boundaries
of their respective domains. Furthermore, by [Pom92, Chapter 3.4, Exercise 1],

$\displaystyle\widetilde{k}(z)$
$\displaystyle=\beta+a_{k}(z-1)^{2-4q^{\prime}}+O(|z-1|^{2-4q^{\prime}+(2-4q^{\prime})\gamma_{k}})$
$\displaystyle\widetilde{m}^{-1}(z)$
$\displaystyle=\beta+a_{m}(z-1)^{4p}+O(|z-1|^{4p+4p\gamma_{k}})$

for some $\gamma_{k},\gamma_{m}\in(0,1),a_{k},a_{m}\in\mathbb{C}$.

It then follows that

$\displaystyle\widetilde{h}(z)=\beta+a_{h}\big{(}z-\beta\big{)}^{\frac{1-2q^{\prime}}{2p}}+O\Big{(}\big{|}z-\beta\big{|}^{\frac{1-2q^{\prime}}{2p}+\big{(}\frac{1-2q^{\prime}}{2p}\big{)}\gamma_{h}}\Big{)}$

for some $\gamma_{h}\in(0,1),a_{h}\in\mathbb{C}$. Since $\varphi_{c}$ does not
distort angles, conjugating $\tilde{h}$ by $\varphi_{c}$ should not change
this equation, and we have the following:

######  Proposition 3.2.

For all $z\in\exp(\Delta_{p})=\varphi_{c}(T_{1}(0))$,

(6)
$\displaystyle\varphi_{c}\circ\widetilde{h}\circ\varphi_{c}^{-1}(z)=1+a^{\prime}\big{(}z-1\big{)}^{\frac{1-2q^{\prime}}{2p}}+O\Big{(}\big{|}z-1\big{|}^{\frac{1-2q^{\prime}}{2p}+(\frac{1-2q^{\prime}}{2p})\gamma}\Big{)}$

for some $\gamma\in(0,1),a^{\prime}\in\mathbb{C}\setminus\\{0\\}$.

Let $G$ be a connected component of
$\Delta_{q^{\prime}}\setminus\overline{\Delta_{p}}$, say the one bounded by
$y=2\pi q^{\prime}x$ and $y=2\pi px$. The polynomial $f_{c}$ induces the map
$\mu_{d+1}(z)=(d+1)z$ on the part of its boundary where $y=2\pi q^{\prime}x$.
The map
$\hat{h}=(\log\circ\varphi_{c})\circ\widetilde{h}\circ(\varphi_{c}^{-1}\circ\exp)$
extends to a continuous map on the part of the boundary where $y=2\pi px$.

Let $\Delta$ be the set
$\Delta_{q^{\prime}}\cap\\{x^{2}+y^{2}<\eta^{2}\\}$(see Figure 10 for an
illustration of $\Delta$). $S(0):=\varphi_{c}^{-1}\circ\exp(\Delta)$ is an
open subset of $\widetilde{S}(0)$. For $\ell\in\\{1,2,...,d\\}$, let
$S(\frac{\ell}{d+1})=\omega^{\ell}S(0)$. The sector
$S\big{(}\frac{\ell}{d+1}\big{)}$ contains
$S_{1}\big{(}\frac{\ell}{d+1}\big{)}$.

The following is a crucial lemma.

Figure 11. The map $H$ is defined by mapping vertical lines to lines joining
the images of their endpoints

######  Lemma 3.3.

There exists a quasiconformal map from $G$ to $\Delta$ that restricts to
$\mu_{d+1}$ on one boundary, and to $\hat{h}$ on the other boundary.

###### Proof.

$G$ has a positive angle at the vertex $0$. In $\log$ coordinates,
$G^{\prime}=\log G$ is a half-infinite horizontal strip with
$\hat{\mu}_{d+1}(z)=z+\log(d+1)$ induced by $\mu_{d+1}$ on the part of the
boundary where $y=\arctan(2\pi q^{\prime})$, and $H(z)=\log\hat{h}(e^{z})$ on
the part of the boundary where $y=\arctan(2\pi p)$.

We will interpolate between $\hat{\mu}_{d+1}$ and $H$ by mapping vertical
lines in $G^{\prime}$ to lines joined by the images of the endpoints. If we
can show that these image lines have uniformly bounded slope, the resulting
map will be quasiconformal. We explain this in detail below.

Set $\theta_{p}=\arctan(2\pi p),\theta_{q^{\prime}}=\arctan(2\pi q^{\prime}).$
We will define $H$ on $\overline{G^{\prime}}$ by extending along vertical
lines:

$\displaystyle
H\Big{(}x+i\big{(}(1-t)\theta_{p}+t\theta_{q^{\prime}}\big{)}\Big{)}$
$\displaystyle=(1-t)H\big{(}x+i\theta_{p}\big{)}+t\hat{\mu}_{d+1}\big{(}x+i\theta_{q^{\prime}}\big{)}$

######  Proposition 3.4.

There exists $R>0$ such that for all $z\in\\{Im(z)=\arctan(2\pi
p)\\}\cap\overline{G^{\prime}}$,

(7) $\displaystyle|H(z)-z|\leq R$

###### Proof.

We will prove this by showing that both $z\mapsto z-H(z)$ and $z\mapsto
H(z)-z$ are bounded above.

Suppose $z-H(z)$ is not bounded above, then for each natural number $n$, there
exists $z_{n}$ such that

$\displaystyle z_{n}-H(z_{n})$ $\displaystyle>n$

and upto a subsequence, the $z_{n}$ tend to $-\infty$. But this implies that

$\displaystyle|\hat{h}(u_{n})|$ $\displaystyle<\frac{|u_{n}|}{e^{n}}$

where $u_{n}=\exp z_{n}$. Furthermore,

$\displaystyle Re(\hat{h}(u_{n}))$ $\displaystyle\leq|\hat{h}(u_{n})|$
$\displaystyle\leq\frac{|u_{n}|}{e^{n}}=\frac{1}{e^{n}}\sqrt{Re(u_{n})^{2}(1+4\pi^{2}p^{2})}$
$\displaystyle\leq C\frac{|Re(u_{n})|}{e^{n}}=C\frac{Re(u_{n})}{e^{n}}$

for some constant $C>0$.

Set $w_{n}=\exp{u_{n}}$, and note that
$\exp\hat{h}(u_{n})=\varphi_{c}\circ\widetilde{h}\circ\varphi_{c}^{-1}(w_{n})$.
Thus

(8)
$\displaystyle|\varphi_{c}\circ\widetilde{h}\circ\varphi_{c}^{-1}(w_{n})|<b|w_{n}|^{\frac{1}{e^{n}}}$

for some $b>0$.

But Equation 8 implies that
$|\varphi\circ\widetilde{h}\circ\varphi_{c}^{-1}(w_{n})|$ converges much
faster to $1$ than allowed by Equation 6, and forms a contradiction. This
proves that $z-H(z)$ is bounded above.

Similarly, suppose $H(z)-z$ is not bounded above as $z\rightarrow-\infty$,
there exists a sequence $z_{n}\rightarrow-\infty$ such that

$\displaystyle H(z_{n})-z_{n}$ $\displaystyle>n$

Thus

$\displaystyle|\hat{h}(u_{n})|$ $\displaystyle>|u_{n}|e^{n}$

Consequently, we have

$\displaystyle Re\hat{h}(u_{n})$
$\displaystyle=\frac{|\hat{h}(u_{n})|}{\sqrt{1+4\pi^{2}q^{\prime 2}}}$
$\displaystyle>\frac{|u_{n}|e^{n}}{\sqrt{1+4\pi^{2}q^{\prime 2}}}$
$\displaystyle=\frac{\sqrt{1+4\pi^{2}p^{2}}}{\sqrt{1+4\pi^{2}q^{\prime
2}}}Re(u_{n})e^{n}$

But this gives us

$\displaystyle|\varphi_{c}\circ\widetilde{h}\circ\varphi_{c}^{-1}(w_{n})|>\iota|w_{n}|^{e^{n}}$

for some constant $\iota>0$, which also contradicts Equation 6.

