# Contrasting chaotic and stochastic forcing: tipping windows and attractor
crises

Peter Ashwin, Julian Newman and Raphael Römer
Department of Mathematics and Statistics,
University of Exeter, Exeter EX4 4QF, UK

###### Abstract

Nonlinear dynamical systems subjected to a combination of noise and time-
varying forcing can exhibit sudden changes, critical transitions or tipping
points where large or rapid dynamic effects arise from changes in a parameter
that are small or slow. Noise-induced tipping can occur where extremes of the
forcing causes the system to leave one attractor and transition to another. If
this noise corresponds to unresolved chaotic forcing, there is a limit such
that this can be approximated by a stochastic differential equation (SDE) and
the statistics of large deviations determine the transitions. Away from this
limit it makes sense to consider tipping in the presence of chaotic rather
than stochastic forcing. In general we argue that close to a parameter value
where there is a bifurcation of the unforced system, there will be a chaotic
tipping window outside of which tipping cannot happen, in the limit of
asymptotically slow change of that parameter. This window is trivial for a
stochastically forced system. Entry into the chaotic tipping window can be
seen as a boundary crisis/non-autonomous saddle-node bifurcation and
corresponds to an exceptional case of the forcing, typically by an unstable
periodic orbit. We discuss an illustrative example of a chaotically forced
bistable map that highlight the richness of the geometry and bifurcation
structure of the dynamics in this case. If a parameter is changing slowly we
note there is a dynamic tipping window that can also be determined in terms of
unstable periodic orbits.

###### Contents

  1. 1 Introduction
  2. 2 Fast chaos limits and stochastic forcing
    1. 2.1 Tipping points and the chaotic tipping window
  3. 3 Dynamics and geometry for a chaotically forced bistable map
    1. 3.1 Parameter dependence for the map
    2. 3.2 Trajectories and UPOs
    3. 3.3 Geometry and dimension of the chaotic saddle
      1. 3.3.1 Box-counting dimension
      2. 3.3.2 Lyapunov dimension
  4. 4 Slowly varying parameters and the dynamic tipping window
  5. 5 Discussion
  6. A Proofs of dynamical properties
    1. A.1 Preliminaries
    2. A.2 Construction of $\mu_{\beta,\sigma}^{+}$
    3. A.3 Continuity of $a^{+}$ and $\Pi^{+}$
    4. A.4 Invariant-graph property of $a_{\beta,\sigma}^{+}$
    5. A.5 Ergodicity and continuity properties of $\mu_{\beta,\sigma}^{+}$
    6. A.6 Bounds on the $x$-projection of $\mathcal{A}^{+}_{\beta,\sigma}$
    7. A.7 Lyapunov exponent of $\mu^{+}_{\beta,\sigma}$
    8. A.8 Attractivity of $\mathcal{A}^{+}_{\beta,\sigma}$
    9. A.9 Construction and properties of $\mu_{\beta,\sigma}^{-}$
    10. A.10 Construction of $r_{\beta,\sigma}$
    11. A.11 Properties of $r_{\beta,\sigma}$ analogous to Secs. A.2–A.7
    12. A.12 Intersections of $\mathcal{C}_{\beta,\sigma}$ with $\mathcal{A}^{\pm}_{\beta,\sigma}$
    13. A.13 Basin of attraction of $\mathcal{A}^{\pm}_{\beta,\sigma}$ for $(\beta,\sigma)\in\mathrm{A}$
    14. A.14 Basin of attraction of $\mathcal{A}^{+}_{\beta,\sigma}$ for $(\beta,\sigma)\in S^{+}\setminus\mathrm{A}$
    15. A.15 Basin of attraction of $\mu^{\pm}_{\beta,\sigma}$
    16. A.16 The $x$-projection of invariant-measure supports in $\mathrm{D}$
    17. A.17 Globally contractive dynamics
    18. A.18 Continuous dependence of $a$ and $\mu$
    19. A.19 Basin of $\mathcal{A}_{\beta,\sigma}$ and $\mu_{\beta,\sigma}$ in $\tilde{\mathrm{D}}_{\lambda<0}$

##  1 Introduction

For complex nonlinear systems subjected to a time-varying input (that we call
forcing in general), there may be critical transitions (or tipping points)
where a small change in input leads to a large change in system state. In many
contexts both intrinsic noisy variability and slow drift variability in
parameters have a strong deterministic component and either of these may give
rise to abrupt changes via noise-induced or bifurcation-induced tipping [21,
22]. Moreover, if the slow drift is not slow enough, rate-induced tipping may
also appear [4, 3, 34].

Indeed, such effects may already appear within a subsystem of an autonomous
(unforced) nonlinear multiscale system because of other timescales present
within the system. As an example, noise in climate models is often seen to be
result of fast forcing by turbulent fluid dynamics of ocean and atmosphere,
while variation appears due to slow forcing such as changes in sea level or
land ice. Each of these fast noise and slow drift processes has a distinct
effect on the dynamics of the climate. This relates to Hasselmann’s programme
to understand weather as “noise” added to climate: see for example [27] for a
recent review, and is particularly relevant in case where critical
transitions/tipping points and chaotic variability are both present - see for
example [26, 8, 2]. In many problems of practical importance there are no
clear timescale separations, meaning that we are not at the limit where
stochastic forcing is appropriate.

The paper is organized as follows. In Sec. 2 we briefly review relations
between stochastic and chaotic forced models with tipping, and describe the
chaotic tipping window that appears in the chaotically forced case. Sec. 3
discusses a discrete-time chaotically forced bistable system where we analyse
the structure of the dynamics and geometry within phase and parameter space.
Implications of slow variation of parameters in this model are discussed in
Sec. 4, where we introduce the dynamic tipping window. In Sec. 5 discuss
possible implications and lessons for more general cases.

##  2 Fast chaos limits and stochastic forcing

Let us consider a system modelled by a multiscale ODE of the form

$\displaystyle\frac{dx}{dt}=$ $\displaystyle
f(x,y,\beta(t))+\frac{1}{\varepsilon_{x}}f_{0}(x,y,\beta(t))$ (1)
$\displaystyle\frac{dy}{dt}=$
$\displaystyle\frac{1}{\varepsilon_{\mathrm{fast}}}g(x,y,\beta(t))$

where $x\in\mathbb{R}^{d}$ is the system we wish to characterise,
$y\in\mathbb{R}^{f}$ is a chaotic subsystem representing the variability,
$\beta(t)\in\mathbb{R}^{p}$ are parameters that may also be varying.

The quantity $\varepsilon_{\mathrm{fast}}>0$ represents a ratio between the
timescales fast and slow processed. The terms multiplied by
$1/\varepsilon_{x}$ represent the amplitude of fast fluctuations on $x$, with
an assumption that
$1\gg\varepsilon_{\mathrm{x}}\gg\varepsilon_{\mathrm{fast}}$. Assume that
$f_{0}$ averages to zero over the fast $y$ dynamics. For fixed $\beta$,
reduction of systems derived from (1) to (3) is possible for some cases of
large forcing by fast chaos. For example consider
$\varepsilon_{\mathrm{x}}=\varepsilon^{1/2}$ and
$\varepsilon_{\mathrm{fast}}=\varepsilon$ then

$\displaystyle\frac{dx}{dt}=$ $\displaystyle
f(x,\beta)+\frac{1}{\sqrt{\varepsilon}}f_{0}(x,y,\beta)$ (2)
$\displaystyle\frac{dy}{dt}=$ $\displaystyle\frac{1}{\varepsilon}g(y,\beta)$

where $y$ is chaotic and the average of $f_{0}$ over this attractor $\langle
f_{0}\rangle_{y}=O(\varepsilon)$. Recent limit theorem results [12, 20] (see
the review [13]) can be applied to show that trajectories of $x$ of (2) are
well-approximated in the limit $\varepsilon_{f}\rightarrow 0$ as solutions of
the Itô SDE

$dx=f(x,\beta(t))dt+\sigma(x)dW_{t}$ (3)

where $(W_{t})_{t\in\mathbb{R}}$ is a Wiener process. If the SDE (3) is
written informally as

$\frac{dx}{dt}=f(x,\beta(t))+\sigma(x)\eta(t)$ (4)

then it can be viewed as a deterministic nonlinear system

$\frac{dx}{dt}=f(x,\beta)$ (5)

subjected to white noise forcing $\eta(t)$ (a “time derivative” of $W_{t}$)
and slow variation of parameters $\beta(t)$.

###  2.1 Tipping points and the chaotic tipping window

Systems of the form (1) and (3) allow one to discuss the variables $x$ of
interest in terms how they are forced by the other variables. Tipping events
(large or rapid changes in state) can arise in (3) for various reasons [4].
These reasons include _bifurcation-induced_ and _noise-induced_ tipping that
can be attributed to loss of stability on slowly changing $\beta$ or because
of extremes of $\eta$. More rapid changes of parameters $\beta(t)$ can give
rise to _rate-induced_ tipping and in general abrupt transitions may be
attributable to a mixture of these effects [4].

Because the white noise in (3) is unbounded, with small but finite probability
the noise can drive any system to exceed an arbitrarily large magnitude in an
arbitrarily short-duration time-interval. This means that in this context
noise-induced tipping between any attractor is always possible on any
timescale. The theory of large deviations for such systems [7, 11] gives a
powerful set of tools to approximate the statistics of such rare events,
especially in the case of asymptotically small $\sigma\ll 1$ but the presence
of white noise _destroys_ any multistability of (5): there is a non-zero
chance of transition between neighbourhoods of any pairs of attractors.

However, except in the limit of asymptotically fast chaotic forcing, the noise
component of (6) is never truly a Wiener process. Indeed, it is often bounded:
the extremes that drive tipping events cannot exceed certain levels. Consider
a model (1) with bounded noise and forcing by a deterministic chaotic system.
We find that tipping event induced by the bounded noise can still occur, but
there the boundedness means that there are limited regions where tipping is
possible. The bifurcations to tipping associated with bounded noise can be
understood from the viewpoint of set-valued dynamics [25, 23]. We consider a
chaotically forced system to the dynamics:

$\displaystyle\frac{dx}{dt}=$ $\displaystyle\ f(x,\beta(t))+\sigma f_{0}(y)$
(6) $\displaystyle\frac{dy}{dt}=$ $\displaystyle\ g(y).$

The dynamical behaviour of the $x$ dynamics of (6) is forced by chaotic
behaviour $y$ and a slow drift $\beta$ both assumed independent from $x$ and
from each other. For fixed $\beta$ we have the frozen system with chaotic
forcing

$\displaystyle\frac{dx}{dt}=$ $\displaystyle\ f(x,\beta)+\sigma f_{0}(y)$ (7)
$\displaystyle\frac{dy}{dt}=$ $\displaystyle\ g(y)$

where we assume the initial $y(0)$ is chosen to be typical with respect to
some ergodic invariant measure $m$ for the $y$ dynamics: we can view this as a
non-autonomous system

$\displaystyle\frac{dx}{dt}=$ $\displaystyle\ f(x,\beta)+\sigma f_{0}(y(t)).$
(8)

Suppose this system with $\sigma=0$ has a branch of stable equilibria for
$\beta<\beta^{*}$ and a bifurcation at some $\beta^{*}$ such that there is no
nearby attractor for $\beta>\beta^{*}$. The addition of chaotic forcing with
$\sigma>0$ will create a chaotic tipping window $(\beta_{-},\beta_{+})$
depending on $\sigma$ such that the chaotic forcing outside the window cannot
influence the tipping, such that this window limits to $\beta^{*}$ as
$\sigma\rightarrow 0$.

The entry to the chaotic tipping window at $\beta^{-}$ is a boundary crisis of
an attractor of (7). Note that in the originally a crisis is an attractor
hitting the stable manifold of an unstable periodic orbit on the basin
boundary [15] but a more general notion of boundary crisis when attractor
hitting its basin boundary can occur at a chaotic saddle [17, 24]. Beyond the
exit of the chaotic tipping window at $\beta^{+}$ the dynamics of the chaotic
system $y$ can no longer delay tipping and there will be rapid escape from a
neighbourhood of where the stable branch previously existed. Within the
chaotic tipping window the behaviour will depend on the precise details of the
chaotic trajectory $y(t)$. In particular, the chaotic attractor will typically
contain many unstable periodic orbits (UPOs) and other exceptional measures
whose statistics do not reflect that of the invariant measure $m$.

Let us consider the set $\mathcal{S}(m)$ of invariant measures for the $y$
dynamics that are supported within the support of $m$. For our system we can
characterise the chaotic tipping window as the smallest window that contains

$\beta_{c}(s):=\sup\\{\beta~{}:~{}\mbox{the system has more than one
attractor}\mbox{ when forced by }s\\}.$

for all $s\in\mathcal{S}(m)$. Determining the end points of this window is
hence a problem of ergodic optimization: we need to find

$\beta_{-}=\inf\\{\beta_{c}(s)~{}:~{}s\in\mathcal{S}(m)\\},~{}~{}~{}\beta_{+}=\sup\\{\beta_{c}(s)~{}:~{}s\in\mathcal{S}(m)\\}.$

For typical systems we expect the measures realising these extremes will be
periodic [19]. However, because $\beta_{c}(s)$ is not simply an average but
the boundary of a region of multistability, it does not seem to be obvious how
to directly apply these results. We note that the threshold on entry to the
chaotic tipping window can also be understood in terms of a _nonautonomous
saddle-node bifurcation_ [1] of the system (8).

##  3 Dynamics and geometry for a chaotically forced bistable map

A noise-forced tipping of an ODE forced by a chaotic component (2) (as for
example in [29]) will be very hard to understand in detail and so for
illustration we examine an iterated bistable map forced by a prototypical
chaotic map. The example we consider is the skew product dynamical system on
$(x,\theta)\in\mathbb{R}\times[0,2\pi)$ defined by

$\displaystyle x_{n+1}=$ $\displaystyle f(x_{n},\theta_{n},\beta,\sigma)$ (9)
$\displaystyle\theta_{n+1}=$ $\displaystyle g(\theta_{n})$

where $g(\theta)=3\theta~{}(\bmod~{}2\pi)$ and $f$ can have bistability for
fixed parameters. We choose

$f(x,\theta,\beta,\sigma)=\alpha\operatorname{\mathrm{arctan}}(x)+\beta+\sigma\cos\theta.$
(10)

For the numerical explorations we fix $\alpha=2$ and vary the real parameters
$\beta,\sigma$. Most of the statements go through for any $\alpha>1$ though we
note that, for example, the regularity of the invariant graph that separates
the attractor basins will decrease for $\alpha$ close to $1$.

Note that the map (9)–(10) is monotonic in $x$: that is, if
$(x_{n},\theta_{n})$ and $(y_{n},\theta_{n})$ are two forward trajectories
with the same $\theta$-component, then

$x_{n}<y_{n}~{}~{}\Rightarrow~{}~{}x_{n+1}<y_{n+1}.$ (11)

Moreover, all points $x$ throughout the whole real line $\mathbb{R}$ are
mapped after one iteration into the finite-width interval

$f(x,\theta,\beta,\sigma)\in\mathrm{I}_{\beta,\sigma}:=\left(\beta-\alpha\frac{\pi}{2}-\sigma,\beta+\alpha\frac{\pi}{2}+\sigma\right).$
(12)

For the $\theta$-component of the state space of the map (9), we regard
$[0,2\pi)$ topologically as wrapped round as a circle. A non-empty compact
subset of the state space $\mathbb{R}\times[0,2\pi)$ is called _invariant_ if
it is mapped surjectively onto itself by (9). The _basin_ (_of attraction_) of
a compact invariant set $\mathcal{A}$ is the set of initial conditions
$(x_{0},\theta_{0})$ for which the distance of the trajectory
$(x_{n},\theta_{n})$ from $\mathcal{A}$ tends to $0$ as $n\to\infty$. A set of
initial conditions $U\subset\mathbb{R}\times[0,2\pi)$ is said to be _uniformly
attracted_ to a compact invariant set $\mathcal{A}$ if there exists a sequence
of values $\varphi_{n}\geq 0$ tending to $0$ such that for every initial
condition $(x_{0},\theta_{0})\in U$, the distance of $(x_{n},\theta_{n})$ from
$\mathcal{A}$ is at most $\varphi_{n}$ for all $n$. A (_local_) _attractor_ is
a compact invariant set $\mathcal{A}$ admitting a neighbourhood $U$ that is
uniformly attracted to $\mathcal{A}$. An attractor’s basin of attraction is
always an open set. A _global attractor_ is an attractor whose basin is the
whole of $\mathbb{R}\times[0,2\pi)$.

Observe that the map (9) is (assuming periodicity in $\theta$) a smooth skew
product map over the chaotic map $g$ for which Lebesgue measure is ergodic.

The special case $\beta=\sigma=0$ corresponds to a product map with two
attractors

$\mathcal{A}^{\pm}=\\{\pm a_{0}\\}\times[0,2\pi)$

where $a_{0}$ is the positive solution of
$x=\alpha\operatorname{\mathrm{arctan}}x$; for $\alpha=2$, we have
$a_{0}\approx 2.33112$. The attractors share a basin boundary, namely

$\mathcal{C}=\\{0\\}\times[0,2\pi),$

which is an edge state of the system (i.e. it is an attractor for the system
restricted to the basin boundary). Note that $\beta\neq 0$ breaks the symmetry
of the system $x\mapsto-x$ while $\sigma>0$ will provide bounded chaotic
forcing to the dynamics of $x$. For the noise-unforced case $\sigma=0$ we can
say more – that (9) reduces to

$x_{n+1}=\alpha\arctan(x_{n})+\beta$ (13)

which has a double-fold hysteresis of attractors on varying $\beta$, as shown
in Figure 1. Namely, letting $x^{*}$ be the unique positive value at which
$\left.\frac{\partial f}{\partial
x}\right|_{x=x^{*}}=\frac{\alpha}{1+(x^{*})^{2}}=1$, there is111Note that
$x^{*}=\sqrt{\alpha-1}$ and so
$\beta^{*}=\alpha\arctan(\sqrt{\alpha-1})-\sqrt{\alpha-1}$. In the particular
case $\alpha=2$ we have $x^{*}=1$ and $\beta^{*}=\frac{\pi}{2}-1\approx
0.5708$. a $\beta^{*}>0$ and continuously $\beta$-parameterised real numbers
$(a^{+}_{\beta})_{\beta\in[-\beta^{*},\infty)}$,
$(a^{-}_{\beta})_{\beta\in(-\infty,\beta^{*}]}$ and
$(r_{\beta})_{\beta\in[-\beta^{*},\beta^{*}]}$, with $a^{\pm}_{\beta}$
strictly increasing in $\beta$ and $r_{\beta}$ strictly decreasing in $\beta$,
such that the following hold for $\sigma=0$:

  1. 1.

For $-\beta^{*}<\beta<\beta^{*}$ there are two attractors
$\mathcal{A}^{\pm}_{\beta}=\\{a^{\pm}_{\beta}\\}\times[0,2\pi)$, and these two
attractors share the same basin boundary, namely the edge state
$\mathcal{C}_{\beta}=\\{r_{\beta}\\}\times[0,2\pi)$.

  2. 2.

The attractor $\mathcal{A}^{+}$ and edge state collide at a saddle-node in the
$x$ variable at $-\beta^{*}$, where we have
$a^{+}_{-\beta^{*}}=r_{-\beta^{\ast}}=x^{*}$. For $\beta<-\beta^{*}$,
$\mathcal{A}^{-}_{\beta}=\\{a^{-}_{\beta}\\}\times[0,2\pi)$ is a global
attractor.

  3. 3.

The attractor $\mathcal{A}^{-}$ and edge state collide at a saddle-node in the
$x$-variable at $\beta^{*}$, where we have
$a^{-}_{\beta^{*}}=r_{\beta^{*}}=-x^{*}$. For $\beta>\beta^{*}$,
$\mathcal{A}^{+}_{\beta}=\\{a^{+}_{\beta}\\}\times[0,2\pi)$ is a global
attractor.

Figure 1: Plots of $x$ vs $f(x)$ for $\alpha=2$ and $\sigma=0$, varying
$\beta$. There are two stable fixed points in cases where $|\beta|<\beta^{*}$
but only one stable fixed point for $|\beta|>\beta^{*}$.