This proves Proposition 3.4. ∎

Figure 12. An illustration of $h$ on
$\pi\big{(}S_{1}\big{(}\frac{1}{3}\big{)}^{+}\big{)}$; the dark components
above are quasiconformally mapped to the dark components below, the lightly
shaded region maps by the Riemann map $\omega^{-1}\widetilde{h}$, and the
white region maps by $f$

It is clear that $H$ interpolates between the maps on the two horizontal
boundaries, and that
$H(G^{\prime}\cap\\{Re(z)=\log(\frac{\eta}{3})\\})=\\{\log\eta+iy:|y|<\arctan(2\pi
q^{\prime})\\}$. Furthermore, $H$ is a quasiconformal map whose dilitation is
bounded above by some $M\geq 1$ (see Figure 11): this is because vertical
lines in the domain are mapped by $H$ to lines whose slopes are bounded below
by some uniform constant, by Equation 7. We conjugate $H$ by the exponential
map to obtain a quasiconformal map $\hat{h}$ from $G$ that satisfies the
properties in the statement of Lemma 3.3. ∎

$G$ could be taken to be either connected component of
$\Delta_{q^{\prime}}\setminus\overline{\Delta_{p}}$. On the dynamical plane,
it corresponds to a component $\mathcal{G}$ of
$Q_{1}(0)\setminus\overline{T_{1}(0)}$. We shall henceforth denote the copy of
$S(0)$ in $\widetilde{A}$ as $A$, and the copy in $\widetilde{B}$ as $B$. With
this in mind, we will take $h_{\mathcal{G},A}$ to mean the map
$(\omega^{\ell}\circ\varphi_{c}^{-1}\circ\exp)\circ\hat{h}\circ(\log\circ\varphi_{c}\circ\omega^{-\ell})$
from the component $\mathcal{G}$ of
$Q_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T_{1}\big{(}\frac{\ell}{d+1}\big{)}$ to $A$, and $h_{\mathcal{G},B}$ to mean
the same map, but from $\mathcal{G}$ to $B$. We will use the same names for
the extended maps from $\overline{\mathcal{G}}$.

####  3.1.4. Constructing a quasiregular map $g$

Let $S$ be a sector of the form
$\pi\Big{(}S^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$, where
$\ell\in\\{1,2,...,d+1\\}$. The map $f$ has a line of discontinuities in $S$
along the ray $\pi\Big{(}\mathcal{R}^{+}_{\frac{\ell}{d+1}}\Big{)}$\- on one
side of this ray, $f$ maps into $\widetilde{A}$ and approaches
$\pi(\mathcal{R}_{0}^{\widetilde{A}})$, whereas on the other side, $f$ maps
into $\widetilde{B}$ and approaches $\pi(\mathcal{R}_{0}^{\widetilde{B}})$.
Define a map $h$ on $\overline{S}$ as follows:

  * •

On $\pi\Big{(}\overline{T^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}}\Big{)}$, let
$h(\pi(z^{+}))=\pi(\widetilde{h}(\omega^{-\ell}z)^{-})\in\pi(V_{0}^{-})$.

  * •

On $\pi\Big{(}\overline{Y_{1}^{+}\big{(}\frac{\ell}{d+1}\big{)}}\Big{)}$, let
$h(\pi(z^{+}))=f(\pi(z^{+}))$, where $f$ is defined as in Section 3.1.2.

  * •

On the connected component $\mathcal{G}$ of
$\pi\Big{(}Q^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ with $\ell=1,2,...,d$ part of
whose boundary $f$ maps into $\partial\widetilde{A}$, let
$h(\pi(z^{+}))=h_{\mathcal{G},A}(z)$.

  * •

On the connected component $\mathcal{G}$ of
$\pi\Big{(}Q^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ with $\ell=1,2,...,d$ part of
whose boundary $f$ maps into $\partial\widetilde{B}$, let
$h(\pi(z^{+}))=h_{\mathcal{G},B}(z)$.

(a) The Riemann surfaces $X_{1}$, $X$

(b) The quasiregular map $g$

Figure 13. The dynamics of a quasiregular model for $\Phi_{d}(c)$

The map $h$ so defined is a quasiconformal homeomorphism from $S$ to
$\pi(V_{0}^{-})\cup A\cup B$ (see Figure 12 for an illustration of $h$ on
$\pi\Big{(}S\big{(}\frac{1}{3}\big{)}^{+}\Big{)}$ when $d=2$), and restricts
to an analytic map on $\pi\Big{(}T_{1}\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}$.

Furthermore, the latter set has smooth boundary at the points
$\pi(\widetilde{z}_{1}^{-})$ and $\pi(\widetilde{z_{2}}^{-})$ in Figure 12:
consider $\pi(\widetilde{z}_{1}^{-})$ for instance. In Böttcher coordinates,
the boundary in a neighborhood of $\widetilde{z}_{1}$ looks like
$f_{c}(\varphi_{c}^{-1}\circ\exp(\gamma))$, where $\gamma$ is a neighborhood
of the boundary of $\Delta_{q}$ at the point $\log\circ\varphi_{c}(z_{1})$;
$\gamma$ is clearly smooth.

On $\pi\Big{(}S_{1}^{-}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$, we define $h$
the same way, except with the following change: on
$\pi\Big{(}\overline{T_{1}^{-}\big{(}\frac{\ell}{d+1}\big{)}}\Big{)}$, let
$h(\pi(z^{-}))=\pi(\widetilde{h}(\omega^{-\ell}z)^{+})\in\pi(V_{0}^{+})$. This
$h$ is a quasiconformal hoemoemorphism from $S$ to $\pi(V_{0}^{+})\cup A\cup
B$. Finally, we construct a quasiregular map on newly defined subsets of
$\widetilde{X}_{1},\widetilde{X}$.
Let

$\displaystyle X$ $\displaystyle=\pi(V_{0}^{+})\cup A\cup B\cup\pi(V_{0}^{-})$

Also let $X_{1}$ be the subset of $\widetilde{X_{1}}$ where all sectors of the
form $\pi\Big{(}\widetilde{S}^{\pm}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$,
$\pi\Big{(}\widetilde{S}^{-}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ are
replaced by $\pi\Big{(}S_{1}^{\pm}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ for
$\ell=1,2,..,d$, and let $X$ be the open subset of $\widetilde{X}$ where
$\widetilde{A}$ and $\widetilde{B}$ are replaced by $A$, $B$ respectively. See
Figure 13(a) for details. Clearly, $X_{1}$ is an open subset of the Riemann
surface $X$ compactly contained in $X$. We define

$\displaystyle g$ $\displaystyle:X_{1}\longrightarrow X$ $\displaystyle
g(\pi(z))$
$\displaystyle=\begin{cases}f(\pi(z))&z\in\Big{(}W^{+}_{1}\setminus\bigcup_{\ell}S^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}\cup\Big{(}W^{-}_{1}\setminus\bigcup_{\ell}S^{-}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)},\ell=1,2,...,d\\\
h(\pi(z))&z\in S^{\pm}_{1}\big{(}\frac{\ell}{d+1}\big{)},\ell=1,2,...,d\\\
\end{cases}$

See Figure 13(b) for an illustration of $g$.

$g$ is quasiregular. Furthermore, any $g$ orbit visits
$\pi\Big{(}Q^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ or
$\pi\Big{(}Q^{-}_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T^{-}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ at most once, and these are
the only regions where $g$ is not analytic. We will use this fact to define a
new complex structure $\sigma$ (given by an ellipse field $E_{x}$ for $x\in
X$) by setting

  * •

$E_{x}=\mathbb{S}^{1}$ if $x\in X\setminus X_{1}$ or if the orbit of $x$ never
visits $\pi\Big{(}Q^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ or
$\pi\Big{(}Q^{-}_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T^{-}_{1}(\frac{\ell}{d+1}\big{)}\Big{)}$

  * •

$E_{x}=(T_{x}g)^{-1}(\mathbb{S}^{1})$ for
$x\in\pi\Big{(}Q^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T^{+}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ or
$x\in\pi\Big{(}Q^{-}_{1}\big{(}\frac{\ell}{d+1}\big{)}\setminus
T^{-}_{1}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$ for some
$\ell\in\\{1,2,...,d\\}$

  * •

$E_{x}=(T_{x}g^{n})^{-1}(E_{g^{n}(x)})$ if $g^{n}(x)$ is the first point in
the $g-$orbit of $x$ that is in one of the regions above.

The complex structure $\sigma$ thus defined has bounded dilitation, and
$g^{*}\sigma=\sigma$.

####  3.1.5. Obtaining a polynomial

Define the map $\tau:X\longrightarrow X$ by sending $\pi(z^{+})$ to
$\pi(z^{-})$. $\tau$ satisfies $\tau^{*}\sigma=\sigma$, $\tau(X_{1})=X_{1}$,
and $\tau^{\circ 2}=id$. We note that

$\displaystyle g\circ\tau=\tau\circ g$

We find an integrating map $\psi$ for $\sigma$ sending $\pi(\beta^{\pm})$ to
$0$, and satisfying $\frac{\psi(z)}{z}\longrightarrow 1$ as
$z\longrightarrow\infty$. The map $G=\psi\circ
g\circ\psi^{-1}:U^{\prime}\longrightarrow U$ is polynomial-like, and has two
critical points with local degree $d+1$. $\kappa=\psi\circ\tau\circ\psi^{-1}$
an analytic involution of $U$, and commutes with $G$ on $U$. We can further
conjugate by a Riemann map taking the pair $(U,0)$ to $(\mathbb{D},0)$. By the
Schwarz lemma, we can assume without loss of generality that
$\kappa\big{|}_{U}(z)=-z$; in particular, $\kappa$ has a global extension.

$G$ is hybrid equivalent to a degree $2d+1$ polynomial $p$ with two critical
points by the straightening theorem. We may choose this hybrid equivalence
$h:\mathbb{C}\longrightarrow\mathbb{C}$ such that $h(0)=0$. Then
$\delta=h\circ\kappa\circ h^{-1}$ is an affine map of $\mathbb{C}$ with
$\delta(0)=0,\delta^{\circ 2}=id,\delta\neq id$. Therefore, $\delta(z)=-z$.
$p$ commutes with $\delta$, and can now be normalized to the form
$a{\displaystyle\int_{0}^{z}}\Big{(}1-\frac{w^{2}}{d}\Big{)}^{d}dw$ for a
unique $a\in\mathbb{C}^{*}$. The choice of $a$ does not depend on our initial
choice of $q,\eta,\psi$ ($h$ is determined by $c,p,q,q^{\prime},\eta$) - we
can show that different choices give rise to hybrid equivalent polynomials.

###  3.2. The image of $\Phi_{d}$

Clearly, $\Phi_{d}(\mathcal{M}_{d+1})\subset\mathcal{CBO}_{d}$. Let
$a=\Phi_{d}(c)$. By our construction, $0$ is a fixed point of $p_{a}$
belonging to the Julia set, and it disconnects the Julia set into two
components.

Under our surgery, the original dynamical ray $\mathcal{R}_{0}$ landing at
$\beta$ gets transformed into an arc $\Gamma$ from $0$ to $\infty$ in the
dynamical plane of $p_{\Phi_{d}(c)}$ whose interior is contained in the
escaping set. In the monic representation $P_{s(a)}$ of $p_{a}$, $\Gamma$ has
the same access as $\mathcal{R}_{s(a)}(0)$. The union $\Gamma\cup-\Gamma$, and
indeed $\mathcal{R}_{0}(s(a))\cup\mathcal{R}_{\frac{1}{2}}(s(a))$, separates
the orbits of the two critical points of $P_{s(a)}$. That is,
$\Phi_{d}(\mathcal{M}_{d+1})\subset\mathcal{CBO}^{(+,-)}_{d}$. We will show in
Section 5 that the image under $\Phi_{d}$ is equal to this set.

##  4\. Continuity of $\Phi_{d}$

To show continuity of $\Phi_{d}$, we will follow the strategy laid out in
[BD88, Chapter II.8], and show it separately when $c$ is on the boundary, or
in the interior of $\mathcal{M}_{d+1}$. Throughout this section, we shall
index all sets and functions in Section 3 in constructing $\Phi_{d}(c)$ by the
subscript $c$. For example, the projecton $\pi$ is referred to as $\pi_{c}$,
the quasiregular map $g$ as $g_{c}$, the domain of $g_{c}$ as $(X_{1})_{c}$
and so on.

######  Lemma 4.1.

If $p_{a},p_{a^{\prime}}$ with $a,a^{\prime}\in\mathcal{CBO}_{d}$ are hybrid
equivalent, then they are affine conjugate.

###### Proof.

This follows from [DH85b, Chapter I.6, Corollary 2]. ∎

###  4.1. The interior case

If $c\in\mathcal{M}_{d+1}^{\circ}$, then proof is based on the proof of
[DH85b, Chapter II.5, Proposition 12].

######  Definition 4.2.

Given $f_{\lambda}:U_{\lambda}^{\prime}\longrightarrow U_{\lambda}$, for
$\lambda\in\Lambda$, let

$\displaystyle\mathcal{U}^{\prime}$ $\displaystyle=\\{(\lambda,z)|z\in
U_{\lambda}^{\prime}\\}$ $\displaystyle\mathcal{U}$
$\displaystyle=\\{(\lambda,z)|z\in U_{\lambda}\\}$

and define $f:\mathcal{U}^{\prime}\longrightarrow\mathcal{U}$ as
$f(\lambda,z)=f_{\lambda}(z)$. If

  1. (1)

$\mathcal{U}^{\prime},\mathcal{U}$ are homeomorphic over $\Lambda$ to
$\Lambda\times\mathbb{D}$.

  2. (2)

projection of $\overline{\mathcal{U}^{\prime}}$ in $\mathcal{U}$ to $\Lambda$
is proper

  3. (3)

$f$ is holomorphic and proper

then $f_{\lambda}$ is called an analytic family.

Let us go back to the construction of $\Phi_{d}(c)$ from $f_{c}$. We first
construct a quasiregular map $g_{c}:(X_{1})_{c}\longrightarrow X_{c}$. This
map is built from $f_{c}$ away from certain escaping sectors, and from the
Riemann map $h_{c}$ on other sectors. Then we find an invariant complex
structure $\sigma_{c}$ for $g_{c}$ and find integrating maps $\psi_{c}$. This
gives us the polynomial-like family $G_{c}:U^{\prime}_{c}\longrightarrow
U_{c}$.

######  Proposition 4.3.

On a connected component $\Lambda$ of $\mathcal{M}^{\circ}_{d+1}$,
$(c,z)\mapsto(c,G_{c}(z))$ is an analytic family of structurally stable
polynomials.

###### Proof.

We show that $G_{c}$ satisfies the three properties of Definition 4.2.