(a) (b)

(c) (d)

(e) (f)

Figure 2: Phase portraits showing numerical approximations of
$\mathcal{A}^{+}$ (yellow), $\mathcal{A}^{-}$ (red) and $\mathcal{C}$ (blue)
for $\sigma=0.24$ and (a) $\beta=0.0$ (b) $\beta=0.1$ (c) $\beta=0.2$ (d)
$\beta=0.3$ (e) $\beta=0.33079$ (f) $\beta=0.5$. In case (f) the numerically
approximated blue “saddle” are in fact only a transient - forward orbits end
up at $\mathcal{A}^{+}$ if $\beta>\beta^{*}-\sigma=0.3307$. There is a non-
autonomous saddle node of the forced system at $\beta=\pi/2-1\approx
0.330796$.

This picture persists in many regards for $\sigma\neq 0$, as we describe in
the next section.

###  3.1 Parameter dependence for the map

Due to the skew-product structure, the set of tangent vectors parallel to the
$x$-axis in the $(x,\theta)$-state space is an invariant subbundle of the
tangent bundle for each $\beta$ and $\sigma$. Accordingly, for any $\beta$ and
$\sigma$, for any ergodic invariant probability measure $\mu$ on
$\mathbb{R}\times[0,2\pi)$, we write $\lambda_{\beta,\sigma}(\mu)$ for the
Lyapunov exponent of $\mu$ in this invariant subbundle. The basin of
attraction of an invariant probability measure $\mu$ refers to the set of
initial conditions $(x_{0},\theta_{0})$ for which the empirical measure
$\frac{1}{N}\sum_{n=0}^{N-1}\delta_{(x_{n},\theta_{n})}$ converges weakly to
$\mu$ as $N\to\infty$; if the basin of attraction of $\mu$ has positive
Lebesgue measure, then we say that $\mu$ is a _physical measure_ (see e.g.
[35]).

We consider how the dynamics varies qualitatively over the
$(\beta,\sigma)$-parameter space $\mathbb{R}\times[0,\infty)$ shown in Figure
4. Note that in this parameter space, a line of gradient $+1$ corresponds to a
constant value of $\beta-\sigma$ and a line of gradient $-1$ corresponds to a
constant value of $\beta+\sigma$ (with these constant values being equal to
the $\beta$-value at the point where the line intersects $\\{\sigma=0\\}$). We
define the following regions of the parameter space, as depicted in Figure 4:

$\begin{array}[]{rl}S^{-}&=\\{(\beta,\sigma)\in\mathbb{R}\times[0,\infty):\beta+\sigma\leq\beta^{*}\\}\\\
S^{+}&=\\{(\beta,\sigma)\in\mathbb{R}\times[0,\infty):\beta-\sigma\geq-\beta^{*}\\}\\\
\mathrm{A}&=S^{+}\cap
S^{-}=\mathrm{hull}\Big{(}(-\beta^{*},0)\,,\,(0,\beta^{*})\,,\,(\beta^{*},0)\Big{)}\\\
\mathrm{B}^{-}&=\\{(\beta,\sigma)\in\mathbb{R}\times[0,\infty):\beta+\sigma<-\beta^{*}\\}\subset
S^{-}\setminus S^{+}\\\
\mathrm{B}^{+}&=\\{(\beta,\sigma)\in\mathbb{R}\times[0,\infty):\beta-\sigma>\beta^{*}\\}\subset
S^{+}\setminus S^{-}\\\
\mathrm{C}^{-}&=S^{-}\setminus(S^{+}\cup\mathrm{B}^{-})\\\
\mathrm{C}^{+}&=S^{+}\setminus(S^{-}\cup\mathrm{B}^{+})\\\
\mathrm{D}&=\big{(}\mathbb{R}\times[0,\infty)\big{)}\setminus\big{(}S^{+}\cup
S^{-}\big{)}.\end{array}$

By construction, the six regions $\mathrm{A}$, $\mathrm{B}^{-}$,
$\mathrm{C}^{-}$, $\mathrm{D}$, $\mathrm{C}^{+}$, $\mathrm{B}^{+}$ (as seen
going clockwise through Figure 4) are mutually disjoint and form a partition
of the whole parameter space $\mathbb{R}\times[0,\infty)$. Note that the pairs
$S^{\pm}$, $\mathrm{B}^{\pm}$ and $\mathrm{C}^{\pm}$ are respectively each
others reflection in the $\sigma$-axis. The sets $\mathrm{A}$ and $\mathrm{D}$
are both symmetric in the $\sigma$-axis.

Let us also define the closed line-segments:

$L^{-}=\\{(\beta,\beta^{*}+\beta):\beta\in[-\beta^{*},0]\\}\qquad
L^{+}=\\{(\beta,\beta^{*}-\beta):\beta\in[0,\beta^{*}]\\}.$

In other words, $L^{-}$ is the intersection of the boundaries of $\mathrm{A}$
and $\mathrm{C}^{-}$; and $L^{+}$ is the intersection of the boundaries of
$\mathrm{A}$ and $\mathrm{C}^{+}$. So the boundary of $\mathrm{A}$ (relative
to $\mathbb{R}\times[0,\infty)$) is precisely $L^{-}\cup L^{+}$. Note that
$S^{\pm}$ and $\mathrm{A}$ are closed sets; and $\mathrm{B}^{\pm}$ and
$\mathrm{D}$ are (relative to $\mathbb{R}\times[0,\infty)$) open sets. The
sets $\mathrm{C}^{\pm}$ are neither open nor closed: they exclude the side of
their boundary constituted by $L^{-}$ and $L^{+}$ respectively, but they each
include the open half-lines forming the remaining two sides of their three-
sided boundary. Proofs of the statements below (except those regarding region
$\mathrm{D}$) are given in Appendix A.

#### Attractors and physical measures in $S^{\pm}$

In each of the sentences in the bullet points below, two statements are being
made: one where $\pm$ and $\mp$ are consistently substituted for $+$ and $-$
respectively throughout, and one where $\pm$ and $\mp$ are consistently
substituted for $-$ and $+$ respectively throughout.

  * •

For all $(\beta,\sigma)\in S^{\pm}$, there is a unique (so in particular,
ergodic) invariant probability measure $\mu_{\beta,\sigma}^{\pm}$ whose
support $\mathcal{A}_{\beta,\sigma}^{\pm}$ is contained in
$[a^{\pm}_{\beta-\sigma},a^{\pm}_{\beta+\sigma}]\times[0,2\pi)$ and whose
$\theta$-marginal is the normalised Lebesgue measure on $[0,2\pi)$.

(See Secs. A.2, A.5, A.6 and the end of A.15 for statements about $\mu^{+}$;
analogous statements for $\mu^{-}$ hold by Sec. A.9.)

  * •

For $(\beta,\sigma)$ in the interior of $\mathrm{S}^{\pm}$,
$\mathcal{A}_{\beta,\sigma}^{\pm}$ is an attractor.

(See Corollary 3 in Sec. A.8.)

  * •

For $(\beta,\sigma)\in S^{\pm}$, Lebesgue-almost every point in the basin of
$\mathcal{A}_{\beta,\sigma}^{\pm}$ is in the basin of
$\mu_{\beta,\sigma}^{\pm}$; in particular, $\mu_{\beta,\sigma}^{\pm}$ is a
physical measure.

(See Proposition 7 in Sec. A.15.)

  * •

For $(\beta,\sigma)\in S^{\pm}$ with $\sigma>0$, the intersection of
$\mathcal{A}_{\beta,\sigma}^{\pm}$ with the line
$\mathcal{A}_{\beta-\sigma}^{\pm}$ is the fixed point
$(a^{\pm}_{\beta-\sigma},\pi)$ and the intersection of
$\mathcal{A}_{\beta,\sigma}^{\pm}$ with the line
$\mathcal{A}_{\beta+\sigma}^{\pm}$ is the fixed point
$(a^{\pm}_{\beta+\sigma},0)$.222Since $\mathcal{A}_{\beta,\sigma}^{\pm}$ is an
invariant set, and $0$ and $\pi$ are fixed points of $g$, the
$(\theta=0)$-section and the $(\theta=\pi)$-section of
$\mathcal{A}_{\beta,\sigma}^{\pm}$ must be _positively_ invariant sets (i.e.
they map into themselves, but do not have to map onto the whole of
themselves); one can show that they contract in forward time towards the
singletons $\\{(a^{\pm}_{\beta+\sigma},0)\\}$ and
$\\{(a^{\pm}_{\beta-\sigma},\pi)\\}$ respectively, but that if $\sigma>0$ then
they include more than just those singletons and therefore are not invariant
sets.

(See Sec. A.6.)

  * •

For $(\beta,\sigma)\in S^{+}\setminus\\{(\mp\beta^{*},0)\\}$, the Lyapunov
exponent $\lambda_{\beta,\sigma}(\mu_{\beta,\sigma}^{\pm})$ is strictly
negative.

(See Sec. A.7.)

  * •

Over $(\beta,\sigma)\in S^{\pm}$: the dependence of $\mu_{\beta,\sigma}^{\pm}$
on $(\beta,\sigma)$ is continuous in the topology of weak convergence; the
dependence of $\mathcal{A}_{\beta,\sigma}^{\pm}$ on $(\beta,\sigma)$ is
continuous in Hausdorff distance; and the dependence of
$\lambda_{\beta,\sigma}(\mu_{\beta,\sigma}^{\pm})$ on $(\beta,\sigma)$ is
continuous.

(See Sec. A.5.)

#### Basin-boundary dynamics in $\mathrm{A}$

  * •

For $(\beta,\sigma)\in\mathrm{A}$, the basin of attraction of
$\mathcal{A}_{\beta,\sigma}^{-}$ and the basin of attraction of
$\mathcal{A}_{\beta,\sigma}^{+}$ share the same boundary
$\mathcal{C}_{\beta,\sigma}$. This basin boundary is contained in
$[r_{\beta+\sigma},r_{\beta-\sigma}]\times[0,2\pi)$.

(See Sec. A.13, which is contingent on definitions in Sec. A.10.)

  * •

There exists a continuous map $(\beta,\sigma,\theta)\mapsto
r_{\beta,\sigma}(\theta)$ from $\mathrm{A}\times[0,2\pi)$ to
$[-x^{\ast},x^{\ast}]$ such that for each $(\beta,\sigma)\in\mathrm{A}$, the
basin boundary $\mathcal{C}_{\beta,\sigma}$ is precisely the graph
$\\{(r_{\beta,\sigma}(\theta),\theta):\theta\in[0,2\pi)\\}$ of
$r_{\beta,\sigma}$.

(See Secs. A.10, A.11 and A.13.)

  * •

For $(\beta,\sigma)\in\mathrm{A}$ with $\sigma>0$, $r_{\beta,\sigma}(\cdot)$
attains the value $r_{\beta+\sigma}$ uniquely at $\theta=0$, and
$r_{\beta,\sigma}(\cdot)$ attains the value $r_{\beta-\sigma}$ uniquely at
$\theta=\pi$.

  * •

For $(\beta,\sigma)\in\mathrm{A}$, the pushforward $\nu_{\beta,\sigma}$ of the
normalised Lebesgue measure under
$\theta\mapsto(r_{\beta,\sigma}(\theta),\theta)$ is an ergodic invariant
probability measure.

(See Sec. A.11.)

  * •

For
$(\beta,\sigma)\in\mathrm{A}\setminus\\{(-\beta^{\ast},0),(\beta^{\ast},0)\\}$,
the Lyapunov exponent $\lambda_{\beta,\sigma}(\nu_{\beta,\sigma})$ is strictly
positive.

(See Sec. A.11.)

  * •

Due to the continuity of $r_{\beta,\sigma}(\theta)$ over
$(\beta,\sigma,\theta)\in\mathrm{A}\times[0,2\pi)$: the support of
$\nu_{\beta,\sigma}$ is $\mathcal{C}_{\beta,\sigma}$; the dependence of
$\nu_{\beta,\sigma}$ on $(\beta,\sigma)$ is continuous in the topology of weak
convergence; the dependence of $\mathcal{C}_{\beta,\sigma}$ on
$(\beta,\sigma)$ is continuous in Hausdorff distance; and the dependence of
$\lambda_{\beta,\sigma}(\nu_{\beta,\sigma})$ on $(\beta,\sigma)$ is
continuous.

(See Sec. A.11.)

#### Intersections of $\mathcal{A}^{\pm}_{\beta,\sigma}$ with
$\mathcal{C}_{\beta,\sigma}$

Combining some of the points made so far:

  * •

For $(\beta,\sigma)$ in the interior of $\mathrm{A}$,
$\mathcal{C}_{\beta,\sigma}$ has empty intersection with
$\mathcal{A}_{\beta,\sigma}^{-}$ and with $\mathcal{A}_{\beta,\sigma}^{+}$.

  * •

For $(\beta,\sigma)\in L^{+}$, $\mathcal{C}_{\beta,\sigma}$ has non-empty
intersection with $\mathcal{A}_{\beta,\sigma}^{-}$; except at
$(\beta,\sigma)=(\beta^{\ast},0)$, this intersection is the single point
$(-x^{\ast},0)$.

  * •

For $(\beta,\sigma)\in L^{-}$, $\mathcal{C}_{\beta,\sigma}$ has non-empty
intersection with $\mathcal{A}_{\beta,\sigma}^{+}$; except at
$(\beta,\sigma)=(-\beta^{\ast},0)$, this intersection is the single point
$(x^{\ast},\pi)$.

(See Sec. A.12 for all three statements.)

#### Attractor basins in $S^{\pm}\setminus\mathrm{A}$

  * •

For $(\beta,\sigma)\in\mathrm{C}^{+}$, the basin of
$\mathcal{A}_{\beta,\sigma}^{+}$ has full Lebesgue measure and includes
$(r_{\beta-\sigma},\infty)\times[0,2\pi)$.

  * •

For $(\beta,\sigma)\in\mathrm{C}^{-}$, the basin of
$\mathcal{A}_{\beta,\sigma}^{-}$ has full Lebesgue measure and includes
$(-\infty,r_{\beta+\sigma})\times[0,2\pi)$.

  * •

For $(\beta,\sigma)\in\mathrm{B}^{\pm}$, $\mathcal{A}_{\beta,\sigma}^{\pm}$ is
a global attractor.

(See Sec. A.14 for all three statements.)

In all three cases, $\mu_{\beta,\sigma}^{\pm}$ is the only invariant measure
whose $\theta$-projection is the normalised Lebesgue measure. (See the end of
Sec. A.15.)

#### Dynamics in and near $\mathrm{D}$

For each $(\beta,\sigma)\in\mathrm{D}$:

  * •

For every initial condition $(x_{0},\theta_{0})\in\mathbb{R}\times[0,2\pi)$,
the distance of the trajectory $(x_{n},\theta_{n})$ from the set
$[a_{\beta-\sigma}^{-},a_{\beta+\sigma}^{+}]\times[0,2\pi)$ tends to $0$ as
$n\to\infty$.

(See Proposition 8 in Sec. A.16.)

  * •

Every invariant probability measure has its support contained in
$[a_{\beta-\sigma}^{-},a_{\beta+\sigma}^{+}]\times[0,2\pi)$.

(See Sec. A.16, after Proposition 8.)

  * •

Every invariant measure whose $\theta$-marginal is the normalised Lebesgue
measure has the points $(a_{\beta-\sigma}^{-},\pi)$ and
$(a_{\beta+\sigma}^{+},0)$ in its support.

(See Proposition 9 in Sec. A.16.)

  * •

There exists at least one ergodic invariant measure whose $\theta$-marginal is
the normalised Lebesgue measure.

(See Sec. A.16, after Proposition 8.)

Now for all $(\beta,\sigma)\in\mathbb{R}\times[0,\infty)$, the following
statements are equivalent:

  * (i)

There is only one invariant measure whose $\theta$-marginal is the normalised
Lebesgue measure.

  * (ii)

For a uniformly distributed $[0,2\pi)$-valued random variable $\theta$, the
width $X_{\beta,\sigma,n}$ of the $n$-th image of
$\mathbb{R}\times\\{\theta\\}$ under the map (9) converges in probability to
$0$ as $n\to\infty$.

(See Proposition 12 in Sec. A.17.) Note that $X_{\beta,\sigma,n}$ is indeed
finite for each $n\geq 1$, due to Eqs. (11) and (12).

Let $\tilde{\mathrm{D}}$ be the set of all
$(\beta,\sigma)\in\mathbb{R}\times[0,\infty)$ for which the equivalent
statements (i) and (ii) above hold. For each
$(\beta,\sigma)\in\tilde{\mathrm{D}}$, we will write $\mu_{\beta,\sigma}$ for
the unique invariant measure whose $\theta$-marginal is the normalised
Lebesgue measure, and we will write $\mathcal{A}_{\beta,\sigma}$ for the
support of $\mu_{\beta,\sigma}$. Let
$\tilde{\mathrm{D}}_{\lambda<0}\subset\tilde{\mathrm{D}}$ be the set of all
$(\beta,\sigma)\in\tilde{\mathrm{D}}$ for which
$\lambda_{\beta,\sigma}(\mu_{\beta,\sigma})<0$.

We have already established that $\tilde{\mathrm{D}}_{\lambda<0}$ includes
$(S^{-}\cup S^{+})\setminus\mathrm{A}$, and that $\tilde{\mathrm{D}}$ has
empty intersection with $\mathrm{A}$. So to determine the sets
$\tilde{\mathrm{D}}$ and $\tilde{\mathrm{D}}_{\lambda<0}$, all that would
remain is to determine their intersection with $\mathrm{D}$.

  * •

Over $(\beta,\sigma)\in\tilde{\mathrm{D}}$, the dependence of
$\mu_{\beta,\sigma}$ on $(\beta,\sigma)$ is continuous in the topology of weak
convergence.

(See Sec. A.18.)

  * •

For $(\beta,\sigma)\in\tilde{\mathrm{D}}_{\lambda<0}$, $X_{\beta,\sigma,n}$
converges exponentially to $0$ almost surely as $n\to\infty$.

(See Proposition 13 in Sec. A.17.)

  * •

Nevertheless, there exist $(\beta,\sigma)\in\mathrm{D}$ (such as $(0,\sigma)$
for any $\sigma>\beta^{*}$) for which there is no deterministic sequence
$c_{n}>0$ tending to $0$ such that $X_{\beta,\sigma,n}$ is almost surely less
than $c_{n}$.

(See Proposition 14 in Sec. A.17.)

  * •

For $(\beta,\sigma)\in\tilde{\mathrm{D}}_{\lambda<0}$, Lebesgue-almost every
point in $\mathbb{R}\times[0,2\pi)$ is in the basin of attraction of
$\mathcal{A}_{\beta,\sigma}$ and in the basin of attraction of
$\mu_{\beta,\sigma}$; so in particular, $\mu_{\beta,\sigma}$ is a physical
measure.

(See Corollary 16 and Proposition 17 in Sec. A.19.)

We conjecture that $\tilde{\mathrm{D}}_{\lambda<0}$ contains the whole of
$\mathrm{D}$; this is equivalent to saying that

$\tilde{\mathrm{D}}=\tilde{\mathrm{D}}_{\lambda<0}=(\mathbb{R}\times[0,\infty))\setminus\mathrm{A}.$

If this conjecture is true, then the physical measure $\mu_{\beta,\sigma}$ has
continuous dependence in the topology of weak convergence over
$(\beta,\sigma)$ varying throughout the complement of $\mathrm{A}$. We note,
however, that while the dependence in Hausdorff distance of
$\mathcal{A}_{\beta,\sigma}$ is continuous over $(\beta,\sigma)\in(S^{-}\cup
S^{+})\setminus\mathrm{A}$, it cannot extend continuously from $(S^{-}\cup
S^{+})\setminus\mathrm{A}$ into $\mathrm{D}$. Numerical support for this
conjecture is presented in Figure 3. Note that by Eqs. (11) and (12), given
any finite-width interval
$[x_{1}^{-},x_{1}^{+}]\supset\mathrm{I}_{\beta,\sigma}$, any forward
trajectory $(x_{n},\theta_{n})$ of (9) starting at $n=0$ has its $x$-component
lying between the “bounding trajectories”

$x_{n}^{-}\leq x_{n}\leq x_{n}^{+}\quad\forall\,n\geq 1$ (14)

where $(x_{n}^{\pm},\theta_{n})$ are forward trajectories starting at $n=1$
with the pre-fixed $x$-component $x_{1}^{\pm}$. So then, the random variable
$X_{\beta,\sigma,n}$ is bounded above by the width of the $(n-1)$-th image of
$[x_{1}^{-},x_{1}^{+}]\times\\{g(\theta)\\}$ under the map (9). So since the
normalised Lebesgue measure on $[0,2\pi)$ is $g$-invariant, if we fix any
finite-width interval $[x_{0}^{-},x_{0}^{+}]\supset\mathrm{I}_{\beta,\sigma}$,
we have that $(\beta,\sigma)\in\tilde{\mathrm{D}}$ if and only if the width of
the $n$-th image of $[x_{0}^{-},x_{0}^{+}]\times\\{\theta\\}$ under the map
(9) converges in probability to $0$ as $n\to\infty$. So in the left plot of
Figure 3, fixing such values $x_{0}^{-},x_{0}^{+}$ and selecting a value
$\theta_{0}$ from the uniform distribution on $[0,2\pi)$, we simulate
subsequent trajectories $(x_{n}^{\pm},\theta_{n})$ starting at
$(x_{0}^{\pm},\theta_{0})$ and plot over $(\beta,\sigma)$-space the value of
$x_{n}^{+}-x_{n}^{-}$ for a large value of $n$. We see values close to $0$
everywhere except in and near $\mathrm{A}$, supporting the conjecture that
$\tilde{\mathrm{D}}=(\mathbb{R}\times[0,\infty))\setminus\mathrm{A}$. In the
right plot of Figure 3, we plot the fibre LE for the attractor (or the mximum
of the fibre LE for the two attractors) found in the left plot. We see
strictly negative values throughout the whole $(\beta,\sigma)$-parameter
space, supporting the conjecture that $\tilde{\mathrm{D}}_{\lambda<0}$ is the
whole of $\tilde{\mathrm{D}}$.

Figure 3: Left: Parameter space $(\beta,\sigma)$ showing number of attractors
in each region, where blue indicates monostability. Bistability occurs in the
yellow triangular region (see region $\mathrm{A}$ in Figure 4. Right: Largest
fibre Lyapunov exponent (LE) for any attractor shown for each point in
parameter space, computed for a randomly chosen $\theta_{0}$. Note the non-
autonomous saddle-node bifurcations on the boundary of the region of
bistability.

#### Non-autonomous saddle-node bifurcation

The boundary $L^{\pm}$ between regions $\mathrm{A}$ and $\mathrm{C}^{\pm}$ can
be seen as a _non-autonomous saddle-node bifurcation_ [1] where one of the
attractors $\mathcal{A}^{\pm}_{\beta,\sigma}$ collides with the repeller
$\mathcal{C}_{\beta,\sigma}$, albeit just at a single fixed point. Let us
emphasise that in contrast to autonomous saddle-node bifurcations (where
$\sigma=0$), for $\sigma>0$ the Lyapunov exponent
$\lambda^{\pm}_{\beta,\sigma}$ is strictly negative even at the critical
boundary in the $(\beta,\sigma)$-parameter space.

###  3.2 Trajectories and UPOs

Figure 2 illustrates typical dynamical behaviours one can find for the map
(9)–(10). The yellow and red trajectories are attractors found by iterating
forwards respectively from $(\pm 2.3,0.1)$ and plotting $N=10^{6}$ points
after omitting the first $T=50$ points as transient: they approximate
$\mathcal{A}^{\pm}_{\beta,\sigma}$. The blue trajectory is found by backwards
iteration of the $x$ dynamics: this approximates the saddle/repeller
$\mathcal{C}_{\beta,\sigma}$. A forwards orbit $\theta_{n}=g^{n}(\theta_{0})$
is generated then we choose $x_{N+T}=0$ and then define

$x_{n-1}=h(x_{n},\theta_{n-1})$

where $h(x,\theta)=\tan((x-\beta-\sigma\cos\theta)/\alpha)$: note that this is
the inverse of $f$ in the sense that

$h(f(x,\theta),\theta)=x.$

The trajectory $(x_{i},\theta_{i})$ is plotted for $i=1,\ldots,N$; the saddle-
node bifurcation lines forced by unstable periodic orbits (UPOs) in Figure 4.

We construct UPOs of period $N$ for $g$ symbolically by choosing a sequence
$a_{i}\in\\{0,1,2\\}$ for $i=1..N$ and noting that the unique periodic point
with coding $\\{a_{i}\\}_{i=1}^{N}$ is

$\theta=2\pi\sum_{i=1}^{N}a_{i}\frac{3^{N-i}}{3^{N}-1}.$

We force with the orbit of this point to numerically determine the saddle-node
bifurcations for the return map forced over one period of the UPO; this is
shown in Figure 4 for all orbits of period $N\leq 3$ and selected orbits with
$N=4$.

Figure 4: Division of the $(\beta,\sigma)$-parameter space
$\mathbb{R}\times[0,\infty)$ for (9) and $\alpha=2$, into regions exhibiting
qualitatively different dynamics. We discuss regions
$S^{-}=\mathrm{B}^{-}\cup\mathrm{C}^{-}\cup\mathrm{A}$ and
$S^{+}=\mathrm{B}^{+}\cup\mathrm{C}^{+}\cup\mathrm{A}$ in the text. Note that
the system is bistable in region $\mathrm{A}$ and monostable elsewhere (see
Figure 3). For parameters in regions $\mathrm{C}^{\pm}$ the attractor is
localised to one of the branches of attractors for $\sigma=0$ but there is
non-uniform approach to the attractor. In region $\mathrm{D}$ the attractors
have fused into one large attractor. The solid/dashed/dotted/dash-dotted lines
correspond to saddle-node bifurcations forced respectively by unstable fixed
points/period 2 points/period 3 points/selected period 4 points of the
$\theta$ dynamics.

###  3.3 Geometry and dimension of the chaotic saddle

In cases of bistability, the basin boundary contains a chaotic saddle
$\mathcal{C}_{\beta,\sigma}$ (actually a repeller) that occupies all of the
boundary. As observed in Figure 2, this invariant set is a repeller of the
skew product system whose geometry we remark on here in relation to the
literature of repellers in skew affine systems; see for example [33, 32]. In
this section we give some numerical estimates of the dimension of
$\mathcal{C}_{\beta,\sigma}$ both via a box counting algorithm [10] and using
the Lyapunov exponents and escape time from $\mathcal{C}_{\beta,\sigma}$ to
give a Lyapunov dimension [18], as estimates of the Hausdorff dimension of
$\mathcal{C}_{\beta,\sigma}$.

Note that for the affine skew product

$x_{n+1}=[\lambda(\theta_{n})]^{-1}x_{n}+\sigma\cos(\theta_{n})$

with $\lambda(\theta)>1$ there will be a repelling invariant graph [6] whose
box dimension is given in terms of the thermodynamic formalism by [6] as a $t$
that is a zero of a pressure function that in our case is

$\mathcal{P}\left((1-t)\ln|\partial_{\theta}g|-\ln|\partial_{x}f|\right)=0$

also called Bowen’s formula [6, 33]. In the case that the expansion in the
$\theta$ direction is uniform at rate $B>1$ and in the $x$ direction is
uniform at rate $A>1$ this reduces to

$\ln B+(1-t)\ln B-\ln A=0$

and so it is possible to show that the invariant graph has Hausdorff dimension
$2-\ln(A)/\ln(B)$. These results do not obviously extend to nonlinear skew
products so we explore the dimensions of the chaotic saddle numerically.

####  3.3.1 Box-counting dimension

Making use of the skew product structure of the system we approximate the
shape of the saddle via forward and backward iterations as follows: We choose
a box $B_{\beta,\sigma}=[x_{l},x_{u}]\times[\theta_{l},\theta_{u}]$ that
contains a part of $\mathcal{C}_{\beta,\sigma}$, and randomly choose $k$
values of $\theta\in[\theta_{l},\theta_{u}]$. Iterating the base map (i.e.
$\theta_{n+1}=3\theta_{n}\mod 2$) forward for $N$ iterations and storing the
trajectories, allows evaluation of the inverted fibre map under $N$ iterations
backward in time. Note that under time inversion, the fibre map becomes
attracting towards $\mathcal{C}_{\beta,\sigma}$. This procedure results in an
approximation of the saddle with $k$ points.

Then, we compute the saddle’s box-counting dimension. This is done by
considering the number $T(\kappa)$ of non-intersecting boxes with edge-length
$\kappa$ that is needed to cover $\mathcal{C}_{\beta,\sigma}$. The dimension
is then given by:

$\displaystyle d_{box}(\mathcal{C}_{\beta,\sigma})$
$\displaystyle=\lim_{\kappa\to 0}\frac{\ln T(\kappa)}{\ln(1/\kappa)}$ (15)

Rearranging this formula for a finite value $\kappa$ gives

$\displaystyle\ln T(\kappa)$ $\displaystyle\propto
d_{box}(\mathcal{C})\cdot\ln T(\kappa).$ (16)

Thus, plotting $\ln T(\kappa)$ vs. $\ln T(\kappa)$, and assuming that this
relation correctly describes the behaviour for small $\kappa$ allows an
estimation of the box counting dimension via a linear fit[31]. The results are
presented in Table 1.

####  3.3.2 Lyapunov dimension

Using results from [18], we can calculate the Lyapunov dimension of
$\mathcal{C}_{\beta,\sigma}$ from the Lyapunov exponents as

$\displaystyle D_{L}$ $\displaystyle=J+\frac{H-(h_{1}^{+}+\cdot\cdot\cdot
h_{J}^{+})}{h_{J+1}^{+}},$ (17)

with $h_{1}^{+}\leq h_{2}^{+}$ the two (positive) Lyapunov exponents, and
$J=1$ given by

$\displaystyle h_{1}^{+}+\cdot\cdot\cdot h_{J}^{+}+h_{J+1}^{+}\geq H\geq
h_{1}^{+}+\cdot\cdot\cdot h_{J}^{+}.$ (18)

$H$ is defined as

$\displaystyle H$ $\displaystyle=\sum_{i=1}^{2}h_{i}^{+}-\tau^{-1},$ (19)

and $\tau$ is the mean escape time from $\mathcal{C}_{\beta,\sigma}$. This is
defined by
$\tau^{-1}=-\lim_{n\rightarrow\infty}\frac{1}{n}\ln\left(\frac{\ell(R^{(n)})}{\ell(R)}\right)$
where $R$ is a box that contains $\mathcal{C}_{\beta,\sigma}$ but no other
invariant sets. $\ell(R)$ is the Lebesgue measure of the set $R$, and
$\ell(R^{(n)})$ the Lebesgue measure of the set of trajectories that are still
in $R$ after $n$ iterations. A numerical approximation can be done with a
linear fit analogously to the procedure described after equation 16 for the
box-counting dimension.

To compute the Lyapunov exponents of $\mathcal{C}_{\beta,\sigma}$, we make use
of the skew product structure of the system (9) and Birkhoff’s ergodic
theorem. These allow us to compute the Lyapunov exponents as ensemble averages
over the logarithm of the directional derivatives of sample points drawn from
the invariant measure on the saddle $\mathcal{C}_{\beta,\sigma}$. We
approximate this invariant measure via the forward and backward evolution
algorithm as described in Section 3.3.1 to compute the Lyapunov exponents:

$\displaystyle\lambda_{1}$
$\displaystyle=\left\langle\ln\bigg{|}\frac{\partial
f(x,\theta,\beta,\sigma)}{\partial x}\bigg{|}\right\rangle$ (20)
$\displaystyle\lambda_{2}$ $\displaystyle=\ln(3)=1.0986$ (21)

where $\langle\cdot\rangle$ denotes the average over an ensemble of $10^{4}$
points chosen randomly with respect to the invariant measure on the saddle.
Using these Lyapunov exponents gives the Lyapunov dimensions shown in Table 1.
The parameters used for the sampling of $\mathcal{C}_{\beta,\sigma}$ are shown
in Table 2. We note that the numerically estimated dimensions are limited by
various factors, in particular the sampling of $\mathcal{C}_{\beta,\sigma}$
may be quite sparse in places.

$\sigma$ | $\beta$ | $\lambda_{1}(\mathcal{C}_{\beta,\sigma})$ | $\tau^{-1}$ | $d_{box}(\mathcal{C}_{\beta,\sigma})$ | $d_{L}(\mathcal{C}_{\beta,\sigma})$
---|---|---|---|---|---
0.24 | 0.0 | 0.684 | 0.684 | 1.326 | 1.378
0.24 | 0.1 | 0.673 | 0.673 | 1.336 | 1.388
0.24 | 0.2 | 0.640 | 0.636 | 1.360 | 1.421
0.24 | 0.3 | 0.580 | 0.572 | 1.429 | 1.480
Table 1: Numerically estimated fibre Lyapunov exponent $\lambda_{1}$, mean escape time $\tau$, box counting dimension $d_{box}$ and Lyapunov dimensions $d_{L}$ of $\mathcal{C}_{\beta,\sigma}$ for a range of $\beta$ and $\sigma=0.24$ such that there is bistability. Observe the monotonic increase in both dimension estimates on increasing $\beta+\sigma$ towards to the non-autonomous saddle-node bifurcation at $\beta^{*}$. $\sigma$ | $\beta$ | $n_{\theta}$ | $B_{[x_{l},x_{u}]}$ | $B_{[\theta_{l},\theta_{u}]}$ | $R_{[x_{l},x_{u}]}$ | $R_{[\theta_{l},\theta_{u}]}$
---|---|---|---|---|---|---
0.24 | 0.0 | 476 | [0.155, 0.176] | [2.925, 2.95] | [-0.3, 0.3] | [0.0, $2\pi$]
0.24 | 0.1 | 441 | [0.0523, 0.075] | [2.925, 2.95] | [-0.4, 0.2] | [0.0, $2\pi$]
0.24 | 0.2 | 333 | [-0.055, -0.025] | [2.925, 2.95] | [-0.55, 0.05] | [0.0, $2\pi$]
0.24 | 0.3 | 200 | [-0.175, -0.125] | [2.925, 2.95] | [-0.75, -0.05] | [0.0, $2\pi$]
Table 2: Parameters chosen to compute the box counting dimensions of
$\mathcal{C}_{\beta,\sigma}$ for different values of $\sigma$ and $\beta$.
$n_{\theta}$ is the number of grid points in $\theta-$direction (resolution of
the saddle in $\theta$-direction), $B_{[x_{l},x_{u}]}$ and
$B_{[\theta_{l},\theta_{u}]}$ give the size of the box $B_{\beta,\sigma}$ in
which we approximated the saddle, and $R_{[x_{l},x_{u}]}$ and
$R_{[\theta_{l},\theta_{u}]}$ give the size of the box $R_{\beta,\sigma}$ that
contains the saddle but no other invariant set. The following parameters are
the same for all values of $\beta$ and $\sigma$: the number of considered
orbits is $k=10^{6}$, the time of forward and backward iterations is
$N=10^{3}$, and the number of grid points in $x$-direction (resolution of the
saddle in $x$-direction) is $n_{x}=400$.

##  4 Slowly varying parameters and the dynamic tipping window

Returning to a discrete version of (7) where we allow the parameters to slowly
vary in addition to chaotic forcing, one can study the interaction between
bifurcation and noise induced effects (B- and N-tipping in the terminology of
[4]). We investigate this by ramping the parameter beta

$\beta_{n}=\beta_{0}+n\epsilon$ (22)

for small $\epsilon$ and simulate a non-autonomous version of (9), namely:

$\displaystyle x_{n+1}=$ $\displaystyle f(x_{n},\theta_{n},\beta_{n})$ (23)
$\displaystyle\theta_{n+1}=$ $\displaystyle g(\theta_{n}).$

We give an initial condition with negative $x_{0}$, typical $\theta_{0}$ and
$\beta_{0}=0$ and fix $\alpha=2$ as above and find that the chaotic tipping
window at $\epsilon=0$ is the limit of a dynamic tipping window for
$\epsilon>0$.

Figure 5 illustrates trajectories of this non-autonomous map (23): these
quickly converge to a neighbourhood of $\mathcal{A}^{-}_{\beta,\sigma}$ for
slow ramping ($\epsilon=0.001$) and various noise amplitudes $\sigma$. We plot
$x$ against $\beta$ and show is the location of the hysteresis curve for
$\epsilon=\sigma=0$, namely the unstable fixed points of (13) which satisfy
$\beta=x-\alpha\arctan(x)$. Note that in the absence of noise $\sigma=0$ the
system tracks the deterministic lower attractor to the tipping point and only
transitions after that. For progressively larger amplitude noise and fixed
$\epsilon$ we find progressively earlier transitions.

Figure 5: Plots of typical trajectories of (23) with $\beta_{n}$ ramped at a
constant rate $\epsilon=0.001$ and a range of $\sigma$, for $\alpha=2$. The
fixed points of (13) are also shown. Observe the delay of tipping to after the
bifurcation in the smallest $\sigma$, but an advance tipping in the larger
$\sigma$.

Figure 6 explores the dependence of the response of (23) on varying rate of
ramping $\epsilon$ for a fixed noise amplitude $\sigma=0.2$. Considering a
variety of UPOs of $g$ with periods up to six we plot (in colours) the
trajectories of the ramped system corresponding to the lower attractor as
$\beta$ increases. We show a typical trajectory in bold black, and the
hysteresis curve of the unforced fixed points $\beta(x)$ in non-bold black.
The dashed black lines correspond to $\beta(x)\pm\sigma$.

There is a non-autonomous saddle-node bifurcation at the turning point of the
leftmost black dotted line, where $\beta=\beta^{*}-\sigma$. This is the start
of the chaotic tipping window and for typical $\theta$ the tipping can occur
close to there by choosing arbitrarily small $\epsilon$. This corresponds also
to the case of forcing by the extreme cases of the UPO that determine start
and end of the chaotic tipping window. More precisely, the monotonicity (11)
and boundedness of the $x$ dynamics holds for (23) even in the case where
$\beta_{n}$ varies333Or indeed $\sigma$! with $n$. In consequence, for any
given trajectory $\theta_{n}$ of $g$ we can find bounding trajectories
$(x^{+}_{n},\theta_{n})$ and $(x^{-}_{n},\theta_{n})$ for $n\geq 1$ of (23)
such that for any trajectory $(x_{n},\theta_{n})$ of (23) we have bounds as in
(14).

The sequence Figure 6(a)-(c) shows the effect of progressively faster ramping:
all of the UPO forced trajectories tip over a range of values of $\beta$
around the unforced value $\beta^{*}$, and this range becomes biased to larger
$\beta$ as $\epsilon$ increases. The typical trajectory (bold black) tips at
some point that also becomes progressively later for $\epsilon$ larger. This
means we can define a dynamic tipping window; the smallest interval that
contains the set of all $\beta$ such that tipping for the non-autonomous
system occurs at this $\beta$ with chaotic forcing typical for some measure
$s\in\mathcal{S}(m)$. Figure 7 shows a numerical approximation of this dynamic
tipping window for fixed $\sigma$ and varying $\epsilon$. Unlike the chaotic
tipping window, the dynamic tipping window will depend on a chosen threshold
to identify tipping. In our case we say there has been tipping from the lower
branch at time $n$ if $x_{n}\geq 1$. However, the dynamic tipping window
clearly limits to the chaotic tipping window for $\epsilon\rightarrow 0$.

(a)

(b)

(c)

Figure 6: Plots of trajectories of (23) with $\beta_{n}$, $\sigma=0.2$ and
$\alpha=2$, for a variety of UPOs of period up to period 6. The fixed points
of (13) are also shown. A typical trajectory with chaotic forcing and these
parameters is also shown (bold black line). We use $\beta_{n}=n\epsilon$ for
(a) $\epsilon=0.0001$, (b) $\epsilon=0.001$ and (c) $\epsilon=0.01$. Figure 7:
The dynamic tipping window for $\sigma=0.2$ and $\alpha=2$ for a range of
values of $\epsilon\geq 10^{-5}$. The values of $\beta$ at tipping are shown
for a number of UPOs of period up to period 5. The chaotic tipping window for
$\epsilon=0$ is bounded by the values of $\beta$ at tipping for fixed points
of (13) shown with coloured stars. The distribution of tipping $\beta$ for an
ensemble of $300$ values of $\theta$ from a uniform distribution is shown in
terms of median and lower/upper quartiles (black stars). Note that the median
passes the unforced tipping $\beta$ (blue star) at around $\epsilon=0.001$
(cf. Figure 9).