  1. (1)

$U^{\prime}_{c},U_{c}$ are homeomorphic to $\mathbb{D}$ and
$c^{\prime}\mapsto{U}^{\prime}_{c},c\mapsto U_{c}$ are both continuous maps in
the Hausdorff topology

  2. (2)

Let $\Pi$ be this projection. Given any compact set $K$ in $\Lambda$, and a
sequence $(c_{n},z_{n})\in\Pi^{-1}(K)$, upto a subsequence,
$c_{n}\longrightarrow c\in K$. We note that
$z_{n}\in\overline{U^{\prime}_{c_{n}}}$.
$\overline{U^{\prime}}_{c_{n}}\longrightarrow\overline{U^{\prime}}_{c}$, and
hence, there exists a sequence $\widetilde{z}_{n}\in\overline{U^{\prime}}_{c}$
such that $|z_{n}-\widetilde{z}_{n}|\longrightarrow 0$. Since
$\overline{U^{\prime}}_{c}$ is compact, upto a subsequence,
$\widetilde{z}_{n}\longrightarrow z\in\overline{U^{\prime}}_{c}$. So
$z_{n}\longrightarrow z$ upto the same subsequence. This shows that
$\Pi^{-1}(K)$ is compact.

  3. (3)

By [MnSS83], every parameter $c\in\Lambda$ is structurally stable. More
particularly, given $c\in\Lambda$, there exists a holomorphic motion
$L:\Lambda\times\hat{\mathbb{C}}\longrightarrow\hat{\mathbb{C}}$ such that
$L_{c}=id$, and for all $\widetilde{c}\in\Lambda$, $L_{\widetilde{c}}$ is
quasiconformal and satisfies $L_{\widetilde{c}}\circ f_{c}\circ
L_{\widetilde{c}}^{-1}=f_{\widetilde{c}}$.

Figure 14. Dynamics of $P_{s(a)}$ for $a\in\mathcal{CBO}_{d}^{(+,-)}$

But this also means that $h_{\widetilde{c}}=L_{\widetilde{c}}\circ h_{c}\circ
L_{\widetilde{c}}^{-1}$ on
$\varphi^{-1}_{\widetilde{c}}\circ\exp(\Delta_{p})$, and by definition of the
quasiconformal extension, $h_{\widetilde{c}}=L_{\widetilde{c}}\circ h_{c}\circ
L_{\widetilde{c}}^{-1}$ on the sector $S_{1}\big{(}\frac{\ell}{d+1}\big{)}$ in
the dynamical plane of $f_{\widetilde{c}}$. Therefore, $g_{c}$, and
consequently, $\sigma_{c}$ depend analytically on $c$. By the measurable
Riemann mapping theorem, the integrating maps $\psi_{c}$ depend
holomorphically on $c$.

For a fixed $z\in U^{\prime}_{c}$, when $\widetilde{c}$ is close to $c$,
$G_{\widetilde{c}}(z)$ is well-defined, and $\widetilde{c}\mapsto
G_{\widetilde{c}}(z)=\psi_{\widetilde{c}}\circ
g_{\widetilde{c}}\circ\psi_{\widetilde{c}}^{-1}(z)$ is holomorphic in
$\widetilde{c}$. For a fixed $c$, $z\mapsto G_{c}(z)$ is holomorphic by
definition. Thus $G_{c}(z)$ is holomorphic in both $c$ and $z$; by Hartog’s
theorem, it is holomorphic as a function of $(c,z)$. Proof that $G_{c}(z)$ is
proper is similar to Point 2 above.

∎

In Proposition 4.3, we showed that $(c,z)\mapsto G_{c}(z)$ is an analytic
family over every connected component $\Lambda$ of
$\mathcal{M}^{\circ}_{d+1}$. Given $c\in\Lambda$, let us pick the hybrid
equivalence $k_{{c}}$ conjugating $G_{c}(z)$ to a polynomial in such a way
that it fixes $0$ and satisfies $k_{{c}}(z)/z\longrightarrow 1$. Then, by
[DH85b, Chapter II.5, Proposition 12], the polynomials $k_{{c}}\circ
G_{{c}}\circ k_{{c}}^{-1}$ form a continuous family over $\Lambda$. As proved
in Section 3, these are affine conjugate to bicritical odd polynomials, and
their critical points vary continuously with respect to ${c}$. Hence there
exists a continuous family of scaling maps $M_{{c}}$ that map these critical
points to $\pm\sqrt{d}$. But then

$\displaystyle p_{\Phi_{d}(\widetilde{c})}=M_{\widetilde{c}}\circ
k_{\widetilde{c}}\circ G_{\widetilde{c}}\circ k_{\widetilde{c}}^{-1}\circ
M_{\widetilde{c}}^{-1},$

is clearly continuous in $\widetilde{c}$.

###  4.2. The Boundary case

The following lemma and its proof are similar to [BD88, Chapter II.8, Lemma
3].

######  Lemma 4.4.

If $p_{a}$ and $p_{a^{\prime}}$ are quasiconformally equivalent via $\psi$,
with $a\in\partial\mathcal{CBO}_{d}$, such that $\psi$ satisfies the
conditions below:

$\displaystyle\psi(0)$ $\displaystyle=0$ $\displaystyle\psi(\sqrt{d})$
$\displaystyle=\sqrt{d}$
$\displaystyle\lim_{z\longrightarrow\infty}\frac{\psi(z)}{z}$
$\displaystyle=1$

then $a=a^{\prime}$.

###### Proof.

We first note that any $\psi$ as above also satisfies
$\psi(-\sqrt{d})=-\sqrt{d}$. If $K_{p_{a}}$ has measure $0$, then $\psi$ is a
hybrid equivalence and the result follows.

Otherwise, our strategy is to build a hybrid equivalence between the two
polynomials, similar to [DH85b, Chapter I.6, Corollary 2] and use Lemma 4.1.
Consider the Beltrami form $\mu=\frac{\overline{\partial}\psi}{\partial\psi}$
and let $\mu_{0}$ be the form that agrees with $\mu$ on $K_{p_{a}}$ and equals
$0$ on $\mathbb{C}\setminus K_{p_{a}}$. Set $k=||\mu_{0}||_{\infty}$.

We note that $k<1$. By the measurable Riemann mapping theorem, for every
$t\in\mathbb{D}_{\frac{1}{k}}$, there exists a unique quasiconformal
homeomorphism $\psi_{t}:\mathbb{C}\longrightarrow\mathbb{C}$ such that

$\displaystyle\frac{\overline{\partial}\psi_{t}}{\partial\psi_{t}}$
$\displaystyle=t\mu_{0}$ $\displaystyle\psi_{t}(0)$ $\displaystyle=0$
$\displaystyle\lim_{z\rightarrow\infty}\frac{\psi_{t}(z)}{z}$
$\displaystyle=1$

We note that $\kappa_{t}(z)=\psi_{t}(-\psi_{t}^{-1}(z))$ is a family of affine
maps that satisfy

$\displaystyle\kappa_{t}(0)$ $\displaystyle=0$
$\displaystyle\lim_{z\mapsto\infty}\frac{\kappa_{t}(z)}{z}$ $\displaystyle=-1$

Therefore, $\kappa_{t}(z)=-z$.