In principle, typical trajectories can tip arbitrarily close to the non-
autonomous saddle node, but because of super persistent chaotic transients
that will exist close to a boundary crisis [16, 14, 24] this will only be for
$\epsilon$ that is exponentially small with respect to $\sigma$.

This is explored further in Figure 8 which shows numerically computed cdfs of
the value of $\beta$ at tipping, measured as the value of $\beta$ where $x=1$
is first exceeded. Observe this cannot occur before the red line which
indicates the start of a dynamic tipping window computed with forcing that is
an extremal UPO. There is a competition between the rate of ramping $\epsilon$
and the amplitude of the noise $\sigma$: when $\epsilon$ is small then the
median tipping location occurs before the unforced threshold, while when it
exceed some rate the tipping location can be delayed. The relationship between
$\sigma$, $\epsilon$ and the median time of tipping is shown in Figure 9,
together with the corresponding cdfs for the value of $\beta$ at tipping.

Analogous to the scaling for the “strong noise” and “weak noise” cases that
are distinguished in [7, Section 3.3], for $\epsilon\ll\sigma^{2}$ the median
tipping will be before $\beta^{*}$ and so the chaotic forcing dominates, while
for $\epsilon\gg\sigma^{2}$ the median tipping will be after $\beta^{+}$ and
the bifurcation dominates. It is somewhat surprising that the dividing line of
the scaling is so precise when considering the median tipping $\beta$ though
this is almost certainly due to the special form of the map.

Figure 8: Cumulative density functions (blue) of $\beta$ where tipping occurs
for an ensemble of $N=300$ initial conditions, for $\sigma=0.2$ $\alpha=2$ and
$\beta_{n}$ increasing by $\epsilon$ per timestep, as in Figure 6. We say the
tipping occurs when $x$ first exceeds $1$. The vertical lines show the
locations of the chaotic tipping window (first tipping for UPO forcing) in
red, and last tipping for UPO forcing in purple. The tipping for $\sigma=0$ is
shown as vertical yellow line. Observe that for the smallest $\epsilon$ the
median is close to the start of the chaotic tipping window.

Figure 9: Left: relationship between $\sigma$ and $\epsilon$ such that
$\beta_{m}$, the median of distribution of chaos-forced tipping, is equal to
$\beta^{*}$. Note the close fit to $\epsilon\sim\sigma^{2}$ for the case where
$\beta_{m}$ is an unbiased estimator of $\beta^{*}$. Right: the cdfs of four
ensembles of $N=300$ initial conditions are shown within the dynamic tipping
windows (as in Figure 8). Note that the steps become discretized for large
$\epsilon$ because only $\beta_{n}=n\epsilon$ are considered here.

##  5 Discussion

In this paper we have explored the effect of bounded chaotic forcing as a
model for noise that we argue may in many applications be closer to the truth
than the stochastic forcing, i.e. unbounded Wiener noise that can be obtained
in a limit of fast chaos.

By analysing an explicit example of a discrete time bistable map with chaotic
forcing we are able to understand the interaction between multiple attractors
and bounded noise forcing and the appearance of a chaotic tipping window. The
start of this window can be seen as a boundary crisis [24] of an attractor of
the skew product system or as a non-autonomous saddle-node bifurcation of the
forced system. This becomes a dynamic tipping window if there is a slow
variation of parameters in addition to chaotic forcing.

The problem of precisely which situations of forcing give rise to tipping and
at what time is likely to be quite complicated in the presence of non-uniform
parameter variation, especially when the timescale of change of $\beta$ is
faster than the timescales associated with attraction to and mixing on the
chaotic attractor. The chaos-forced transitions discussed can be related to
bifurcations in set-valued dynamical systems such as investigated in [25] as a
model for bounded noise forced systems.

In this paper, although we initially discuss continuous time models with
chaotic forcing, we do not consider any explicit examples for continuous time
models with chaos. These have been considered elsewhere, for example [2, 5]
consider a bistable (Stommel) system forced by a chaotic (Lorenz) system as a
conceptual model for tipping of the Atlantic Meridional Overturning
Circulation (AMOC) in response to fast chaotic variability, looking at cases
where the system limits to an autonomous system in the past and future limits.

Some of the results here can presumably be generalized to more general cases
of non-autonomous parameter variation. For example, considering
$\beta_{n}=\max(0,\min(n\epsilon,\beta^{\max}))$ can be investigated using
examples as in Section 4. Such a case is analogous to a discrete “parameter
shift” between $\beta=0$ and $\beta=\beta^{\max}>0$ at a rate $\epsilon$ [4,
3]. By picking $\theta_{0}$ and $x_{0}$ on one of the attractors for $\beta=0$
and then fixing $\beta^{\max}$ we expect to find different behaviours
depending on where the chaotic tipping window lies in relation to
$[0,\beta^{\max}]$.

The tipping thresholds in general will involve considering extreme measures of
the forcing map that may not be so simple as in this example: see [19] for a
recent review. For the example considered, one can include chaotic forcing of
arbitrary amplitude by varying the parameter $\sigma$. Nonetheless, the large
positive Lyapunov exponent $\ln(3)$ of the forcing chaos is always fast with
respect to the dynamics of forced map unless $|\beta|$ is large. There are
likely to be other effects present when the chaotic forcing becomes comparable
or weaker than the attraction for the forced map, and it may be interesting to
investigate such cases. We do not discuss here the problem of finding early
warnings of an impending tipping point in this context. However, the
observation that the fibre LE does not need to be zero at the crisis suggests
that methods based purely on LEs are likely to miss prediction of an impending
tipping point. See also for example [21, 23]. Conversely, as suggested in
[30], it may be possible to use the detailed dynamics of the forced system to
control the location of the tipping point or the behaviour after tipping. This
will effectively require coupling from the forced system back to the forcing,
and we do not consider this here.

### Acknowledgements

PA and JN received funding from the European Union’s Horizon 2020 research and
innovation programme under grant agreement No. 820970 (TiPES). RR received
funding from the European Union’s Horizon 2020 Research and Innovation
Programme under the Marie Skłodowska-Curie Grant Agreement No. 956170
(CriticalEarth).

### Data availability

The Matlab and Julia code to generate the figures in this paper is available
from https://github.com/peterashwin/tipping-windows-2024.

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##  Appendix A Proofs of dynamical properties

The system (9) that we consider exhibits non-autonomous saddle-node
bifurcations as studied in [1]. However, due to the special structure of our
system, we can give more precise conclusions than in the general setup of [1],
using elementary proofs.

###  A.1 Preliminaries

By way of general notational convention, given an injective function $f$, we
will write $f^{-1}$ to denote the inverse of $f$ defined on the range of $f$.
For $x\in\mathbb{R}$ and $\delta>0$, $B_{\delta}(x):=(x-\delta,x+\delta)$.

Recall that $[0,2\pi)$ is endowed with the topology of a circle, and that
$g\colon[0,2\pi)\to[0,2\pi)$ is the tripling map,
$g(\theta)=3\theta~{}(\bmod~{}2\pi)$. We define the notations below where we
fix $\alpha>1$.

$\displaystyle f_{\beta}\colon\overline{\mathbb{R}}$
$\displaystyle\to\mathbb{R}$ $\displaystyle f_{\beta}(x)$
$\displaystyle=\alpha\operatorname{\mathrm{arctan}}(x)+\beta\qquad\forall\,\beta\in\mathbb{R},\,x\in\overline{\mathbb{R}}$
$\displaystyle f^{\prime}\colon\mathbb{R}$ $\displaystyle\to\mathbb{R}$
$\displaystyle f^{\prime}(x)$
$\displaystyle=\frac{\alpha}{1+x^{2}}=f_{\beta}^{\prime}(x)\qquad\textrm{(indep.\
of $\beta$)}$ $\displaystyle f_{\beta,\sigma,\theta}$
$\displaystyle=f_{\beta+\sigma\cos(\theta)}\qquad\forall\,\beta\in\mathbb{R},\,\sigma\geq
0,\,\theta\in[0,2\pi)$ $\displaystyle f_{\beta,\sigma,\theta,n}$
$\displaystyle=f_{\beta,\sigma,g^{n-1}(\theta)}\circ\ldots\circ
f_{\beta,\sigma,g(\theta)}\circ
f_{\beta,\sigma,\theta}\qquad\forall\,n\in\mathbb{N}_{0}$ $\displaystyle
G_{\beta,\sigma}\colon\mathbb{R}\times[0,2\pi)$
$\displaystyle\to\mathbb{R}\times[0,2\pi)$ $\displaystyle
G_{\beta,\sigma}(x,\theta)$
$\displaystyle=(f_{\beta,\sigma,\theta}(x),g(\theta)).$

So $G_{\beta,\sigma}$ is precisely the function mapping $(x_{n},\theta_{n})$
to $(x_{n+1},\theta_{n+1})$ in the difference equation (9). Note that for all
$n\in\mathbb{N}_{0}$ and $(x,\theta)\in\mathbb{R}\times[0,2\pi)$,

$G_{\beta,\sigma}^{n}(x,\theta)=(f_{\beta,\sigma,\theta,n}(x),g^{n}(\theta)).$

Note that the function $f_{0}=\alpha\operatorname{\mathrm{arctan}}$ is an odd
function, and that for each $\beta$, the function $f_{\beta}=f_{0}+\beta$ is
strictly increasing on $\overline{\mathbb{R}}$, strictly convex on
$(-\infty,0]$, and strictly concave on $[0,\infty)$, and that for all $y$ in
the range of $f_{\beta}$,

$f_{\beta}^{-1}(y)=\tan\\!\left(\tfrac{y-\beta}{\alpha}\right).$ (24)

Recall the $\beta$-parameterised families $a^{\pm}_{\beta}$ and $r_{\beta}$
described in Sec. 3:

  * •

for $\beta\in[-\beta^{*},\infty)$, $f_{\beta}$ has a unique fixed point
$a^{+}_{\beta}$ in $[x^{*},\infty)$;

  * •

for $\beta\in(-\infty,\beta^{*}]$, $f_{\beta}$ has a unique fixed point
$a^{-}_{\beta}$ in $(-\infty,-x^{*}]$;

  * •

for $\beta\in[-\beta^{*},\beta^{*}]$, $f_{\beta}$ has a unique fixed point
$r_{\beta}$ in $[-x^{*},x^{*}]$;

where $x^{*}>0$ is the unique nonnegative root of $f^{\prime}=1$ (so $-x^{*}$
is the unique nonpositive root of $f^{\prime}=1$), and $\beta^{*}>0$ is the
critical $\beta$-value for the saddle-node bifurcation at which
$a^{-}_{\beta}$ and $r_{\beta}$ collide (so $-\beta^{*}$ is the critical
$\beta$-value for the saddle-node bifurcation at which $a^{+}_{\beta}$ and
$r_{\beta}$ collide). The fixed points of $f_{\beta}$ and their stability can
be visualised from Fig. 1. Note that $x^{*}$ and $\beta^{*}$ depend on
$\alpha>1$.

Now $x^{*}$ a fixed point of $f_{-\beta^{*}}$, and so $[x^{*},\infty]$ is
mapped into itself by $f_{-\beta^{*}}$; and so we can define a function
$\varphi\colon[0,\infty]\to[0,\infty]$ by

$\displaystyle\varphi(x)$ $\displaystyle=f_{-\beta^{*}}(x^{*}+x)-x^{*}$
$\displaystyle=\alpha\operatorname{\mathrm{arctan}}(x^{*}+x)-\alpha\operatorname{\mathrm{arctan}}(x^{*})$
$\displaystyle=f_{\beta}(x^{*}+x)-f_{\beta}(x^{*})\qquad\textrm{for all
}\beta.$

Note that $\varphi$ is increasing on $[0,\infty]$; and since $x^{*}$ is
attracting from the right under $f_{-\beta^{*}}$, we have

$\varphi^{n}(\infty)=f_{-\beta^{*}}^{n}(\infty)-x^{*}\to 0\quad\textrm{ as
}n\to\infty.$

Similarly, since $a_{-\beta^{*}}^{-}$ and $x^{*}$ are fixed points of
$f_{-\beta^{*}}$, $[a_{-\beta^{*}}^{-},x^{*}]$ is mapped bijectively onto
itself by $f_{-\beta^{*}}$; and so we can define a function
$\psi\colon[0,x^{*}-a_{-\beta^{*}}^{-}]\to[0,x^{*}-a_{-\beta^{*}}^{-}]$ by

$\displaystyle\psi(y)$ $\displaystyle=x^{*}-f_{-\beta^{*}}^{-1}(x^{*}-y)$
$\displaystyle=x^{*}-\tan\\!\left(\operatorname{\mathrm{arctan}}(x^{*})-\tfrac{y}{\alpha}\right)$
$\displaystyle=x^{*}-f_{\beta}^{-1}(f_{\beta}(x^{*})-y)\qquad\textrm{for all
}\beta.$

Note that $\psi$ is increasing on $[0,x^{*}-a_{-\beta^{*}}^{-}]$; and since,
under $f_{-\beta^{*}}$, there is a heteroclinic connection from the left-
sided-repelling point $x^{*}$ to the right-sided-attracting point
$a_{-\beta^{*}}^{-}$, we have that for any $y\in[0,x^{*}-a_{-\beta^{*}}^{-})$,

$\psi^{n}(y)=x^{*}-(f_{-\beta^{*}}^{n})^{-1}(x^{*}-y)\to 0\quad\textrm{ as
}n\to\infty.$

Let $\mathbb{P}$ be the normalised Lebesgue measure on $[0,2\pi)$. We have
that $\mathbb{P}$ is a $g$-ergodic invariant probability measure. Hence in
particular, $\mathbb{P}$ cannot be expressed non-trivially as a weighted
average of $g$-invariant probability measures; consequently, since the ergodic
decomposition theorem gives that every $G_{\beta,\sigma}$-invariant
probability measure can be expressed as a weighted average of
$G_{\beta,\sigma}$-ergodic invariant probability measures, we have the
following:

######  Lemma 1.

For any Borel set $A\subset\mathbb{R}\times[0,2\pi)$,

  * •

if there exists a $G_{\beta,\sigma}$-invariant probability measure with
$\theta$-marginal $\mathbb{P}$ that assigns full measure to $A$, then there
exists a $G_{\beta,\sigma}$-ergodic invariant probability measure with
$\theta$-marginal $\mathbb{P}$ that assigns full measure to $A$;

  * •

if there exists more than one $G_{\beta,\sigma}$-invariant probability measure
with $\theta$-marginal $\mathbb{P}$ that assigns full measure to $A$, then
there exists more than one $G_{\beta,\sigma}$-ergodic invariant probability
measure with $\theta$-marginal $\mathbb{P}$ that assigns full measure to $A$.

Let $\Theta$ denote the set of all two-sided orbits of $g$, that is

$\Theta=\\{(\theta_{i})_{i\in\mathbb{Z}}\in[0,2\pi)^{\mathbb{Z}}\,|\,\forall\,i\in\mathbb{Z},\theta_{i+1}=g(\theta_{i})\\}.$

Note that since $[0,2\pi)$ is compact under its topology as a circle and $g$
is continuous, $\Theta$ is compact. Let $\tilde{\mathbb{P}}$ be the unique
probability measure on $\Theta$ with the property that for every
$i\in\mathbb{Z}$, the $i$-th marginal of $\tilde{\mathbb{P}}$ is $\mathbb{P}$.

We define the $(\beta,\sigma)$-parameter regions $\mathrm{A}$,
$\mathrm{B}^{\pm}$, $\mathrm{C}^{\pm}$, $\mathrm{D}$ and $S^{\pm}$ as in Sec.
3.1.

###  A.2 Construction of $\mu_{\beta,\sigma}^{+}$

For all $(\beta,\sigma,\theta)\in S^{+}\times[0,2\pi)$, since

$\beta+\sigma\cos(\theta)\geq\beta-\sigma\geq-\beta^{*},$

we have

$f_{\beta,\sigma,\theta}(x^{*})\geq f_{\beta-\sigma}(x^{*})\geq
f_{-\beta^{*}}(x^{*})=x^{*}$

and so

$f_{\beta,\sigma,\theta}([x^{*},\infty])\subset[x^{*},\infty).$

Therefore, for all $(\beta,\sigma)\in S^{+}$ and
$\boldsymbol{\theta}=(\theta_{i})_{i\in\mathbb{Z}}\in\Theta$, noting that

$f_{\beta,\sigma,\theta_{-(n+1)},n+1}=f_{\beta,\sigma,\theta_{-n},n}\circ
f_{\beta,\sigma,\theta_{-(n+1)}}\quad\forall\,n\geq 0,$

we have (by induction) that the sequence of sets
$f_{\beta,\sigma,\theta_{-n},n}([x^{*},\infty])$ is nested, with diameter
bounded above by $\varphi^{n}(\infty)$; and since $\varphi^{n}(\infty)\to 0$
as $n\to\infty$, the intersection of this sequence of sets is a singleton
$\\{a^{+}_{\beta,\sigma}(\boldsymbol{\theta})\\}$.

For each $(\beta,\sigma)\in S^{+}$, we define the probability measure
$\mu_{\beta,\sigma}^{+}$ on $\mathbb{R}\times[0,2\pi)$ to be the pushforward
of $\tilde{\mathbb{P}}$ under the map
$\Pi^{+}_{\beta,\sigma}\colon\boldsymbol{\theta}\mapsto(a^{+}_{\beta,\sigma}(\boldsymbol{\theta}),\theta_{0})$,
where $\theta_{0}$ denotes the $0$-coordinate of $\boldsymbol{\theta}$. Note
that the $\theta$-marginal of $\mu_{\beta,\sigma}^{+}$ is the same as the
$\theta_{0}$-marginal of $\tilde{\mathbb{P}}$, namely $\mathbb{P}$.

###  A.3 Continuity of $a^{+}$ and $\Pi^{+}$

We now show that the map $(\beta,\sigma,\boldsymbol{\theta})\mapsto
a^{+}_{\beta,\sigma}(\boldsymbol{\theta})$ is continuous on
$S^{+}\times\Theta$; from this it immediately follows that the map
$(\beta,\sigma,\boldsymbol{\theta})\mapsto\Pi^{+}_{\beta,\sigma}(\boldsymbol{\theta})$
is also continuous on $S^{+}\times\Theta$. Fix
$(\beta,\sigma,\boldsymbol{\theta})\in S^{+}\times\Theta$, and fix
$\varepsilon>0$. Let $N$ be such that

$f_{\beta,\sigma,\theta_{-N},N}([x^{*},\infty])\subset
B_{\varepsilon}\big{(}a^{+}_{\beta,\sigma}(\boldsymbol{\theta})\big{)}.$

We can find a neighbourhood $U\subset\mathbb{R}\times[0,\infty)\times[0,2\pi)$
of $(\beta,\sigma,\theta_{-N})$ small enough that every
$(\tilde{\beta},\tilde{\sigma},\tilde{\theta})\in U$ has

$f_{\tilde{\beta},\tilde{\sigma},\tilde{\theta},N}([x^{*},\infty])\subset
B_{\varepsilon}\big{(}a^{+}_{\beta,\sigma}(\boldsymbol{\theta})\big{)}.$

So then, every $(\tilde{\beta},\tilde{\sigma},\tilde{\boldsymbol{\theta}})\in
S^{+}\times\Theta$ with $(\tilde{\beta},\tilde{\sigma},\tilde{\theta}_{-N})\in
U$ has

$a^{+}_{\tilde{\beta},\tilde{\sigma}}(\tilde{\boldsymbol{\theta}})\in
B_{\varepsilon}\big{(}a^{+}_{\beta,\sigma}(\boldsymbol{\theta})\big{)}.$

###  A.4 Invariant-graph property of $a_{\beta,\sigma}^{+}$

For all $(\beta,\sigma)\in S^{+}$ and $(\theta_{i})_{i}\in\Theta$,

$\displaystyle a^{+}_{\beta,\sigma}((\theta_{i+1})_{i})$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,\theta_{-(n-1)},n}(\infty)$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,\theta_{0}}\Big{(}f_{\beta,\sigma,\theta_{-(n-1)},n-1}(\infty)\Big{)}$
$\displaystyle=f_{\beta,\sigma,\theta_{0}}\\!\left(\lim_{n\to\infty}f_{\beta,\sigma,\theta_{-(n-1)},n-1}(\infty)\right)\qquad\textrm{since
$f_{\beta,\sigma,\theta_{0}}$ is continuous}$
$\displaystyle=f_{\beta,\sigma,\theta_{0}}\Big{(}a^{+}_{\beta,\sigma}((\theta_{i})_{i})\Big{)}.$

###  A.5 Ergodicity and continuity properties of $\mu_{\beta,\sigma}^{+}$

Let $\tilde{g}\colon\Theta\to\Theta$ denote the shift map
$(\theta_{i})_{i}\mapsto(\theta_{i+1})_{i}$. Since $\mathbb{P}$ is
$g$-ergodic, $\tilde{\mathbb{P}}$ is $\tilde{g}$-ergodic. By the invariant-
graph property of $a_{\beta,\sigma}^{+}$ established in Sec. A.4, we have

$\Big{(}a^{+}_{\beta,\sigma}((\theta_{i+1})_{i})\,,\,\theta_{1}\Big{)}=G_{\beta,\sigma}\Big{(}a^{+}_{\beta,\sigma}((\theta_{i})_{i})\,,\,\theta_{0}\Big{)}$

for all $(\theta_{i})_{i}\in\Theta$, i.e.

$\Pi^{+}_{\beta,\sigma}\circ\tilde{g}=G_{\beta,\sigma}\circ\Pi^{+}_{\beta,\sigma}.$

In other words, $\Pi^{+}_{\beta,\sigma}$ is a morphism from $\tilde{g}$ to
$G_{\beta,\sigma}$; consequently, since $\tilde{\mathbb{P}}$ is a
$\tilde{g}$-ergodic invariant measure, it follows that
$\mu_{\beta,\sigma}^{+}$ is a $G_{\beta,\sigma}$-ergodic invariant measure.

Due to the continuous dependence of
$\Pi^{+}_{\beta,\sigma}(\boldsymbol{\theta})$ on $(\beta,\sigma)$ for each
$\boldsymbol{\theta}$, the dominated convergence theorem gives that
$\mu^{+}_{\beta,\sigma}$ depends continuously on $(\beta,\sigma)$ in the
topology of weak convergence.

Since $\mu^{+}_{\beta,\sigma}$ is $G_{\beta,\sigma}$-invariant and
$G_{\beta,\sigma}$ is continuous, we have that the support
$\mathcal{A}^{+}_{\beta,\sigma}$ of $\mu^{+}_{\beta,\sigma}$ is a
$G_{\beta,\sigma}$-invariant set. Moreover, due to the facts that $\Theta$ is
compact and $\tilde{\mathbb{P}}$ has full support in $\Theta$ and
$\Pi^{+}_{\beta,\sigma}$ is continuous, we have that
$\mathcal{A}^{+}_{\beta,\sigma}$ is precisely the image of $\Theta$ under
$\Pi^{+}_{\beta,\sigma}$. Due to the continuous dependence of
$\Pi^{+}_{\beta,\sigma}(\boldsymbol{\theta})$ on
$(\beta,\sigma,\boldsymbol{\theta})$, it then follows that
$\mathcal{A}^{+}_{\beta,\sigma}$ has continuous dependence on $(\beta,\sigma)$
in Hausdorff distance.

###  A.6 Bounds on the $x$-projection of $\mathcal{A}^{+}_{\beta,\sigma}$

Recall that we have shown that $\mathcal{A}^{+}_{\beta,\sigma}$ is the image
of $\Theta$ under $\Pi^{+}_{\beta,\sigma}$. We will show that

  * •

$\mathcal{A}^{+}_{\beta,\sigma}$ is contained in
$[a^{+}_{\beta-\sigma},a^{+}_{\beta+\sigma}]\times[0,2\pi)$, i.e. the range of
$a^{+}_{\beta,\sigma}$ over $\Theta$ is contained in
$[a^{+}_{\beta-\sigma},a^{+}_{\beta+\sigma}]$;

  * •

if $\sigma>0$ then the intersection of $\mathcal{A}_{\beta,\sigma}^{+}$ with
$\mathcal{A}_{\beta-\sigma}^{+}=\\{a_{\beta-\sigma}^{+}\\}\times[0,2\pi)$ is
the point $(a^{+}_{\beta-\sigma},\pi)$ and the intersection of
$\mathcal{A}_{\beta,\sigma}^{+}$ with
$\mathcal{A}_{\beta+\sigma}^{+}=\\{a_{\beta+\sigma}^{+}\\}\times[0,2\pi)$ is
the point $(a^{+}_{\beta+\sigma},0)$.

First note that for all $\beta\in[-\beta^{*},\infty)$,

$a^{+}_{\beta}=\lim_{n\to\infty}f_{\beta}^{n}(\infty).$

Now fix $(\beta,\sigma)\in S^{+}$, i.e. $\beta-\sigma\in[-\beta^{*},\infty)$.
So, since

$f_{\beta-\sigma}\leq f_{\beta,\sigma,\theta}\leq f_{\beta+\sigma}$

for all $\theta$, we have that for every $\boldsymbol{\theta}\in\Theta$,

$\displaystyle
a^{+}_{\beta-\sigma}=\lim_{n\to\infty}f_{\beta-\sigma}^{n}(\infty)\leq\lim_{n\to\infty}f_{\beta,\sigma,\theta_{-n},n}(\infty)$
$\displaystyle=a^{+}_{\beta,\sigma}(\boldsymbol{\theta})$ $\displaystyle
a^{+}_{\beta+\sigma}=\lim_{n\to\infty}f_{\beta+\sigma}^{n}(\infty)\geq\lim_{n\to\infty}f_{\beta,\sigma,\theta_{-n},n}(\infty)$
$\displaystyle=a^{+}_{\beta,\sigma}(\boldsymbol{\theta}).$

Hence $\mathcal{A}^{+}_{\beta,\sigma}$ is contained in
$[a^{+}_{\beta-\sigma},a^{+}_{\beta+\sigma}]\times[0,2\pi)$.

Now note that $0$ and $\pi$ are fixed points of $g$. So the points
$(0)_{i\in\mathbb{Z}}$ and $(\pi)_{i\in\mathbb{Z}}$ are members of $\Theta$,
and we have

$\displaystyle a^{+}_{\beta,\sigma}((0)_{i\in\mathbb{Z}})$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,0,n}(\infty)=\lim_{n\to\infty}f_{\beta+\sigma}^{n}(\infty)=a^{+}_{\beta+\sigma}$
$\displaystyle a^{+}_{\beta,\sigma}((\pi)_{i\in\mathbb{Z}})$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,\pi,n}(\infty)=\lim_{n\to\infty}f_{\beta-\sigma}^{n}(\infty)=a^{+}_{\beta-\sigma}.$

Hence the points $(a^{+}_{\beta+\sigma},0)$ and $(a^{+}_{\beta-\sigma},\pi)$
belong to $\mathcal{A}^{+}_{\beta,\sigma}$. Now supposing that $\sigma>0$, we
will show that these points are the only points in the intersection of
$\mathcal{A}^{+}_{\beta,\sigma}$ respectively with
$\mathcal{A}_{\beta+\sigma}^{+}$ and with $\mathcal{A}_{\beta-\sigma}^{+}$.
For this, it is sufficient to show that

  * (i)

for every $\boldsymbol{\theta}\in\Theta$ with $\theta_{0}\neq\pi$,
$a^{+}_{\beta,\sigma}(\boldsymbol{\theta})>a^{+}_{\beta-\sigma}$;

  * (ii)

for every $\boldsymbol{\theta}\in\Theta$ with $\theta_{0}\neq 0$,
$a^{+}_{\beta,\sigma}(\boldsymbol{\theta})<a^{+}_{\beta+\sigma}$.

We show (i); (ii) is similar. Fix any $(\theta_{i})_{i}\in\Theta$ with
$\theta_{0}\neq\pi$. Since $\pi$ is a fixed point of $g$, it follows that
$\theta_{-1}\neq\pi$. So since $\sigma>0$ and $\theta_{-1}\neq\pi$, we have
that $f_{\beta,\sigma,\theta_{-1}}$ is strictly greater than
$f_{\beta-\sigma}$ everywhere. So, using the invariant-graph property of
$a^{+}_{\beta,\sigma}$ applied at the point $(\theta_{i-1})_{i}$ we have

$a^{+}_{\beta,\sigma}((\theta_{i})_{i})=f_{\beta,\sigma,\theta_{-1}}\Big{(}a^{+}_{\beta,\sigma}((\theta_{i-1})_{i})\Big{)}\geq
f_{\beta,\sigma,\theta_{-1}}(a^{+}_{\beta-\sigma})>f_{\beta-\sigma}(a^{+}_{\beta-\sigma})=a^{+}_{\beta-\sigma}.$
(25)

###  A.7 Lyapunov exponent of $\mu^{+}_{\beta,\sigma}$

Note that $f^{\prime}$ is continuous and always takes strictly positive
values. So, since $\mu_{\beta,\sigma}^{+}$ is compactly supported, Birkhoff’s
ergodic theorem gives that the Lyapunov exponent
$\lambda_{\beta,\sigma}(\mu_{\beta,\sigma}^{+})$ is well-defined and given by

$\lambda_{\beta,\sigma}(\mu_{\beta,\sigma}^{+})=\int_{\mathbb{R}\times[0,2\pi)}\ln
f^{\prime}(x)\,\mu_{\beta,\sigma}^{+}(d(x,\theta))=\int_{\Theta}\ln
f^{\prime}(a_{\beta,\sigma}^{+}(\boldsymbol{\theta}))\,\tilde{\mathbb{P}}(d\boldsymbol{\theta}).$

Suppose $\sigma>0$. Then as in (25), every $\boldsymbol{\theta}\in\Theta$
outside the $\tilde{\mathbb{P}}$-null set $\\{\theta_{-1}=\pi\\}$ has
$a_{\beta,\sigma}^{+}(\boldsymbol{\theta})>a^{+}_{\beta-\sigma}\geq x^{*}$ and
therefore

$f^{\prime}(a_{\beta,\sigma}^{+}(\boldsymbol{\theta}))<1;$

and so it follows that

$\lambda_{\beta,\sigma}(\mu_{\beta,\sigma}^{+})<0.$

Since $\ln f^{\prime}$ is continuous and the support of
$\mu^{+}_{\beta,\sigma}$ is uniformly bounded over bounded subsets of $S^{+}$,
the continuous dependence of $\mu^{+}_{\beta,\sigma}$ on $(\beta,\sigma)$ in
the topology of weak convergence implies that
$\lambda_{\beta,\sigma}(\mu_{\beta,\sigma}^{+})$ has continuous dependence on
$(\beta,\sigma)$.

###  A.8 Attractivity of $\mathcal{A}^{+}_{\beta,\sigma}$

######  Lemma 2.

Fix $(\beta,\sigma)\in S^{+}$. The set $[x^{*},\infty)\times[0,2\pi)$ is
uniformly attracted to $\mathcal{A}^{+}_{\beta,\sigma}$.

###### Proof.

For notational convenience, given $x,\tilde{x}\in\mathbb{R}$ and
$\theta\in[0,2\pi)$, we write

$\mathrm{dist}\big{(}(x,\theta),(\tilde{x},\theta)\big{)}=|x-\tilde{x}|.$

To show the desired result, we will show that for every
$(x,\theta)\in[x^{*},\infty)\times[0,2\pi)$ there exists $\tilde{x}$ with
$(\tilde{x},\theta)\in\mathcal{A}^{+}_{\beta,\sigma}$ such that for all $n\geq
0$,

$\mathrm{dist}\big{(}G_{\beta,\sigma}^{n}(x,\theta),G_{\beta,\sigma}^{n}(\tilde{x},\theta)\big{)}\leq\varphi^{n}(\infty).$

Since $\mathcal{A}^{+}_{\beta,\sigma}$ is $G_{\beta,\sigma}$-invariant and
$\varphi^{n}(\infty)\to 0$ as $n\to\infty$, this gives the desired result.

Fix $(x,\theta)\in[x^{*},\infty)\times[0,2\pi)$. Pick a point
$\boldsymbol{\theta}\in\Theta$ whose $0$-coordinate is equal to $\theta$, and
set $\tilde{x}=a^{+}_{\beta,\sigma}(\boldsymbol{\theta})$; so
$(\tilde{x},\theta)\in\mathcal{A}^{+}_{\beta,\sigma}$. We have that both $x$
and $\tilde{x}$ belong to $[x^{*},\infty)$; and so for all $n\geq 0$, since
the diameter of $f_{\beta,\sigma,\theta,n}([x^{*},\infty])$ is at most
$\varphi^{n}(\infty)$, we have the desired result. ∎

Now if $\beta-\sigma$ is strictly greater than $-\beta^{*}$ then the minimal
$x$-coordinate of $\mathcal{A}^{+}_{\beta,\sigma}$, namely
$a^{+}_{\beta-\sigma}$ (as shown in Sec. A.6), is strictly greater than
$x^{*}$; and so we immediately have the following corollary:

######  Corollary 3.

If $\beta-\sigma$ is strictly greater than $-\beta^{*}$, then
$\mathcal{A}^{+}_{\beta,\sigma}$ is a local attractor.

###  A.9 Construction and properties of $\mu_{\beta,\sigma}^{-}$

Let us first verify that the involution
$\iota\colon(x,\theta)\mapsto(-x,\theta+\pi)$ serves as a conjugacy between
$G_{\beta,\sigma}$ and $G_{-\beta,\sigma}$:

$\displaystyle\big{(}G_{\beta,\sigma}\circ\iota\big{)}(x,\theta)$
$\displaystyle=G_{\beta,\sigma}(-x,\theta+\pi)$
$\displaystyle=(\alpha\operatorname{\mathrm{arctan}}(-x)+\beta+\sigma\cos(\theta+\pi)\,,\,3(\theta+\pi))$
$\displaystyle=(-\alpha\operatorname{\mathrm{arctan}}(x)+\beta-\sigma\cos(\theta)\,,\,3\theta+3\pi)$
$\displaystyle=\big{(}-(\alpha\operatorname{\mathrm{arctan}}(x)-\beta+\sigma\cos(\theta))\,,\,3\theta+\pi\big{)}$
$\displaystyle=\iota\big{(}\alpha\operatorname{\mathrm{arctan}}(x)-\beta+\sigma\cos(\theta)\,,\,3\theta\big{)}$
$\displaystyle=\big{(}\iota\circ G_{-\beta,\sigma}\big{)}(x,\theta).$

Since

$S^{-}=\\{(\beta,\sigma):(-\beta,\sigma)\in S^{+}\\},$

for every $(\beta,\sigma)\in S^{-}$ we can define $\mu_{\beta,\sigma}^{-}$ to
be the pushforward of $\mu_{-\beta,\sigma}^{+}$ under $\iota$. Since $\iota$
is a conjugacy between $G_{\beta,\sigma}$ and $G_{-\beta,\sigma}$, and
$\mu_{-\beta,\sigma}^{+}$ is a $G_{-\beta,\sigma}$-ergodic measure, we have
that $\mu_{\beta,\sigma}^{-}$ is a $G_{\beta,\sigma}$-ergodic measure; and for
all the other properties that we have established for
$\mu_{\beta,\sigma}^{+}$, the analogous properties for
$\mu_{\beta,\sigma}^{-}$ hold.

###  A.10 Construction of $r_{\beta,\sigma}$

Rather than seeking to define $\mathcal{C}_{\beta,\sigma}$ directly in terms
of the basins of attraction of $\mathcal{A}^{\pm}_{\beta,\sigma}$, we will
first present a construction of the function
$r_{\beta,\sigma}\colon[0,2\pi)\to\mathbb{R}$, and we will define
$\mathcal{C}_{\beta,\sigma}$ to be the graph of $r_{\beta,\sigma}$; with these
definitions, we will prove all the required properties, including that when
both $\mathcal{A}^{-}_{\beta,\sigma}$ and $\mathcal{A}^{+}_{\beta,\sigma}$
exist, $\mathcal{C}_{\beta,\sigma}$ is the basin boundary of both
$\mathcal{A}^{-}_{\beta,\sigma}$ and $\mathcal{A}^{+}_{\beta,\sigma}$.

Recall that the parameter region $\mathrm{A}$ is defined to be $S^{+}\cap
S^{-}$. For all $(\beta,\sigma,\theta)\in\mathrm{A}\times[0,2\pi)$, since
$\beta-\sigma\geq-\beta^{*}$ we have as in Sec. A.2 that

$f_{\beta,\sigma,\theta}(x^{*})\geq x^{*};$

and similarly, since $\beta+\sigma\leq\beta^{*}$, we have

$f_{\beta,\sigma,\theta}(-x^{*})\leq f_{\beta+\sigma}(-x^{*})\leq
f_{\beta^{*}}(-x^{*})=-x^{*}.$

Combining these two facts, we have that

$[-x^{*},x^{*}]\subset f_{\beta,\sigma,\theta}([-x^{*},x^{*}])$

and so

$f_{\beta,\sigma,\theta}^{-1}([-x^{*},x^{*}])\subset[-x^{*},x^{*}].$

Therefore, noting that for all $n\geq 0$,

$f_{\beta,\sigma,\theta,n+1}^{-1}=f_{\beta,\sigma,\theta,n}^{-1}\circ
f_{\beta,\sigma,g^{n}(\theta)}^{-1}$

on the range of $f_{\beta,\sigma,\theta,n+1}$, we have (by induction) that the
sequence of sets $f_{\beta,\sigma,\theta,n}^{-1}([-x^{*},x^{*}])$ is nested,
with diameter bounded above by $\psi^{n}(2x^{*})$. Since $a_{-\beta^{*}}^{-}$
is strictly less than $-x^{*}$, we have that $2x^{*}$ is strictly less than
$x^{*}-a_{-\beta^{*}}^{-}$ and so $\psi^{n}(2x^{*})$ tends to $0$ as
$n\to\infty$. Hence, the intersection of the nested sequence of sets
$f_{\beta,\sigma,\theta,n}^{-1}([-x^{*},x^{*}])$ is a singleton
$\\{r_{\beta,\sigma}(\theta)\\}$.

###  A.11 Properties of $r_{\beta,\sigma}$ analogous to Secs. A.2–A.7

For all $(\beta,\sigma)\in\mathrm{A}$, let
$\Psi_{\beta,\sigma}\colon[0,2\pi)\to\mathbb{R}\times[0,2\pi)$ be the map
$\Psi_{\beta,\sigma}(\theta)=(r_{\beta,\sigma}(\theta),\theta)$, and let
$\nu_{\beta,\sigma}$ be the pushforward of $\mathbb{P}$ under
$\Psi_{\beta,\sigma}$.

Similarly to in Sec. A.3, we show that the map $(\beta,\sigma,\theta)\mapsto
r_{\beta,\sigma}(\theta)$ is continuous on $\mathrm{A}\times[0,2\pi)$. Fix
$(\beta,\sigma,\theta)\in\mathrm{A}\times[0,2\pi)$ and $\varepsilon>0$. Let
$N$ be such that

$f_{\beta,\sigma,\theta,N}^{-1}([-x^{*},x^{*}])\subset
B_{\varepsilon}\big{(}r_{\beta,\sigma}(\theta)\big{)},$

i.e.