$\psi_{t}\circ p_{a}\circ\psi_{t}^{-1}$ is a polynomial with exactly two
critical points that commutes with $\kappa_{t}$. It has the form
$\widetilde{a}(t){\displaystyle\int_{0}^{z}}\Big{(}1-\frac{w}{x(t)}\Big{)}^{d}\Big{(}1+\frac{w}{x(t)}\Big{)}^{d}dw$.
The functions
$\widetilde{a},x:\mathbb{D}_{\frac{1}{k}}\longrightarrow\mathbb{C}$ are
holomorphic, with $x(0)=\sqrt{d}$, $\widetilde{a}(0)=a$. These polynomials are
odd, therefore by conjugating them by $h_{t}(z)=\frac{z\sqrt{d}}{x(t)}$, we
obtain polynomials of the form
$a(t){\displaystyle\int_{0}^{z}}\Big{(}1-\frac{w^{2}}{d}\Big{)}^{d}dw$, where
$a:\mathbb{D}_{\frac{1}{k}}\longrightarrow\mathbb{C}$ is holomorphic, with
$a(t)\in\mathcal{CBO}_{d}$, and $a(0)=a\in\partial\mathcal{CBO}_{d}$. But this
implies that $a(t)$ is a constant function, and $\psi\circ\psi_{1}^{-1}\circ
h_{1}^{-1}$ is a hybrid equivalence between $p_{a}$ and $p_{a^{\prime}}$.
Lemma 4.4 implies $a=a^{\prime}$.
∎

To show continuity of $\Phi_{d}$ at $c\in\partial\mathcal{M}_{d+1}$, it
suffices to show that its graph is closed, that is, if
$c_{n}\in\mathcal{M}_{d+1}$ converge to $c$ and
$a_{n}=\Phi_{d}(c_{n})\longrightarrow\widetilde{a}$, then
$\widetilde{a}=\Phi_{d}(c)$.

Let

$\displaystyle a$ $\displaystyle=\Phi_{d}(c)$ $\displaystyle f_{n}$
$\displaystyle=f_{c_{n}}$ $\displaystyle g$ $\displaystyle=g_{c}$
$\displaystyle g_{n}$ $\displaystyle=g_{c_{n}}$ $\displaystyle\sigma$
$\displaystyle=\sigma_{c}$ $\displaystyle\sigma_{n}$
$\displaystyle=\sigma_{c_{n}}$ $\displaystyle\psi$ $\displaystyle=\psi_{c}$
$\displaystyle\psi_{n}$ $\displaystyle=\psi_{c_{n}}$ $\displaystyle G$
$\displaystyle=G_{c}=\psi\circ g\circ\psi^{-1}$ $\displaystyle G_{n}$
$\displaystyle=G_{c_{n}}=\psi_{n}\circ g_{n}\circ\psi_{n}^{-1}$
$\displaystyle\varphi$ $\displaystyle=\varphi_{c}$ $\displaystyle\varphi_{n}$
$\displaystyle=\varphi_{c_{n}}$ $\displaystyle V_{n}$
$\displaystyle=(V_{0})_{c_{n}}$ $\displaystyle V$ $\displaystyle=(V_{0})_{c}$
$\displaystyle Q_{n}(\ell)$
$\displaystyle=\Big{(}Q_{1}\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}_{c_{n}}$
$\displaystyle Q(\ell)$
$\displaystyle=\Big{(}Q_{1}\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}_{c}$
$\displaystyle T_{n}(\ell)$
$\displaystyle=\Big{(}T_{1}\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}_{c_{n}}$
$\displaystyle T(\ell)$
$\displaystyle=\Big{(}T_{1}\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}_{c}$

######  Proposition 4.5.

The sequence of quasiregular maps $g_{n}$ converge to $g$.

###### Proof.

On both the $+$ and $-$ copies of $(W_{0})_{c_{n}}$, $g_{n}$ coincides with
$f_{n}$ away from the sectors $Q_{1}(\ell)$, for $\ell=1,2,...,d$. On each of
these sectors, $g_{n}$ has as its components a conformal map
$\widetilde{h}_{n}:T_{n}(\ell)\longrightarrow V_{n}$, chosen uniquely so that
the triple $((z_{1})_{n},(z_{2})_{n},\omega^{\ell}\beta_{n})$ is mapped to the
triple $((\widetilde{z}_{1})_{n},(\widetilde{z}_{2})_{n}),\beta_{n})$, and a
quasiconformal extension to $Q_{n}(\ell)\setminus\overline{T_{n}(\ell)}$.
Similarly, on $T(\ell)$, $g$ agrees with an analytic map
$\widetilde{h}:T(\ell)\longrightarrow V$ chosen so that the triple
$(z_{1},z_{2},\omega^{\ell}\beta)$ is sent to
$(\widetilde{z}_{1},\widetilde{z}_{2},\beta)$. Fix an
$\ell\in\\{1,2,...,d\\}$. We will first show that the $\widetilde{h}_{n}$
converge to $\widetilde{h}$.

Let $\rho_{n}$ be the Riemann map that sends $\mathbb{D}$ to $V_{n}$, with
$\rho_{n}(0)=0$ and $\rho_{n}^{\prime}(0)>0$. Observe that $V_{n}$ converges
to $V$ with respect to the point $0$ in the sense of kernel convergence (see
[Pom92, Section 1.4]). By Carathéodory’s kernel convergence theorem ([Pom92,
Theorem 1.8]), $\rho_{n}\longrightarrow\rho$ uniformly in $\mathbb{D}$, where
$\rho:\mathbb{D}\longrightarrow V$ is a conformal map that sends $0$ to $0$
and satisfies $\rho^{\prime}(0)>0$. Since the boundaries of $V_{n},V$ are
quasicircles, the $\rho_{n}$ extend to $\partial\mathbb{D}$ and these boundary
maps converge uniformly to the boundary extension of $\rho$. Thus, the triples
$(s_{n},t_{n},w_{n})$ in $\mathbb{S}^{1}$ that map under $\rho_{n}$ to
$((\widetilde{z}_{1})_{n},(\widetilde{z}_{2})_{n}),\beta_{n})$ converge to the
triple $(s,t,w)$ in $\mathbb{S}^{1}$ that maps under $\rho$ to
$(\widetilde{z}_{1},\widetilde{z}_{2},\beta)$.

Let $M_{n}:\mathbb{D}\rightarrow\mathbb{D}$ be a sequence of automorphisms
that send $(1,i,-1)$ to $(s_{n},t_{n},w_{n})$, and let $M$ be the automorphism
of $\mathbb{D}$ that sends $(1,i,-1)$ to $(s,t,w)$. Then $M_{n}\longrightarrow
M$ on $\overline{\mathbb{D}}$.

Lastly, for a given $\ell$, note that $\varphi_{n}(T_{n}(\ell))$ is the same
domain $D=\exp(\Delta_{q})=\varphi(T(\ell))$ for each $n$, and furthermore,
$(\varphi_{n}((z_{1})_{n}),\varphi_{n}((z_{2})_{n}),\varphi_{n}(\omega^{\ell}\beta_{n}))=(\varphi(z_{1}),\varphi(z_{2}),\varphi(\omega^{\ell}\beta))$
(we note that $\omega^{\ell}\beta_{n}$ and $\omega^{\ell}\beta$ are tips, ie.
a unique dynamical ray lands at each of these points, so evaluating the
Böttcher chart at these points makes sense). Let
$e:D\longrightarrow\mathbb{D}$ be the Riemann map that takes the triple
$(\varphi(z_{1}),\varphi(z_{2}),\varphi(\omega^{\ell}\beta))$ in $\partial D$
to $(1,i,-1)$. Then

$\displaystyle\widetilde{h}_{n}$ $\displaystyle=\rho_{n}\circ M_{n}\circ
e\circ\varphi_{n}$ $\displaystyle\widetilde{h}$ $\displaystyle=\rho\circ
M\circ e\circ\varphi$

It is clear by our discussion that
$\widetilde{h}_{n}\longrightarrow\widetilde{h}$.

Therefore, on the sectors $T_{n}(\ell)$, the sequence $g_{n}$ converges to
$g$. But note that the quasiconformal extension to $Q_{n}(\ell)$ is done in
the same way for each $n$. Therefore, $g_{n}\longrightarrow g$.