$[-x^{*},x^{*}]\subset\Big{(}f_{\beta,\sigma,\theta,N}(r_{\beta,\sigma}(\theta)-\varepsilon)\,,\,f_{\beta,\sigma,\theta,N}(r_{\beta,\sigma}(\theta)+\varepsilon)\Big{)}.$

So we can find a neighbourhood
$U\subset\mathbb{R}\times[0,\infty)\times[0,2\pi)$ of $(\beta,\sigma,\theta)$
small enough that every $(\tilde{\beta},\tilde{\sigma},\tilde{\theta})\in U$
has

$f_{\tilde{\beta},\tilde{\sigma},\tilde{\theta},N}^{-1}([-x^{*},x^{*}])\subset
B_{\varepsilon}\big{(}r_{\beta,\sigma}(\theta)\big{)}.$

So then, every $(\tilde{\beta},\tilde{\sigma},\tilde{\theta})$ in the
intersection of $\mathrm{A}\times[0,2\pi)$ with $U$ has

$r_{\tilde{\beta},\tilde{\sigma}}(\tilde{\theta})\in
B_{\varepsilon}\big{(}r_{\beta,\sigma}(\theta)\big{)}.$

Thus we have shown that the map $(\beta,\sigma,\theta)\mapsto
r_{\beta,\sigma}(\theta)$ is continuous; and it follows that the map
$(\beta,\sigma,\theta)\mapsto\Psi_{\beta,\sigma}(\theta)$ is likewise
continuous.

Similarly to in Sec. A.4, we show that $r_{\beta,\sigma}$ has the invariant-
graph property: for all $(\beta,\sigma,\theta)\in\mathrm{A}\times[0,2\pi)$,

$\displaystyle r_{\beta,\sigma}(g(\theta))$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,g(\theta),n}^{-1}(x^{*})$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,\theta}\Big{(}f_{\beta,\sigma,\theta,n+1}^{-1}(x^{*})\Big{)}$
$\displaystyle=f_{\beta,\sigma,\theta}\\!\left(\lim_{n\to\infty}f_{\beta,\sigma,\theta,n+1}^{-1}(x^{*})\right)\qquad\textrm{since
$f_{\beta,\sigma,\theta}$ is continuous}$
$\displaystyle=f_{\beta,\sigma,\theta}\big{(}r_{\beta,\sigma}(\theta)\big{)}.$

Similarly to in Sec. A.5, it follows that $\Psi_{\beta,\sigma}$ is a morphism
from $g$ to $G_{\beta,\sigma}$, and hence, since $\mathbb{P}$ is a $g$-ergodic
invariant measure, we have that $\nu_{\beta,\sigma}$ is a
$G_{\beta,\sigma}$-ergodic invariant measure. As in Sec. A.5, we have that:
$\nu_{\beta,\sigma}$ depends continuously on $(\beta,\sigma)$ in the topology
of weak convergence; the graph
$\mathcal{C}_{\beta,\sigma}:=\\{(r_{\beta,\sigma}(\theta),\theta):\theta\in[0,2\pi)\\}$
of $r_{\beta,\sigma}$ is precisely the support of $\nu_{\beta,\sigma}$; and
$\mathcal{C}_{\beta,\sigma}$ has continuous dependence on $(\beta,\sigma)$ in
Hausdorff distance.

Similarly to in Sec. A.6, we show that the range of $r_{\beta,\sigma}$ is
contained in $[r_{\beta+\sigma},r_{\beta-\sigma}]$. Since
$(\beta,\sigma)\in\mathrm{A}$, we have that both $\beta-\sigma$ and
$\beta+\sigma$ lie in the interval $[-\beta^{*},\beta^{*}]$, and so
$r_{\beta+\sigma}$ and $r_{\beta-\sigma}$ both exist. Now note that for all
$\beta\in[-\beta^{*},\beta^{*}]$, since $x^{*}\in[r_{\beta},a_{\beta}^{+}]$,
we have

$r_{\beta}=\lim_{n\to\infty}(f_{\beta}^{n})^{-1}(x^{*}).$

Since (by Eq. (24))

$f_{\beta+\sigma}^{-1}(y)\leq f_{\beta,\sigma,\theta}^{-1}(y)\leq
f_{\beta-\sigma}^{-1}(y)$

for all $\theta\in[0,2\pi)$ and $y\in[-x^{*},x^{*}]$, we have that for all
$\theta\in[0,2\pi)$,

$\displaystyle r_{\beta-\sigma}$
$\displaystyle=\lim_{n\to\infty}(f_{\beta-\sigma}^{n})^{-1}(x^{*})\geq\lim_{n\to\infty}f_{\beta,\sigma,\theta,n}^{-1}(x^{*})=r_{\beta,\sigma}(\theta)$
$\displaystyle r_{\beta+\sigma}$
$\displaystyle=\lim_{n\to\infty}(f_{\beta+\sigma}^{n})^{-1}(x^{*})\leq\lim_{n\to\infty}f_{\beta,\sigma,\theta,n}^{-1}(x^{*})=r_{\beta,\sigma}(\theta).$

Hence, the range of $r_{\beta,\sigma}$ is contained in
$[r_{\beta+\sigma},r_{\beta-\sigma}]$.

Next, similarly to in Sec. A.6, we show that if $\sigma>0$ then
$r_{\beta,\sigma}$ attains the value $r_{\beta-\sigma}$ uniquely at
$\theta=\pi$, and attains the value $r_{\beta+\sigma}$ uniquely at $\theta=0$.
Since $0$ and $\pi$ are fixed points of $g$, we have

$\displaystyle r_{\beta,\sigma}(0)$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,0,n}^{-1}(x^{*})=\lim_{n\to\infty}(f_{\beta+\sigma}^{n})^{-1}(x^{*})=r_{\beta+\sigma}$
$\displaystyle r_{\beta,\sigma}(\pi)$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,\pi,n}^{-1}(x^{*})=\lim_{n\to\infty}(f_{\beta-\sigma}^{n})^{-1}(x^{*})=r_{\beta-\sigma}.$

Now assuming $\sigma>0$, every $\theta$ distinct from $\pi$ has
$f_{\beta,\sigma,\theta}^{-1}$ being strictly less than
$f_{\beta-\sigma}^{-1}$ on $[-x^{*},x^{*}]$, and therefore

$r_{\beta,\sigma}(\theta)=f_{\beta,\sigma,\theta}^{-1}\Big{(}r_{\beta,\sigma}(g(\theta))\Big{)}\leq
f_{\beta,\sigma,\theta}^{-1}(r_{\beta-\sigma})<f_{\beta-\sigma}^{-1}(r_{\beta-\sigma})=r_{\beta-\sigma}.$

A similar argument gives that every $\theta$ distinct from $0$ has
$r_{\beta,\sigma}(\theta)>r_{\beta+\sigma}$.

Similarly to in Sec. A.7, we have

$\lambda_{\beta,\sigma}(\nu_{\beta,\sigma})=\frac{1}{2\pi}\int_{0}^{2\pi}\ln
f^{\prime}(r_{\beta,\sigma}(\theta))\,d\theta.$

Every $\theta$ outside of $\\{0,\pi\\}$ has $r_{\beta,\sigma}(\theta)$ lying
in the open interval $(r_{\beta+\sigma},r_{\beta-\sigma})$, and hence in the
open interval $(-x^{*},x^{*})$, and so

$f^{\prime}(r_{\beta,\sigma}(\theta))>1;$

and so it follows that

$\lambda_{\beta,\sigma}(\nu_{\beta,\sigma})>0.$

As in Sec. A.7, $\lambda_{\beta,\sigma}(\nu_{\beta,\sigma})$ has continuous
dependence on $(\beta,\sigma)$.

###  A.12 Intersections of $\mathcal{C}_{\beta,\sigma}$ with
$\mathcal{A}^{\pm}_{\beta,\sigma}$

Drawing together the results in Sec. A.6 and the analogous results in Sec.
A.11:

For $(\beta,\sigma)$ in the interior of $\mathrm{A}$ relative to
$\mathbb{R}\times[0,\infty)$, we have that

  * •

$\beta-\sigma>-\beta^{*}$ and so
$r_{\beta-\sigma}<x^{*}<a^{+}_{\beta-\sigma}$;

  * •

$\beta+\sigma<\beta^{*}$ and so
$a^{-}_{\beta+\sigma}<-x^{*}<r_{\beta+\sigma}$;

and so

  * •

the $x$-projection of $\mathcal{A}^{+}_{\beta,\sigma}$ (namely,
$[a^{+}_{\beta-\sigma},a^{+}_{\beta+\sigma}]$) lies strictly to the right of
$x^{*}$;

  * •

the $x$-projection of $\mathcal{A}^{-}_{\beta,\sigma}$ (namely,
$[a^{-}_{\beta-\sigma},a^{-}_{\beta+\sigma}]$) lies strictly to the left of
$-x^{*}$;

  * •

the $x$-projection of $\mathcal{C}_{\beta,\sigma}$ (namely,
$[r_{\beta+\sigma},r_{\beta-\sigma}]$) lies strictly between $-x^{*}$ and
$x^{*}$;

and hence $\mathcal{C}_{\beta,\sigma}$ has empty intersection with
$\mathcal{A}^{+}_{\beta,\sigma}$ and with $\mathcal{A}^{-}_{\beta,\sigma}$.

Now note that

$L^{\pm}=\\{(\beta,\sigma)\in\mathrm{A}\,:\,\beta\pm\sigma=\pm\beta^{*}\\}.$

So then:

  * •

For $(\beta,\sigma)\in L^{-}$, we have
$r_{\beta-\sigma}=a^{+}_{\beta-\sigma}=x^{*}$, and so both
$\mathcal{A}^{+}_{\beta,\sigma}$ and $\mathcal{C}_{\beta,\sigma}$ include the
point $(x^{*},\pi)$. If, furthermore, $\sigma>0$, then this is the only point
of intersection of $\mathcal{A}^{+}_{\beta,\sigma}$ or
$\mathcal{C}_{\beta,\sigma}$ with $\\{x^{*}\\}\times[0,2\pi)$, and hence is
the only point of intersection of $\mathcal{A}^{+}_{\beta,\sigma}$ with
$\mathcal{C}_{\beta,\sigma}$.

  * •

For $(\beta,\sigma)\in L^{+}$, we have
$r_{\beta+\sigma}=a^{-}_{\beta+\sigma}=-x^{*}$, and so both
$\mathcal{A}^{-}_{\beta,\sigma}$ and $\mathcal{C}_{\beta,\sigma}$ include the
point $(-x^{*},0)$. If, furthermore, $\sigma>0$, then this is the only point
of intersection of $\mathcal{A}^{+}_{\beta,\sigma}$ or
$\mathcal{C}_{\beta,\sigma}$ with $\\{-x^{*}\\}\times[0,2\pi)$, and hence is
the only point of intersection of $\mathcal{A}^{-}_{\beta,\sigma}$ with
$\mathcal{C}_{\beta,\sigma}$.

###  A.13 Basin of attraction of $\mathcal{A}^{\pm}_{\beta,\sigma}$ for
$(\beta,\sigma)\in\mathrm{A}$

######  Proposition 4.

For $(\beta,\sigma)\in\mathrm{A}$,

  * (i)

every $(x,\theta)$ with $x>r_{\beta,\sigma}(\theta)$ is in the basin of
$\mathcal{A}^{+}_{\beta,\sigma}$;

  * (ii)

every $(x,\theta)$ with $x<r_{\beta,\sigma}(\theta)$ is in the basin of
$\mathcal{A}^{-}_{\beta,\sigma}$;

  * (iii)

hence $\mathcal{A}^{+}_{\beta,\sigma}$ and $\mathcal{A}^{-}_{\beta,\sigma}$
share the same basin boundary, namely $\mathcal{C}_{\beta,\sigma}$.

###### Proof.

Point (iii) follows from points (i) and (ii) due to the continuity of
$r_{\beta,\sigma}$. We will prove (i); the proof of (ii) is similar.

Fix $(\beta,\sigma)\in\mathrm{A}$, and fix $(x,\theta)$ with
$x>r_{\beta,\sigma}(\theta)$. Take $N$ such that
$f_{\beta,\sigma,\theta,N}^{-1}(x^{*})\leq x$. Then $x^{*}\leq
f_{\beta,\sigma,\theta,N}(x)$, i.e.
$G_{\beta,\sigma}^{N}(x,\theta)\in[x^{*},\infty)\times[0,2\pi)$, and so Lemma
2 gives that $(x,\theta)$ is in the basin of $\mathcal{A}^{+}_{\beta,\sigma}$.
∎

###  A.14 Basin of attraction of $\mathcal{A}^{+}_{\beta,\sigma}$ for
$(\beta,\sigma)\in S^{+}\setminus\mathrm{A}$

By construction, the region

$S^{+}\setminus\mathrm{A}\,=\,S^{+}\setminus
S^{-}\,=\,\\{(\beta,\sigma)\in\mathbb{R}\times[0,\infty)\,:\,\beta-\sigma\geq-\beta^{*}\
\textrm{ and }\ \beta+\sigma>\beta^{*}\\}$

partitions into region $\mathrm{B}^{+}$ and region $\mathrm{C}^{+}$. Now

$(\beta,\sigma)\not\in\mathrm{B}^{+}\quad\Longleftrightarrow\quad\beta-\sigma\leq\beta^{*},$

and so $\mathrm{C}^{+}$ is the set of all members of
$S^{+}\setminus\mathrm{A}$ that fulfil this, i.e.

$\mathrm{C}^{+}=\\{(\beta,\sigma)\in\mathbb{R}\times[0,\infty)\,:\,-\beta^{*}\leq\beta-\sigma\leq\beta^{*}<\beta+\sigma\\}.$

######  Proposition 5.

  * (I)

For each $(\beta,\sigma)\in\mathrm{C}^{+}$, the basin of
$\mathcal{A}^{+}_{\beta,\sigma}$ includes both the set
$(r_{\beta-\sigma},\infty)\times[0,2\pi)$ and a set of the form
$\mathbb{R}\times E$ for some $\mathbb{P}$-full measure set
$E\subset[0,2\pi)$.

  * (II)

For each $(\beta,\sigma)\in\mathrm{B}^{+}$, $\mathcal{A}^{+}_{\beta,\sigma}$
is a global attractor.

###### Proof.

(I) Fix $(\beta,\sigma)\in\mathrm{C}^{+}$. Since
$\beta-\sigma\in[-\beta^{*},\beta^{*}]$, we have in particular that
$r_{\beta-\sigma}$ exists. Now for any $(x,\theta)$ with $x>r_{\beta-\sigma}$:

  * •

If $\beta-\sigma=-\beta^{*}$, then $r_{\beta-\sigma}=x^{*}$, so $x>x^{*}$, and
so Lemma 2 gives that $(x,\theta)$ is in the basin of
$\mathcal{A}^{+}_{\beta,\sigma}$.

  * •

If $\beta-\sigma>-\beta^{*}$ then

$x^{*}<a^{+}_{\beta-\sigma}=\lim_{n\to\infty}f_{\beta-\sigma}^{n}(x)\leq\liminf_{n\to\infty}f_{\beta,\sigma,\theta,n}(-\infty),$

and so there are $n$ for which $f_{\beta,\sigma,\theta,n}(x)>x^{*}$; so Lemma
2 gives that $(x,\theta)$ is in the basin of $\mathcal{A}^{+}_{\beta,\sigma}$.

Thus we have shown that the basin of $\mathcal{A}^{+}_{\beta,\sigma}$ includes
$(r_{\beta-\sigma},\infty)\times[0,2\pi)$.

Now since $\beta+\sigma>\beta^{*}$ (and therefore, also,
$\beta+\sigma>-\beta^{*}$), we have

$x^{*}<a^{+}_{\beta+\sigma}=\lim_{n\to\infty}f_{\beta+\sigma}^{n}(-\infty)=\lim_{n\to\infty}f_{\beta,\sigma,0,n}(-\infty).$

Therefore, we can find a positive integer $N$ and a neighbourhood $U$ of $0$
(in the space of angles) such that for all $\theta\in U$,
$f_{\beta,\sigma,\theta,N}(-\infty)\geq x^{*}$. So $G^{N}(\mathbb{R}\times
U)\subset[x^{*},\infty)\times[0,2\pi)$. Therefore, by Lemma 2, any initial
condition whose trajectory under $G$ enters the set $\mathbb{R}\times U$
belongs to the basin of $\mathcal{A}^{+}_{\beta,\sigma}$. In other words,
letting $E:=\bigcup_{n=0}^{\infty}g^{-n}(U)$, we have that $\mathbb{R}\times
E$ is contained in the basin of $\mathcal{A}^{+}_{\beta,\sigma}$. And since
$\mathbb{P}$ is $g$-ergodic and $\mathbb{P}(U)>0$, we have $\mathbb{P}(E)=1$.

(II) Fix $(\beta,\sigma)\in\mathrm{B}^{+}$. Since $\beta-\sigma>\beta^{*}$, we
have

$\lim_{n\to\infty}f_{\beta-\sigma}^{n}(-\infty)=a_{\beta-\sigma}>x^{*}$

and so we can find a positive integer $N$ such that for all
$\theta\in[0,2\pi)$, $f_{\beta,\sigma,\theta,N}(-\infty)\geq
f_{\beta-\sigma}^{N}(-\infty)\geq x^{*}$. So
$G^{N}(\mathbb{R}\times[0,2\pi))\subset[x^{*},\infty)\times[0,2\pi)$, and
hence Lemma 2 gives that $\mathcal{A}^{+}_{\beta,\sigma}$ is a global
attractor. ∎

The analogous results for the basin of $\mathcal{A}^{-}_{\beta,\sigma}$ with
$(\beta,\sigma)\in S^{-}\setminus\mathrm{A}$ also hold.

###  A.15 Basin of attraction of $\mu^{\pm}_{\beta,\sigma}$

Fix $(\beta,\sigma)\in S^{+}$.

######  Lemma 6.

For every $(x,(\theta_{i})_{i})\in\mathbb{R}\times\Theta$ with
$(x,\theta_{0})$ in the basin of $\mathcal{A}^{+}_{\beta,\sigma}$, we have
that

$\left|f_{\beta,\sigma,\theta_{0},n}(x)-a^{+}_{\beta,\sigma}((\theta_{i+n})_{i})\right|\to
0\quad\textrm{as }n\to\infty.$

###### Proof.

Fix $(x,(\theta_{i})_{i})\in\mathbb{R}\times\Theta$ with $(x,\theta_{0})$ in
the basin of $\mathcal{A}^{+}_{\beta,\sigma}$.

First suppose there exists $N\geq 0$ such that
$f_{\beta,\sigma,\theta_{0},N}(x)\geq x^{*}$; then since the range of
$a^{+}_{\beta,\sigma}$ is also contained in $[x^{*},\infty)$, it follows that
for all $n\geq N$,

$\displaystyle\left|f_{\beta,\sigma,\theta_{0},n}(x)-a^{+}_{\beta,\sigma}((\theta_{i+n})_{i})\right|$
$\displaystyle=\left|f_{\beta,\sigma,\theta_{N},n-N}\Big{(}f_{\beta,\sigma,\theta_{0},N}(x)\Big{)}-f_{\beta,\sigma,\theta_{N},n-N}\Big{(}a^{+}_{\beta,\sigma}((\theta_{i+N})_{i})\Big{)}\right|$
$\displaystyle\leq\mathrm{diam}(f_{\beta,\sigma,\theta_{N},n-N}([x^{*},\infty]))$
$\displaystyle\leq\varphi^{n-N}(\infty).$

Hence, since $\varphi^{n-N}(\infty)\to 0$ as $n\to\infty$, we have the desired
conclusion.

Now suppose instead that $f_{\beta,\sigma,\theta_{0},n}(x)<x^{*}$ for all
$n\geq 0$. As in Sec. A.12, if it were the case that
$\beta-\sigma>-\beta^{*}$, then the $x$-projection of
$\mathcal{A}^{+}_{\beta,\sigma}$ would lie strictly to the right of $x^{*}$.
So since $(x,\theta_{0})$ in the basin of $\mathcal{A}^{+}_{\beta,\sigma}$, we
must have that $\beta-\sigma=-\beta^{*}$ (i.e. $a^{+}_{\beta-\sigma}=x^{*}$)
and $\lim_{n\to\infty}f_{\beta,\sigma,\theta_{0},n}(x)=x^{*}$. If $\sigma=0$
then on the whole of $\Theta$ we have

$a^{+}_{\beta,\sigma}=a^{+}_{-\beta^{*},0}=x^{*}$

and so the fact that $\lim_{n\to\infty}f_{\beta,\sigma,\theta_{0},n}(x)=x^{*}$
is precisely the required result. So now assume that $\sigma>0$. So the only
point in $\mathcal{A}^{+}_{\beta,\sigma}$ whose $x$-coordinate is $x^{*}$ is
$(x^{*},\pi)$. So since $(x,\theta_{0})$ in the basin of
$\mathcal{A}^{+}_{\beta,\sigma}$, we have that $\theta_{n}\to\pi$ as
$n\to\infty$. Hence, due to the continuity of $a^{+}_{\beta,\sigma}$,

$\lim_{n\to\infty}a^{+}_{\beta,\sigma}((\theta_{i+n})_{i})=a^{+}_{\beta,\sigma}((\pi)_{i})=a^{+}_{\beta-\sigma}=x^{*}.$

Combining this with the fact that
$\lim_{n\to\infty}f_{\beta,\sigma,\theta_{0},n}(x)=x^{*}$ gives the required
result. ∎

######  Proposition 7.

There is a $\mathbb{P}$-full measure set $R\subset[0,2\pi)$ such that every
initial condition $(x,\theta)$ in the basin of
$\mathcal{A}^{+}_{\beta,\sigma}$ with $\theta\in R$ is in the basin of
$\mu^{+}_{\beta,\sigma}$.

###### Proof.

Define

$\displaystyle\tilde{G}_{\beta,\sigma}\colon\mathbb{R}\times\Theta$
$\displaystyle\to\mathbb{R}\times\Theta$
$\displaystyle\tilde{G}_{\beta,\sigma}(x,(\theta_{i})_{i})$
$\displaystyle=(f_{\beta,\sigma,\theta_{0}}(x),(\theta_{i+1})_{i})$

and

$\displaystyle\tilde{\Pi}^{+}_{\beta,\sigma}\colon\Theta$
$\displaystyle\to\mathbb{R}\times\Theta$
$\displaystyle\tilde{\Pi}^{+}_{\beta,\sigma}(\boldsymbol{\theta})$
$\displaystyle=(a^{+}_{\beta,\sigma}(\boldsymbol{\theta}),\boldsymbol{\theta}).$

Let $\tilde{\mu}^{+}_{\beta,\sigma}$ be the pushforward of
$\tilde{\mathbb{P}}$ under $\tilde{\Pi}^{+}_{\beta,\sigma}$. By the same
argument as in Sec. A.5, we have that $\tilde{\Pi}^{+}_{\beta,\sigma}$ is a
morphism from $\tilde{g}$ to $\tilde{G}_{\beta,\sigma}$ and therefore
$\tilde{\mu}^{+}_{\beta,\sigma}$ is a $\tilde{G}_{\beta,\sigma}$-ergodic
invariant measure. So $\tilde{\mu}^{+}_{\beta,\sigma}$-almost every point is
in the basin of $\tilde{\mu}^{+}_{\beta,\sigma}$ under
$\tilde{G}_{\beta,\sigma}$. Thus, in other words, for
$\tilde{\mathbb{P}}$-almost every $\boldsymbol{\theta}$,
$\tilde{\Pi}^{+}_{\beta,\sigma}(\boldsymbol{\theta})$ is in the basin of
$\tilde{\mu}^{+}_{\beta,\sigma}$ under $\tilde{G}_{\beta,\sigma}$. So let
$\Theta_{1}\subset\Theta$ be a $\tilde{\mathbb{P}}$-full measure set such that
for every $\boldsymbol{\theta}\in\Theta_{1}$,
$\tilde{\Pi}^{+}_{\beta,\sigma}(\boldsymbol{\theta})$ is in the basin of
$\tilde{\mu}^{+}_{\beta,\sigma}$ under $\tilde{G}_{\beta,\sigma}$. Let
$R\subset[0,2\pi)$ be the image of $\Theta_{1}$ under the projection
$(\theta_{i})_{i}\mapsto\theta_{0}$. So $R$ is a $\mathbb{P}$-full measure
subset of $[0,2\pi)$.

Fix $(x,\theta)$ in the basin of $\mathcal{A}^{+}_{\beta,\sigma}$ with
$\theta\in R$, and fix a bounded uniformly continuous function
$h\colon\mathbb{R}\times[0,2\pi)\to\mathbb{R}$. Take
$(\theta_{i})_{i}\in\Theta_{1}$ with $\theta_{0}=\theta$. Write
$\tilde{h}(x^{\prime},(\theta_{i}^{\prime})_{i})=h(x^{\prime},\theta_{0}^{\prime})$
for all $(x^{\prime},(\theta_{i}^{\prime})_{i})\in\mathbb{R}\times\Theta$.
Then for all $N\geq 1$,

$\displaystyle d_{N}:=$ $\displaystyle\
\int_{\mathbb{R}\times[0,2\pi)}h\,d\mu^{+}_{\beta,\sigma}-\frac{1}{N}\sum_{n=0}^{N-1}h(G_{\beta,\sigma}^{n}(x,\theta))$
$\displaystyle=$ $\displaystyle\
\int_{\mathbb{R}\times\Theta}\tilde{h}\,d\tilde{\mu}^{+}_{\beta,\sigma}-\frac{1}{N}\sum_{n=0}^{N-1}h(f_{\beta,\sigma,\theta_{0},n}(x),\theta_{n})$
$\displaystyle=$ $\displaystyle\
\Bigg{(}\int_{\mathbb{R}\times\Theta}\tilde{h}\,d\tilde{\mu}^{+}_{\beta,\sigma}-\frac{1}{N}\sum_{n=0}^{N-1}h(a^{+}_{\beta,\sigma}((\theta_{i+n})_{i}),\theta_{n})\Bigg{)}$
$\displaystyle\hskip
56.9055pt+\frac{1}{N}\sum_{n=0}^{N-1}\bigg{(}h(f_{\beta,\sigma,\theta_{0},n}(x),\theta_{n})-h(a^{+}_{\beta,\sigma}((\theta_{i+n})_{i}),\theta_{n})\bigg{)}$
$\displaystyle=$ $\displaystyle\
\Bigg{(}\underbrace{\int_{\mathbb{R}\times\Theta}\tilde{h}\,d\tilde{\mu}^{+}_{\beta,\sigma}-\frac{1}{N}\sum_{n=0}^{N-1}\tilde{h}(\tilde{G}_{\beta,\sigma}^{n}(\tilde{\Pi}^{+}_{\beta,\sigma}((\theta_{i})_{i})))}_{A_{N}}\Bigg{)}$
$\displaystyle\hskip
56.9055pt+\frac{1}{N}\\!\sum_{n=0}^{N-1}\bigg{(}\\!\underbrace{h(f_{\beta,\sigma,\theta_{0},n}(x),\theta_{n})-h(a^{+}_{\beta,\sigma}((\theta_{i+n})_{i}),\theta_{n})}_{B_{n}}\bigg{)}.$

Since $\tilde{\Pi}^{+}_{\beta,\sigma}((\theta_{i})_{i}))$ is in the basin of
$\tilde{\mu}^{+}_{\beta,\sigma}$ under $\tilde{G}_{\beta,\sigma}$ and
$\tilde{h}$ is bounded and continuous, we have that $A_{N}\to 0$ as
$N\to\infty$. Since $h$ is uniformly continuous, Lemma 6 gives that $B_{n}\to
0$ as $n\to\infty$. Thus $d_{N}\to 0$ as $N\to\infty$, i.e. $(x,\theta)$ is in
the basin of $\mu^{+}_{\beta,\sigma}$. ∎

By Lemma 2, the basin of $\mathcal{A}^{+}_{\beta,\sigma}$ includes
$[x^{*},\infty)\times[0,2\pi)$; hence it follows from Proposition 7 that
$\mu^{+}_{\beta,\sigma}$ is a physical measure and is the only ergodic
invariant measure with support contained in $[x^{*},\infty)\times[0,2\pi)$
whose $\theta$-marginal is $\mathbb{P}$. Moreover, if $(\beta,\sigma)\in
S^{+}\setminus\mathrm{A}$, then it follows from Proposition 7 and Proposition
5(I) that $\mu^{+}_{\beta,\sigma}$ is the only ergodic invariant measure whose
$\theta$-marginal is $\mathbb{P}$.

(Each of the above two statements about uniqueness of an ergodic invariant
measure then translates into a statement about uniqueness of an invariant
measure by Lemma 1.)

The analogous results for $\mu^{-}_{\beta,\sigma}$ holds.

###  A.16 The $x$-projection of invariant-measure supports in $\mathrm{D}$

First observe that $\mathrm{D}$ is contained in the complement of
$\mathrm{B}^{-}\cup\mathrm{B}^{+}$.

Given any $(\beta,\sigma)$, a compact set $S\subset\mathbb{R}\times[0,2\pi)$
is said to be _positively invariant_ under $G_{\beta,\sigma}$ if
$G_{\beta,\sigma}(S)$ is a subset of $S$.

######  Proposition 8.

For each $(\beta,\sigma)$ outside of $\mathrm{B}^{-}\cup\mathrm{B}^{+}$, the
set $[a^{-}_{\beta-\sigma},a^{+}_{\beta+\sigma}]\times[0,2\pi)$ is positively
invariant and for every initial condition
$(x,\theta)\in\mathbb{R}\times[0,2\pi)$, the distance of
$G_{\beta,\sigma}^{n}(x,\theta)$ from
$[a^{-}_{\beta-\sigma},a^{+}_{\beta+\sigma}]\times[0,2\pi)$ tends to $0$ as
$n\to\infty$.

###### Proof.

Since $(\beta,\sigma)\not\in\mathrm{B}^{-}$, we have that
$\beta+\sigma\geq-\beta^{*}$, and so $a^{+}_{\beta+\sigma}$ exists and for all
$\theta\in[0,2\pi)$,

$\limsup_{n\to\infty}f_{\beta,\sigma,\theta,n}(\infty)\leq\lim_{n\to\infty}f_{\beta+\sigma}^{n}(\infty)=a^{+}_{\beta+\sigma}.$
(26)

Similarly, since $(\beta,\sigma)\not\in\mathrm{B}^{+}$, we have that
$\beta-\sigma\leq\beta^{*}$, and so $a^{-}_{\beta-\sigma}$ exists and for all
$\theta\in[0,2\pi)$,

$\liminf_{n\to\infty}f_{\beta,\sigma,\theta,n}(-\infty)\geq\lim_{n\to\infty}f_{\beta-\sigma}^{n}(-\infty)=a^{-}_{\beta-\sigma}.$
(27)

Due to the monotonicity of the maps $f_{\beta,\sigma,\theta,n}$, combining
Eqs. (26) and (27) gives the desired result. ∎

As a consequence of Proposition 8, we have that for all $(\beta,\sigma)$
outside of $\mathrm{B}^{-}\cup\mathrm{B}^{+}$, the support of every
$G_{\beta,\sigma}$-invariant probability measure is contained in
$[a^{-}_{\beta-\sigma},a^{+}_{\beta+\sigma}]\times[0,2\pi)$. Moreover, since
$[a^{-}_{\beta-\sigma},a^{+}_{\beta+\sigma}]\times[0,2\pi)$ is positively
invariant and $[a^{-}_{\beta-\sigma},a^{+}_{\beta+\sigma}]$ is compact, the
random Krylov-Bogolyubov theorem [9, Corollary 6.13] together with Lemma 1
gives that there exists at least one $G_{\beta,\sigma}$-ergodic invariant
measure with $\theta$-marginal $\mathbb{P}$.

######  Proposition 9.

For $(\beta,\sigma)\in\mathrm{D}$, the support of every
$G_{\beta,\sigma}$-invariant measure $\mu$ with $\theta$-marginal $\mathbb{P}$
includes the points $(a^{-}_{\beta-\sigma},\pi)$ and
$(a^{+}_{\beta+\sigma},0)$.

We start with a lemma.

######  Lemma 10.

  * (I)

For any $(\beta,\sigma)$ outside $S^{+}$, for every $\theta_{0}\in[0,2\pi)$
with $\pi\in\overline{\\{g^{i}(\theta_{0})\\}_{i\geq 0}}$ and every
neighbourhood $U\subset\mathbb{R}\times[0,2\pi)$ of
$(a^{-}_{\beta-\sigma},\pi)$, there exists $n\geq 0$ such that
$G_{\beta,\sigma}^{n}(\mathbb{R}\times\\{\theta_{0}\\})$ is contained in $U$.

  * (II)

For any $(\beta,\sigma)$ outside $S^{-}$, for every $\theta_{0}\in[0,2\pi)$
with $0\in\overline{\\{g^{i}(\theta_{0})\\}_{i\geq 0}}$ and every
neighbourhood $U\subset\mathbb{R}\times[0,2\pi)$ of
$(a^{+}_{\beta+\sigma},0)$, there exists $n\geq 0$ such that
$G_{\beta,\sigma}^{n}(\mathbb{R}\times\\{\theta_{0}\\})$ is contained in $U$.

###### Proof.

We prove (I); the proof of (II) is similar. Fix $(\beta,\sigma)$ outside
$S^{+}$. So $\beta-\sigma<-\beta^{*}$, and so the singleton
$\\{a^{-}_{\beta-\sigma}\\}$ is a global attractor of $f_{\beta-\sigma}$. Fix
$\theta_{0}\in[0,2\pi)$ with $\pi\in\overline{\\{g^{i}(\theta_{0})\\}_{i\geq
0}}$. Without loss of generality, take $U$ to be the open square of side-
length $2\varepsilon$ centred on $(a^{-}_{\beta-\sigma},\pi)$ for some
$\varepsilon>0$.

Since $\pi$ is a fixed point of $g$, we can find $m\geq 0$ such that

$f_{\beta,\sigma,\pi,m}(\overline{\mathbb{R}})=f_{\beta-\sigma}^{m}(\overline{\mathbb{R}})\subset\big{(}a^{-}_{\beta-\sigma}-\varepsilon,a^{-}_{\beta-\sigma}+\varepsilon\big{)}.$

So, by the compactness of $\overline{\mathbb{R}}$ and the continuity of the
map $(x,\theta)\mapsto f_{\beta,\sigma,\theta,m}(x)$ from
$\overline{\mathbb{R}}\times[0,2\pi)$ to $\mathbb{R}$, we can then find
$\delta>0$ such that for all $\theta\in(\pi-\delta,\pi+\delta)$,

$f_{\beta,\sigma,\theta,m}(\overline{\mathbb{R}})\subset\big{(}a^{-}_{\beta-\sigma}-\varepsilon,a^{-}_{\beta-\sigma}+\varepsilon\big{)}.$

Using again the fact that $\pi$ is a fixed point of $g$, since $g$ is
continuous we can find $\delta_{1}\in(0,\delta)$ such that for every
$\theta\in(\pi-\delta_{1},\pi+\delta_{1})$,
$g^{m}(\theta)\in(\pi-\varepsilon,\pi+\varepsilon)$. Since
$\pi\in\overline{\\{g^{i}(\theta_{0})\\}_{i\geq 0}}$, we can find $n_{0}\geq
0$ such that $g^{n_{0}}(\theta_{0})\in(\pi-\delta_{1},\pi+\delta_{1})$. So
now, letting $n:=n_{0}+m$, we have

$g^{n}(\theta_{0})=g^{m}(g^{n_{0}}(\theta_{0}))\in(\pi-\varepsilon,\pi+\varepsilon)$

and

$f_{\beta,\sigma,\theta_{0},n}(\mathbb{R})\subset
f_{\beta,\sigma,g^{n_{0}}(\theta_{0}),m}(\mathbb{R})\subset\big{(}a^{-}_{\beta-\sigma}-\varepsilon,a^{-}_{\beta-\sigma}+\varepsilon\big{)},$

and therefore

$G_{\beta,\sigma}^{n}(\mathbb{R}\times\\{\theta_{0}\\})\subset U.\qed$

###### Proof of Proposition 9.

Fix $(\beta,\sigma)\in\mathrm{D}$, and fix a $G_{\beta,\sigma}$-invariant
measure $\mu$ with $\theta$-marginal $\mathbb{P}$. Since $\mathbb{P}$ is a
$g$-ergodic invariant probability measure, the trajectory of
$\mathbb{P}$-almost every point $\theta_{0}$ under $g$ is dense in $[0,2\pi)$.
So taking $(x_{0},\theta_{0})\in\mathrm{supp}\,\mu$ with $\theta_{0}$ having a
dense trajectory, Lemma 10(I) gives that $(a^{-}_{\beta-\sigma},\pi)$ is in
the closure of the $G_{\beta,\sigma}$-trajectory of $(x_{0},\theta_{0})$, and
Lemma 10(II) gives that $(a^{+}_{\beta+\sigma},0)$ is in the closure of the
$G_{\beta,\sigma}$-trajectory of $(x_{0},\theta_{0})$. ∎

###  A.17 Globally contractive dynamics

We will need the following general fact:

######  Lemma 11.

Let $(X_{n})_{n\geq 1}$ be a uniformly bounded sequence of nonnegative-valued
random variables that converges in probability to $0$. Then
$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}X_{n}$ is almost surely equal to
$0$.

###### Proof.

Since $(X_{n})_{n}$ is uniformly bounded, so is
$\big{(}\frac{1}{N}\sum_{n=1}^{N}X_{n}\big{)}_{N}$. Now for a uniformly
bounded sequence of random variables, convergence in probability is equivalent
to convergence in $L^{1}$; so

$0=\lim_{n\to\infty}\mathbb{E}[X_{n}]=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}[X_{n}]=\lim_{N\to\infty}\mathbb{E}\\!\left[\frac{1}{N}\sum_{n=1}^{N}X_{n}\right],$

and therefore $\frac{1}{N}\sum_{n=1}^{N}X_{n}$ converges in probability to $0$
as $N\to\infty$. Since convergence in probability implies the existence of a
subsequence on which we have almost sure convergence, the result follows. ∎

Now for any $(\beta,\sigma)\in\mathbb{R}\times[0,\infty)$, $n\geq 1$ and
$\theta\in[0,2\pi)$, let

$X_{\beta,\sigma,n}(\theta)=f_{\beta,\sigma,\theta,n}(\infty)-f_{\beta,\sigma,\theta,n}(-\infty).$

Similarly to in Sec. A.2, note that for each
$(\theta_{i})_{i\in\mathbb{Z}}\in\Theta$, the sequence of sets
$f_{\beta,\sigma,\theta_{-n},n}(\overline{\mathbb{R}})$ is nested. So let

$\Theta_{\beta,\sigma}=\left\\{(\theta_{i})_{i}\in\Theta\,:\,\bigcap_{n=0}^{\infty}f_{\beta,\sigma,\theta_{-n},n}(\overline{\mathbb{R}})\textrm{
is a singleton}\right\\},$

and for each $(\theta_{i})_{i}\in\Theta_{\beta,\sigma}$, let
$a_{\beta,\sigma}((\theta_{i})_{i})\in\mathbb{R}$ be such that

$\bigcap_{n=0}^{\infty}f_{\beta,\sigma,\theta_{-n},n}(\overline{\mathbb{R}})=\\{a_{\beta,\sigma}((\theta_{i})_{i})\\}.$

Now let us define $\tilde{\mathrm{D}}$ to be the set of all
$(\beta,\sigma)\in\mathbb{R}\times[0,\infty)$ for which
$\tilde{\mathbb{P}}(\Theta_{\beta,\sigma})=1$; and similarly to in Sec. A.2,
for each $(\beta,\sigma)\in\tilde{\mathrm{D}}$, let $\mu_{\beta,\sigma}$ be
the pushforward of $\tilde{\mathbb{P}}$ under the map
$\Pi_{\beta,\sigma}\colon\boldsymbol{\theta}\mapsto(a_{\beta,\sigma}(\boldsymbol{\theta}),\theta_{0})$,
where $\theta_{0}$ denotes the $0$-coordinate of $\boldsymbol{\theta}$. We
will now see that this matches the definitions of $\tilde{\mathrm{D}}$ and
$\mu_{\beta,\sigma}$ given in Sec. 3.1.