By definition of $\sigma_{n}$ and $\sigma$, we must have
$\sigma_{n}\longrightarrow\sigma$, and consequently, by the Measurable Riemann
Mapping Theorem, $\psi_{n}\longrightarrow\psi$.

∎

This discussion tells us that

$\displaystyle G_{n}$ $\displaystyle=\psi_{n}\circ
g_{n}\circ\psi_{n}^{-1}\longrightarrow\psi\circ g\circ\psi^{-1}=G$

Now consider the hybrid equivalences $k_{n}$ that conjugate $G_{n}$ to
$p_{a_{n}}$. These have bounded dilitation ratio and map $0$ to $0$, and hence
form an equicontinuous family. Upto a subsequence, $k_{n}$ converge to a
quasiconformal map $\widetilde{k}$. Thus, $k_{n}\circ G_{n}\circ
k_{n}^{-1}\longrightarrow\widetilde{k}\circ G\circ\widetilde{k}^{-1}$. We will
call the latter map $\widetilde{G}$.

Using [DH85b, Chapter II.7, Lemma, p.313], $\widetilde{G}$ is quasiconformally
equivalent to $p_{\widetilde{a}}$ (not necessarily hybrid equivalent), but it
is also quasiconformally equivalent to $k\circ G\circ k^{-1}$, which in turn
is hybrid equivalent to $p_{a}$.

This shows that $p_{\widetilde{a}}$ is quasiconformally equivalent to $p_{a}$.
We can choose the equivalence so that the conditions of Lemma 4.4 are
satisfied. But in order to use this lemma, we also need to show that
$a\in\partial\mathcal{CBO}_{d}$.

Consider a sequence $c_{n}^{*}$ of Misiurewicz parameters tending to $c$, and
let $a_{n}^{*}=\Phi_{d}(c_{n}^{*})$. Then $a_{n}^{*}$ is Misiurewicz, and
there exists a subsequence $a_{n}^{*}\longrightarrow
a^{*}\in\partial\mathcal{CBO}_{d}$. By the paragraphs above, $a^{*}=a$, and
hence, $a\in\partial\mathcal{CBO}_{d}$. Now we apply Lemma 4.4 again to get
$a=a^{\prime}$.

##  5\. Injectivity of $\Phi_{d}$

In this section will construct an inverse
$\Psi_{d}:\mathcal{CBO}_{d}^{(+,-)}\longrightarrow\mathcal{M}_{d+1}$ of
$\Phi_{d}$.

###  5.1. Dynamics of maps in $\mathcal{CBO}_{d}^{(+,-)}$

Given $a\in\mathcal{CBO}_{d}^{(+,-)}$, let $P_{s(a)}$ be the monic
representative of $p_{a}$ as defined in Section 2. As in the construction of
$\Phi_{d}$, for $\theta=0,\frac{1}{2}$, let $S_{\theta}$ be invariant sectors
at $0$ with same slope. That is,

$\displaystyle S_{0}$ $\displaystyle=\\{\varphi_{s(a)}^{-1}(e^{s+2\pi
it}):s\in(0,\eta),|t|<qs\\}$ $\displaystyle S_{\frac{1}{2}}$
$\displaystyle=\\{\varphi_{s(a)}^{-1}(-e^{s+2\pi it}):s\in(0,\eta),|t|<qs\\}$

Note that $S_{\frac{1}{2}}=-S_{0}$.

We choose $q$ to be small enough so that $S_{0}\cap S_{\frac{1}{2}}=\\{0\\}$,
and the inverse image of each $S_{\theta}$ under $P_{s(a)}$ consists of
exactly $2d+1$ components. The point $0$ has pre-images
$\\{0=x_{0},x_{1},x_{2},...,x_{2d}\\}$ under $P_{s(a)}$, of which $d$ \- say
$x_{1},x_{2},...,x_{d}$, are in $F^{L}_{s(a)}$, and $d$ are in $F^{R}_{s(a)}$.
Let $S_{\theta}(x_{\ell})$ be the inverse image of $S_{\theta}$ based at
$x_{\ell}$ for $\ell\neq 0$.

Let $W$ be the region bounded by an equipotential $\\{z|G_{s}(z)=\eta\\}$ and
define $W_{i}=P_{s(a)}^{-\circ i}(W)$. For a given $\ell\in\\{1,2,...,2d\\}$,
let $S$ be the connected component of $W_{1}\setminus(S_{0}(x_{\ell})\cup
S_{\frac{1}{2}}(x_{\ell}))$ that does not contain $0$. Then $P_{s(a)}$ maps
$S$ to $F^{L}_{s(a)}$ if $S\subset F^{R}_{s(a)}$, and to $F^{R}_{s(a)}$ if
$S\subset F^{L}_{s(a)}$.

We have illustrated this in Figure 14.

###  5.2. Definition of $\Psi_{d}$

With $a$ as above,

Figure 15. Cut and paste surgery on $P_{s(a)}$

construct the Riemann surface $Y$ as follows: let $Y_{0}=W\cap F^{L}_{s(a)}$,
and identify the boundaries $Y_{0}\cap\mathcal{R}_{0}(s(a))$ and
$Y_{0}\cap\mathcal{R}_{0}(s(a))$ by identifying points on either ray with same
speed of escape. Additionally, if necessary, smoothe the boundary of $Y_{0}$
at the point $w$ as shown in Figure 15. $S_{0}\cap F^{L}_{s(a)}$ and
$S_{\frac{1}{2}}\cap F^{L}_{s(a)}$ with this boundary identification become a
single sector which we shall call $\widetilde{S}$. We let
$Y_{1}=P_{s(a)}^{-1}(Y_{0})$ with this boundary identification. Clearly,
$\overline{Y_{1}}\subset Y_{0}$.

Given $\ell\in\\{1,2,...,d\\}$, let $S$ be the connected component of
$Y_{1}\setminus(S_{0}(x_{\ell})\cup S_{\frac{1}{2}}(x_{\ell}))$ that does not
contain $0$, and let $S^{\prime}$ be the component that does. Let
$S_{\ell}=S\cup S_{0}(x_{\ell})\cup S_{\frac{1}{2}}(x_{\ell})$ (see Figure
15). Pick a quasiconformal homeomorphism
$e_{\ell}:S_{\ell}\mapsto\widetilde{S}$ that extends to a homeomorphism from
$\partial S_{\ell}$ to $\partial\widetilde{S}$, and coincides with $P_{s(a)}$
on $\partial S_{\ell}\cap\partial S^{\prime}$. For example, this can be
constructed in a manner similar to
$g_{c}\Big{|}_{\pi_{c}\big{(}S_{1}^{\pm}\big{(}\frac{\ell}{d+1}\big{)}\big{)}}$
in Section 3. Define

$\displaystyle F:Y_{1}$ $\displaystyle\longrightarrow Y_{0}$ $\displaystyle
F(z)$ $\displaystyle=\begin{cases}P_{s(a)}(z)&z\in
Y_{1}\setminus\bigcup_{\ell=1}^{d}S_{\ell}\\\ e_{\ell}(z)&z\in S_{\ell}\text{
for some }\ell\in\\{1,2,...,d\\}\end{cases}$

$F$ is clearly a quasiregular map of degree $d+1$ with a single critical
point. We define an $F-$ invariant complex structure $\sigma$ on $Y_{0}$ as

  * •

$E_{z}=\mathbb{S}^{1}$ if $z\in Y_{0}\setminus Y_{1}$ or the $F-$ orbit of $z$
does not intersect $S_{\ell}$ for any $\theta,\ell$

  * •

$E_{z}=(DF^{n})^{-1}(\mathbb{S}^{1})$ if $F^{n}(z)$ is the first point in the
orbit of $z$ that is in $S_{\ell}$

Figure 16. Alternative construction of $\Psi_{d}(a)$ by renormalization; the
‘$*$’ marks the critical point $-\sqrt{d}s(a)$ of $P_{s(a)}$

Every $F-$ orbit visits $S_{\theta}(x_{i})$ at most once. So, $\sigma$ has
bounded dilitation. Note that $F^{*}\sigma=\sigma$, and thus, $F$ is
quasiconformally equivalent to a polynomial-like map $y:V_{1}\longrightarrow
V$ with degree $d+1$ and a single critical point. The map $y$ is hybrid
equivalent to a polynomial of the form $f_{c}(z)=z^{d+1}+c$. Note that $y$
only determines the affine equivalence class of $f_{c}$, and thus $c$ is not
unique if $d>1$; however, we impose the condition that the identified rays
$\mathcal{R}_{0}(s(a))$ and $\mathcal{R}_{\frac{1}{2}}(s(a))$ are eventually
mapped to the same access as the dynamical ray at angle $0$ to $f_{c}$ (with
respect to the Böttcher chart where $\frac{\varphi_{c}(z)}{z}\rightarrow 1$ as
$z\rightarrow\infty$). This determines $c$ uniquely. It is clear that
$c\in\mathcal{M}_{d+1}$; we therefore define $\Psi_{d}(a)=c$.

######  Remark 5.1.

We may also construct $\Psi_{d}(a)$ by choosing a renormalization of
$P_{s(a)}$.

Choose a neighborhood $\mathcal{E}$ of $0$, in which $P_{s(a)}$ is conjugate
to $z\mapsto rz$ for some $r\in\mathbb{C}$ with $|r|>1$, small enough so that
$\overline{\mathcal{E}}$ does not contain any critical points, and satisfying

$\displaystyle\mathcal{E}\cap S_{0}$ $\displaystyle=S_{0}\cap W_{i}$
$\displaystyle\mathcal{E}\cap S_{\frac{1}{2}}$
$\displaystyle=S_{\frac{1}{2}}\cap W_{i}$

Let $\mathcal{V}$ be an open set defined the union of $W_{i}\cap F^{L}_{s(a)}$
and $\mathcal{E}$. Then, there exists a connected component
$\mathcal{V}^{\prime}$ of $P_{s(a)}^{-1}(\mathcal{V})$ such that
$\overline{\mathcal{V}^{\prime}}\subset\mathcal{V}$, and
$P_{s(a)}\big{|}_{\mathcal{V}^{\prime}}:\mathcal{V}^{\prime}\longrightarrow\mathcal{V}$
is polynomial-like of degree $d+1$ (see Figure 16). This polynomial-like map
has a unique critical point at $-\sqrt{d}$, and by the straightening theorem,
it is hybrid equivalent to a unicritical degree $d+1$ polynomial.

We can show for an appropriate choice of domains, the map $F$ defined above in
the first definition of $\Psi_{d}(a)$ and
$P_{s(a)}\big{|}_{\mathcal{V}^{\prime}}$ are hybrid equivalent.

We may use the same methods as in Section 4 to show that $\Psi_{d}$ is
continuous.

###  5.3. $\Psi_{d}$ is the inverse of $\Phi_{d}$

Given $c\in\mathcal{M}_{d+1}$, let ${c^{\prime}}=\Psi_{d}\circ\Phi_{d}(c)$. We
will follow the construction to show that $f_{c^{\prime}}$ and $f_{c}$ are
hybrid equivalent, and thus, $c^{\prime}=c$.

Let $a=\Phi_{d}(c)$. The construction $a\mapsto\Psi_{d}(a)$ involves picking
the sectors $S_{0}$ and $S_{\frac{1}{2}}$ in the dynamical plane of
$P_{s(a)}$, constructing a Riemann surface $Y$, a quasiregular map $F_{s(a)}$,
and lastly, a polynomial like map $y_{s(a)}$.

On the other hand, the construction $c\mapsto\Phi_{d}(c)$ goes through the
steps $f_{c}\mapsto g_{c}\mapsto G_{c}\mapsto P_{s(a)}$. We will only be
working with the ‘$-$’ copies of $S(\frac{\ell}{d+1}),K_{f_{c}}$, etc., and so
we shall drop the ‘$-$’ superscript. The first step in the construction of
$\Phi_{d}(c)$ uses the quotient map $\pi_{c}$, and we have

$\displaystyle g_{c}(\pi_{c}(z))$ $\displaystyle=\pi_{c}(f_{c}(z))\text{ away
from sectors }\pi_{c}\Big{(}S\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}$
$\displaystyle G_{c}$ $\displaystyle=\psi_{c}\circ g_{c}\circ\psi_{c}^{-1}$
$\displaystyle P_{s(a)}$ $\displaystyle=k_{c}\circ G_{c}\circ k_{c}^{-1}$

where $\psi_{c}$ is quasiconformal and $k_{c}$ is a hybrid equivalence.

Figure 17. Building a conjugacy between $f_{c}$ and $F_{s(a)}$. The wavily
shaded region in the top figure is $S(0)$; it is contained in $\mathcal{S}$
and its two copies map under $k_{c}\circ\psi_{c}\circ\pi_{c}$ to the sectors
$S_{0}$ and $S_{1}$ respectively, which we cut to make $\tilde{S}$. We define
$\hat{\phi}$ on the darkly shaded region on the top to the darkly shaded
region at the bottom.

In the dynamical plane of $P_{s(a)}$, for $\ell\in\\{1,2,...,d\\}$, define

$\displaystyle\widetilde{S}_{0}(x_{\ell})$
$\displaystyle=\Big{\\{}\varphi_{s(a)}^{-1}(e^{r+2\pi
it}):s\in(0,\eta),\Big{|}t-\frac{\ell}{2d}\Big{|}<qs\Big{\\}}$
$\displaystyle\widetilde{S}_{\frac{1}{2}}(x_{\ell})$
$\displaystyle=\Big{\\{}\varphi_{s(a)}^{-1}(-e^{s+2\pi
it}):s\in(0,\eta),\Big{|}t-\frac{\ell}{2d}\Big{|}<qs\Big{\\}}$

and let $\widetilde{S}_{\ell}$ be the union of
$\widetilde{S}_{0}(x_{\ell}),\tilde{S}_{\frac{1}{2}}(x_{\ell})$ and the
connected component of $Y_{0}\setminus$
$\widetilde{S}_{0}(x_{\ell})\cup\tilde{S}_{\frac{1}{2}}(x_{\ell})$ that
contains $S_{\ell}$, as defined in Section 5.2. See Figure 15 for an
illustration of $\widetilde{S}_{\ell}$. Let
$\widetilde{\phi}=k_{c}\circ\psi_{c}\circ\pi_{c}$.

In the dynamical plane of $f_{c}$, let $S_{1}\big{(}\frac{\ell}{d+1}\big{)}$ ,
$\ell=0,1,...,d$, be as defined in Equations 1 to 4 (the equipotential $\eta$
and the slope factor $q$ may be different from the ones used for $P_{s(a)}$).
There are two copies of $S(0)=f_{c}\big{(}S_{1}(0)\big{)}$ in the dynamical
plane of $G_{c}$, but we will pick the copy that eventually gets mapped to a
sector that intersects $S_{0}$. More generally, for a suitable choice of
equipotential and slope factor in the $f_{c}$ \- plane, we may assume that the
open sets $S\big{(}\frac{\ell}{d+1}\big{)}=\omega^{\ell}S(0)$ are eventually
mapped into $\widetilde{S}_{\ell}$, and that $S(0)$ is eventually mapped to
$S_{0}$ (or to $S_{\frac{1}{2}}$). That is,

$\displaystyle S_{\ell}(c)$
$\displaystyle=\widetilde{\phi}\Big{(}S\Big{(}\frac{\ell}{d+1}\Big{)}\Big{)}\subset\widetilde{S}_{\ell}\text{
for }\ell=0,1,...,d$ $\displaystyle\tilde{\phi}(V_{0})$
$\displaystyle=Y_{0}\setminus S_{0}(c)$

where the domain $V_{0}$ is as defined in Equation 5.