######  Proposition 12.

For $(\beta,\sigma)\in\mathbb{R}\times[0,\infty)$, the following statements
are equivalent:

  * (i)

$(\beta,\sigma)\in\tilde{\mathrm{D}}$;

  * (ii)

over the probability space $([0,2\pi),\mathbb{P})$, $X_{\beta,\sigma,n}$
converges in probability to $0$ as $n\to\infty$;

  * (iii)

there is only one $G_{\beta,\sigma}$-invariant measure with $\theta$-marginal
$\mathbb{P}$.

If these statements hold, $\mu_{\beta,\sigma}$ is the unique
$G_{\beta,\sigma}$-invariant measure with $\theta$-marginal $\mathbb{P}$.

###### Proof.

For all $(\theta_{i})_{i\in\mathbb{Z}}\in\Theta$ and $n\geq 1$, let

$\tilde{X}_{\beta,\sigma,n}((\theta_{i})_{i})=f_{\beta,\sigma,\theta_{-n},n}(\infty)-f_{\beta,\sigma,\theta_{-n},n}(\infty).$

Define

$\displaystyle\dot{a}^{-}_{\beta,\sigma}((\theta_{i})_{i})$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,\theta_{-n},n}(-\infty)$
$\displaystyle\dot{a}^{+}_{\beta,\sigma}((\theta_{i})_{i})$
$\displaystyle=\lim_{n\to\infty}f_{\beta,\sigma,\theta_{-n},n}(\infty)$
$\displaystyle\tilde{X}_{\beta,\sigma}$
$\displaystyle=\dot{a}^{+}_{\beta,\sigma}-\dot{a}^{-}_{\beta,\sigma}=\lim_{n\to\infty}\tilde{X}_{\beta,\sigma,n}$

So
$\bigcap_{n=0}^{\infty}f_{\beta,\sigma,\theta_{-n},n}(\overline{\mathbb{R}})$
is precisely
$[\dot{a}^{-}_{\beta,\sigma}((\theta_{i})_{i}),\dot{a}^{+}_{\beta,\sigma}((\theta_{i})_{i})]$.

We first show the equivalence of (i) and (ii). For all $n\geq 1$, since
$\mathbb{P}$ is precisely the $\theta_{-n}$-marginal of $\tilde{\mathbb{P}}$,
the law of $\tilde{X}_{\beta,\sigma,n}$ over $(\Theta,\tilde{\mathbb{P}})$ is
the same as the law of $X_{\beta,\sigma,n}$ over $([0,2\pi),\mathbb{P})$.
Hence, as $n\to\infty$, $X_{\beta,\sigma,n}$ converges in law to the law of
$\tilde{X}_{\beta,\sigma}$. But
$\tilde{X}_{\beta,\sigma}\overset{\tilde{\mathbb{P}}\textrm{-a.s.}}{=}0$ if
and only if
$\dot{a}^{-}_{\beta,\sigma}\overset{\tilde{\mathbb{P}}\textrm{-a.s.}}{=}\dot{a}^{+}_{\beta,\sigma}$,
i.e. if and only if $(\beta,\sigma)\in\tilde{\mathrm{D}}$.

We next show that (iii)$\Rightarrow$(i). Let $\dot{\mu}^{\pm}_{\beta,\sigma}$
be the pushforward of $\tilde{\mathbb{P}}$ under the map
$\dot{\Pi}^{\pm}_{\beta,\sigma}\colon\boldsymbol{\theta}\mapsto(\dot{a}^{\pm}_{\beta,\sigma}(\boldsymbol{\theta}),\theta_{0})$.
Note that the $x$-marginal of $\dot{\mu}^{\pm}_{\beta,\sigma}$ is precisely
the law of $\dot{a}^{\pm}_{\beta,\sigma}$ over $(\Theta,\tilde{\mathbb{P}})$.
Now just as in Sec. A.4, $\dot{a}^{\pm}_{\beta,\sigma}$ fulfils the invariant-
graph property; and therefore, as in Secs. A.2 and A.5,
$\dot{\mu}^{-}_{\beta,\sigma}$ and $\dot{\mu}^{+}_{\beta,\sigma}$ are
$G_{\beta,\sigma}$-invariant measures with $\theta$-marginal $\mathbb{P}$.
Therefore, assuming (iii), we have
$\dot{\mu}^{-}_{\beta,\sigma}=\dot{\mu}^{+}_{\beta,\sigma}$ and so the law of
$\dot{a}^{-}_{\beta,\sigma}$ is the same as the law of
$\dot{a}^{+}_{\beta,\sigma}$. But since
$\dot{a}^{-}_{\beta,\sigma}\leq\dot{a}^{+}_{\beta,\sigma}$ throughout
$\Theta$, it then follows that
$\dot{a}^{-}_{\beta,\sigma}\overset{\tilde{\mathbb{P}}\textrm{-a.s.}}{=}\dot{a}^{+}_{\beta,\sigma}$.

To show (i)$\Rightarrow$(ii), suppose for a contradiction that (i) holds and
(ii) does not. On the basis of Lemma 1, let $\mu_{\beta,\sigma}^{1}$ and
$\mu_{\beta,\sigma}^{2}$ be distinct $G_{\beta,\sigma}$-ergodic invariant
measures with $\theta$-marginal $\mathbb{P}$, and let
$E_{1},E_{2}\subset[0,2\pi)$ be the $\theta$-projection of the basin of
$\mu_{\beta,\sigma}^{1},\mu_{\beta,\sigma}^{2}$ respectively. Now $E_{1}$ and
$E_{2}$ are $\mathbb{P}$-full measure sets; and since
$(X_{\beta,\sigma,n})_{n\geq 1}$ is uniformly bounded by $\alpha\pi$, Lemma 11
gives that
$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}X_{\beta,\sigma,n}\overset{\mathbb{P}\textrm{-a.s.}}{=}0$.
So fix a point $\theta\in E_{1}\cap E_{2}$ such that
$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}X_{\beta,\sigma,n}(\theta)=0$.
Fix $x_{1},x_{2}$ such that $(\theta,x_{1})$ and $(\theta,x_{2})$ are in the
basin of $\mu_{\beta,\sigma}^{1}$ and $\mu_{\beta,\sigma}^{2}$ respectively.
Let $h\colon\mathbb{R}\times[0,2\pi)$ be a $1$-Lipschitz function such that
$\int_{\mathbb{R}\times[0,2\pi)}h\,d\mu_{\beta,\sigma}^{1}\neq\int_{\mathbb{R}\times[0,2\pi)}h\,d\mu_{\beta,\sigma}^{2}$.
We have

$\displaystyle 0$
$\displaystyle<\left|\int_{\mathbb{R}\times[0,2\pi)}h\,d\mu_{\beta,\sigma}^{1}-\int_{\mathbb{R}\times[0,2\pi)}h\,d\mu_{\beta,\sigma}^{2}\right|$
$\displaystyle=\lim_{N\to\infty}\left|\frac{1}{N}\sum_{n=1}^{N}\bigg{(}h(G_{\beta,\sigma}^{n}(\theta,x_{1}))-h(G_{\beta,\sigma}^{n}(\theta,x_{2}))\bigg{)}\right|$
$\displaystyle\leq\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}|f_{\beta,\sigma,\theta,n}(x_{1})-f_{\beta,\sigma,\theta_{0},n}(x_{2})|$
$\displaystyle\leq\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}X_{\beta,\sigma,n}(\theta)$
$\displaystyle=0,$

giving a contradiction.

If the statements (i)–(iii) hold, then as in the proof of the direction
(iii)$\Rightarrow$(i), $\mu_{\beta,\sigma}$ is a $G_{\beta,\sigma}$-invariant
measure with $\theta$-marginal $\mathbb{P}$. ∎

Now let $\tilde{\mathrm{D}}_{\lambda<0}$ be as defined in Sec. 3.1, namely,
$\tilde{\mathrm{D}}_{\lambda<0}\subset\tilde{\mathrm{D}}$ is the set of all
$(\beta,\sigma)\in\tilde{\mathrm{D}}$ for which
$\lambda_{\beta,\sigma}(\mu_{\beta,\sigma})<0$.

######  Proposition 13.

For all $(\beta,\sigma)\in\tilde{\mathrm{D}}_{\lambda<0}$,
$X_{\beta,\sigma,n}$ exponentially converges to $0$ $\mathbb{P}$-a.s. as
$n\to\infty$.

###### Proof.

Fix $(\beta,\sigma)\in\tilde{\mathrm{D}}_{\lambda<0}$. By a stable manifold
theorem [28, Proposition 3.3], for $\mu_{\beta,\sigma}$-almost all
$(x,\theta)\in\mathbb{R}\times[0,2\pi)$, there exists $\delta>0$ such that

$f_{\beta,\sigma,\theta,n}(x+\delta)-f_{\beta,\sigma,\theta,n}(x-\delta)\to
0\textrm{ exponentially as }n\to\infty.$

So take a value $\delta_{1}>0$ sufficiently small that the set

$Y:=\left\\{(x,\theta)\,:\,f_{\beta,\sigma,\theta,n}(x+\delta_{1})-f_{\beta,\sigma,\theta,n}(x-\delta_{1})\to
0\textrm{ exponentially as }n\to\infty\right\\}$

has $\mu_{\beta,\sigma}(Y)>0$. Let

$Z=\left\\{(x,\theta)\in\mathbb{R}\times[0,2\pi)\,:\,\lim_{n\to\infty}\frac{1}{N}\\#\left\\{n\in\\{1,\ldots,N\\}:G_{\beta,\sigma}^{n}(x,\theta)\in
Y\right\\}=\mu_{\beta,\sigma}(Y)\right\\}.$

Since $\mu_{\beta,\sigma}$ is a $G_{\beta,\sigma}$-ergodic probability
measure, we have that $\mu_{\beta,\sigma}(Z)=1$. Now since
$X_{\beta,\sigma,n}$ converges in probability to $0$ as $n\to\infty$, by Lemma
11 (applied to $\mathbbm{1}_{\\{X_{\beta,\sigma,n}>\delta_{1}\\}}$) there is a
$\mathbb{P}$-full measure set $E_{0}\subset[0,2\pi)$ such that for all
$\theta\in E_{0}$,

$\limsup_{n\to\infty}\frac{1}{N}\\#\left\\{n\in\\{1,\ldots,N\\}:X_{\beta,\sigma,n}(\theta)\leq\delta_{1}\right\\}=1.$

Let $E$ be the intersection of $E_{0}$ and the $\theta$-projection of $Z$. So
$E$ is a $\mathbb{P}$-full measure set. We will show that for all $\theta\in
E$, $X_{\beta,\sigma,n}(\theta)\to 0$ exponentially as $n\to\infty$. Fix
$\theta\in E$, and fix $x$ such that $(x,\theta)\in Z$. Since

$\displaystyle\limsup_{n\to\infty}\frac{1}{N}\\#\left\\{n\in\\{1,\ldots,N\\}:X_{\beta,\sigma,n}(\theta)\leq\delta_{1}\right\\}$
$\displaystyle=1\quad\textrm{and}$
$\displaystyle\lim_{n\to\infty}\frac{1}{N}\\#\left\\{n\in\\{1,\ldots,N\\}:G_{\beta,\sigma}^{n}(x,\theta)\in
Y\right\\}$ $\displaystyle>0,$

there must exist $n_{0}\geq 1$ with the properties that
$X_{\beta,\sigma,n_{0}}(\theta)\leq\delta_{1}$ and
$G_{\beta,\sigma}^{n_{0}}(x,\theta)\in Y$. But

  * •

the fact that $X_{\beta,\sigma,n_{0}}(\theta)\leq\delta_{1}$ implies that

$f_{\beta,\sigma,\theta,n_{0}}(\overline{\mathbb{R}})\subset[f_{\beta,\sigma,\theta,n_{0}}(x)-\delta_{1},f_{\beta,\sigma,\theta,n_{0}}(x)+\delta_{1}]=:I_{n_{0}}$

and therefore

$f_{\beta,\sigma,\theta,n_{0}+n}(\overline{\mathbb{R}})\subset
f_{\beta,\sigma,g^{n_{0}}(\theta),n}(I_{n_{0}})$ (28)

for all $n\geq 0$;

  * •

the fact that $G_{\beta,\sigma}^{n_{0}}(x,\theta)\in Y$ implies that

$\mathrm{diam}(f_{\beta,\sigma,g^{n_{0}}(\theta),n}(I_{n_{0}}))\to 0\textrm{
exponentially as }n\to\infty.$ (29)

Combining Eqs. 28 and (29) gives that
$X_{\beta,\sigma,n}(\theta)=\mathrm{diam}(f_{\beta,\sigma,\theta,n}(\overline{\mathbb{R}}))\to
0$ exponentially as $n\to\infty$. ∎

######  Proposition 14.

For all $\sigma\geq 0$, there is no $\mathbb{P}$-full measure subset of
$[0,2\pi)$ on which $X_{0,\sigma,n}\to 0$ uniformly as $n\to\infty$.

###### Proof.

The set $\\{\frac{\pi}{2},\frac{3\pi}{2}\\}$ is a $2$-periodic orbit of $g$ on
which $\cos=0$, and hence

$f_{0,\sigma,\frac{\pi}{2},n}=f_{0}^{n}$

for all $n\geq 0$. Hence

$\lim_{n\to\infty}X_{0,\sigma,n}(\tfrac{\pi}{2})=\left(\lim_{n\to\infty}f_{0}^{n}(\infty)\right)-\left(\lim_{n\to\infty}f_{0}^{n}(-\infty)\right)=2a_{0}>0.$

Since $X_{0,\sigma,n}$ is continuous for each $n$, and $\tfrac{\pi}{2}$
belongs to the closure of every $\mathbb{P}$-full measure set, we then have
the desired result. ∎

###  A.18 Continuous dependence of $a$ and $\mu$

Let
$\mathfrak{S}=\\{(\beta,\sigma,\boldsymbol{\theta})\in\mathbb{R}\times[0,\infty)\times\Theta\,:\,\boldsymbol{\theta}\in\Theta_{\beta,\sigma}\\}$.
Then exactly the same argument as in Sec. A.3 with $\mathfrak{S}$ in place of
$S^{+}\times\Theta$ and with $\overline{\mathbb{R}}$ in place of
$[x^{*},\infty]$ yields that the map
$(\beta,\sigma,\boldsymbol{\theta})\mapsto
a_{\beta,\sigma}(\boldsymbol{\theta})$ is continuous on $\mathfrak{S}$.

Consequently, for each $(\beta,\sigma)\in\tilde{\mathrm{D}}$, the image of
$\Theta_{\beta,\sigma}$ under $\Pi_{\beta,\sigma}$ is contained in the support
$\mathcal{A}_{\beta,\sigma}$ of $\mu_{\beta,\sigma}$.

Moreover (similarly to in Sec. A.5), over
$(\beta,\sigma)\in\tilde{\mathrm{D}}$, $\mu_{\beta,\sigma}$ depends
continuously on $(\beta,\sigma)$ in the topology of weak convergence: Fix a
sequence $(\beta_{n},\sigma_{n})$ in $\tilde{\mathrm{D}}$ converging to a
point $(\beta,\sigma)\in\tilde{\mathrm{D}}$. The set
$\Theta_{\beta,\sigma}\cap\bigcap_{n}\Theta_{\beta_{n},\sigma_{n}}$ is a
$\tilde{\mathbb{P}}$-full measure set on which $\Pi_{\beta_{n},\sigma_{n}}$
converges pointwise to $\Pi_{\beta,\sigma}$, and so the dominated convergence
theorem gives that $\mu_{\beta_{n},\sigma_{n}}$ converges weakly to
$\mu_{\beta,\sigma}$.

###  A.19 Basin of $\mathcal{A}_{\beta,\sigma}$ and $\mu_{\beta,\sigma}$ in
$\tilde{\mathrm{D}}_{\lambda<0}$

For $(\beta,\sigma)\in\tilde{\mathrm{D}}$, we said in the proof of Proposition
12 that $\dot{a}^{+}_{\beta,\sigma}$ and $\dot{a}^{-}_{\beta,\sigma}$ fulfil
the invariant-graph property; hence, since

$\Theta_{\beta,\sigma}=\\{\boldsymbol{\theta}\in\Theta\,:\,\dot{a}^{+}_{\beta,\sigma}(\boldsymbol{\theta})=\dot{a}^{-}_{\beta,\sigma}(\boldsymbol{\theta})\\},$

we have that $g(\Theta_{\beta,\sigma})\subset\Theta_{\beta,\sigma}$ and
$a_{\beta,\sigma}$ fulfils the invariant-graph property: for all
$(\theta_{i})_{i}\in\Theta_{\beta,\sigma}$ and $n\geq 0$,

$a_{\beta,\sigma}((\theta_{i+n})_{i})=f_{\beta,\sigma,\theta_{0},n}\big{(}a_{\beta,\sigma}((\theta_{i})_{i})\big{)}.$
(30)

Hence, a consequence of Proposition 13 is the following.

######  Corollary 15.

Fix $(\beta,\sigma)\in\tilde{\mathrm{D}}_{\lambda<0}$. There is a
$\tilde{\mathbb{P}}$-full measure set $\tilde{E}\subset\Theta_{\beta,\sigma}$
such that for every $(x,(\theta_{i})_{i})\in\mathbb{R}\times\tilde{E}$,

$\left|f_{\beta,\sigma,\theta_{0},n}(x)-a_{\beta,\sigma}((\theta_{i+n})_{i})\right|\to
0\quad\textrm{as }n\to\infty.$

###### Proof.

On the basis of Proposition 13, let $E\subset[0,2\pi)$ be a $\mathbb{P}$-full
measure set on which $X_{\beta,\sigma,n}$ converges pointwise to $0$. Let
$\tilde{E}$ be the set of points in $\Theta_{\beta,\sigma}$ whose
$0$-coordinate belongs to $E$. Then due to (30), for all
$(x,(\theta_{i})_{i})\in\mathbb{R}\times\tilde{E}$ we have

$\left|f_{\beta,\sigma,\theta_{0},n}(x)-a_{\beta,\sigma}((\theta_{i+n})_{i})\right|\leq\mathrm{diam}(f_{\beta,\sigma,\theta_{0},n}(\mathbb{R}))=X_{\beta,\sigma,n}(\theta_{0})\to
0\quad\textrm{as }n\to\infty.\qed$

######  Corollary 16.

Fix $(\beta,\sigma)\in\tilde{\mathrm{D}}_{\lambda<0}$. There is a
$\mathbb{P}$-full measure set $E_{0}\subset[0,2\pi)$ such that
$\mathbb{R}\times E_{0}$ is contained in the basin of
$\mathcal{A}_{\beta,\sigma}$.

###### Proof.

Let $\tilde{E}$ be as in Corollary 15, and let $E_{0}$ be the
$\theta_{0}$-projection of $\tilde{E}$. Fix $(x,\theta)\in\mathbb{R}\times
E_{0}$, and take a point $(\theta_{i})_{i}\in\tilde{E}$ with $0$-coordinate
$\theta$. Since, the image of $\Theta_{\beta,\sigma}$ under
$\Pi_{\beta,\sigma}$ is contained in $\mathcal{A}_{\beta,\sigma}$, we have in
particular that

$\Big{(}a_{\beta,\sigma}((\theta_{i+n})_{i})\,,\,g^{n}(\theta)\Big{)}\in\mathcal{A}_{\beta,\sigma}.$

for all $n\geq 0$. So then, from Corollary 15, we have that the distance of
$G^{n}(x,\theta)$ from $\mathcal{A}_{\beta,\sigma}$ tends to $0$ as
$n\to\infty$. ∎

######  Proposition 17.

Fix $(\beta,\sigma)\in\tilde{\mathrm{D}}_{\lambda<0}$. There is a
$\mathbb{P}$-full measure set $R\subset[0,2\pi)$ such that $\mathbb{R}\times
R$ is contained in the basin of $\mu_{\beta,\sigma}$.

###### Proof.

Construct $\Theta_{1}\subset\Theta$ as in the proof of Proposition 7 with
$a_{\beta,\sigma}$ in place of $a^{+}_{\beta,\sigma}$. Let
$\Theta_{2}=\Theta_{1}\cap\tilde{E}$, where $\tilde{E}$ is as in Corollary 15.
Let $R$ be the $\theta_{0}$-projection of $\Theta_{2}$. Then for any
$(x,\theta)\in\mathbb{R}\times R$, one shows that $(x,\theta)$ is in the basin