Our strategy will be to set up a quasiconformal map $\phi:V_{0}\cup
S(0)\longrightarrow Y_{0}$ that has agrees with $\widetilde{\phi}$ away from
certain sectors, and conjugates $f_{c}$ and $F_{s(a)}$.
Let

$\displaystyle V$
$\displaystyle=\widetilde{\phi}^{-1}(Y_{0}\setminus\widetilde{S})$
$\displaystyle V_{1}$ $\displaystyle=f_{c}^{-1}(V)$ $\displaystyle\mathcal{S}$
$\displaystyle=V_{0}\cup S(0)\setminus V$
$\displaystyle\mathcal{S}\Big{(}\frac{\ell}{d+1}\Big{)}$
$\displaystyle=\widetilde{\phi}^{-1}(\tilde{S}_{\ell})\text{ for
}\ell=1,2,...,d$

See Figure 17 for details. For all $z\in
V_{1}\setminus\Big{(}\mathcal{S}\cup\bigcup_{\ell}\mathcal{S}\big{(}\frac{\ell}{d+1}\big{)}\Big{)}$,

$\displaystyle F_{s(a)}\circ\widetilde{\phi}(z)$
$\displaystyle=\widetilde{\phi}\circ f_{c}(z)$

Furthermore, with degree one,

$\displaystyle f_{c}\Big{(}\mathcal{S}\Big{(}\frac{\ell}{d+1}\Big{)}\cap
V_{1}\Big{)}$ $\displaystyle=\mathcal{S}\text{ for }\ell=1,2,...,d$
$\displaystyle f_{c}(\mathcal{S}\cap V_{1})$ $\displaystyle=\mathcal{S}$

For $z\in\mathcal{S}$, $z=f_{c}(w)$ for $d$ distinct $w\in V_{1}$. We can
assume that $F_{s(a)}\circ\widetilde{\phi}(w)$ does not depend on the choice
of preimage $w$, since $\widetilde{\phi}(w)\in S_{\ell}$ and
$F_{s(a)}\big{|}_{S_{\ell}}$ depends on the homeomorphisms $e_{\ell}$ defined
as in Section 5.2, which we have freedom in choosing.
So we set

$\displaystyle\hat{\phi}(z)$ $\displaystyle=F_{s(a)}\circ\widetilde{\phi}(w)$

Define

$\displaystyle\phi$ $\displaystyle:V_{0}\cup S(0)\longrightarrow Y_{0}$
$\displaystyle\phi(z)$
$\displaystyle=\begin{cases}\widetilde{\phi}(z)&z\not\in\mathcal{S}\\\
\hat{\phi}(z)&z\in\mathcal{S}\end{cases}$

By the discussion above, for all $z\in f_{c}^{-1}(V_{0}\cup S(0))$,

$\displaystyle F_{s(a)}\circ\phi(z)$ $\displaystyle=\phi\circ f_{c}(z)$

We note that $\pi_{c}$ changes the angle at $\beta_{c}$ from $2\pi$ to $\pi$,
and has zero dilitation on $K_{f_{c}}\setminus\\{\beta_{c}\\}$. Also note that
$\psi_{c}$ has zero dilitation on $\pi_{c}(K_{f_{c}})$.

On the other hand, the cutting procedure in Section 5.2 changes the angle
$\pi$ made by the boundary of $F^{L}_{s(a)}$ at $0$ to the angle $2\pi$ in the
plane of $F_{s(a)}$. Lastly, note that $k_{c}$ has zero dilitation on
$\psi_{c}\circ\pi_{c}(K_{f_{c}})\setminus\\{0\\}$.

Combined, this information tells us that we have constructed a quasiconformal
map $\phi:V_{0}\cup S(0)\longrightarrow Y$ that has zero dilitation on
$K_{c}$, and conjugates $f_{c}$ to $F_{s(a)}$.

Now, if $z\in\phi(K_{c})$, a point $F^{\circ n}_{s(a)}$ in the orbit of $z$
cannot be in the interior of $S_{\ell}$ for any $\ell$. Therefore, the
quasiconformal map that conjugates $F_{s(a)}$ to $y_{s(a)}$ has zero
dilitation on $\phi(K_{c})$. That is, $y_{s(a)}$ and $f_{c}$ are hybrid
equivalent. Thus, $f_{c^{\prime}}$ and $f_{c}$ are hybrid equivalent, implying
$c=c^{\prime}$.

In a similar manner, we can show that $p_{\widetilde{a}}$, where
$\widetilde{a}=\Phi_{d}\circ\Psi_{d}(a)$, is hybrid equivalent to $p_{a}$.
That is, $\Phi_{d}\circ\Psi_{d}(a)=a$.

This finishes the proof of Theorem 1.1.

We will end with a discussion of how the image under $\Phi_{d}$ fits inside
$\mathcal{CBO}_{d}$.

######  Lemma 5.2.

$\mathcal{CBO}^{(+,-)}_{d}$ disconnects $\mathcal{CBO}_{d}$ into infinitely
many components.

###### Proof.

Let $f_{c}(z)=z^{d+1}+c$ be a polynomial where the orbit of $c$ contains the
$\beta-$fixed point where the dynamical ray at angle $0$ lands. There are
infinitely many values of $c$ in $\mathcal{M}_{d+1}$ that satisfy this
condition - these are precisely the landing points of parameter rays at angles
$\frac{i}{d^{n}}$ for $n\geq 1$ and $0<i<d^{n}$. These are included in the set
of “tips” of $\mathcal{M}_{d+1}$.

Given such a $c$, let $a=\Phi_{d}(c)$. Then the orbit of both critical points
$\pm\sqrt{d}$ of $p_{a}$ eventually lands on $0$ \- that is, there exists $k$
such that $p_{a}^{\circ k}(\pm\sqrt{d})=0$.

In the dynamical plane of the monic representative $P_{s(a)}$, the dynamical
rays at angles $0,\frac{1}{2}$ land at $0$. Thus there exist two angles
$\theta_{1},\theta_{2}$ such that $(2d+1)^{k-1}\theta_{1}\equiv 0$ and
$(2d+1)^{k-1}\theta_{2}\equiv\frac{1}{2}$, which both land at the critical
value $P_{s(a)}\big{(}-s(a)\sqrt{d}\big{)}$. In the parameter plane of
$\mathcal{MBO}_{d}$, the parameter rays at angle $\theta_{1},\theta_{2}$ both
land at $s(a)$\- which means that $s(a)$ is a cut-point of
$\mathcal{MBO}_{d}$, which is equivalent to saying that $a$ is a cut-point of
$\mathcal{CBO}_{d}$.

Another way to show this is to see that exists
$a^{\prime}\in\mathcal{CBO}_{d}$ close to $a$ such that
$P_{s(a^{\prime})}^{\circ k}(\sqrt{d})\in F^{L}_{s(a^{\prime})}$ and
$P_{s(a^{\prime})}^{\circ k}(-\sqrt{d})\in F^{R}_{s(a^{\prime})}$. That is,
the orbits of both critical points eventually “cross over” to the other side.
So $a^{\prime}\not\in\mathcal{CBO}_{d}^{(+,-)}$. ∎

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