# Periodic boundary points for simply connected Fatou components of
transcendental maps

Anna Jové This work is supported by the Spanish government grant FPI
PRE2021-097372 and PID2020-118281GB-C32. Corresponding author.
<EMAIL_ADDRESS>Departament de Matemàtiques i Informàtica,
Universitat de Barcelona, Barcelona, Spain

###### Abstract

Let $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a transcendental map, and
let $U$ be an attracting or parabolic basin, or a doubly parabolic Baker
domain. Assume $U$ is simply connected. Then, we prove that periodic points
are dense in $\partial U$, under certain hypothesis on the postsingular set.
This generalizes a result by F. Przytycki and A. Zdunik for rational maps
[PZ94]. Our proof uses techniques from measure theory, ergodic theory,
conformal analysis, and inner functions. In particular, a result on the
distortion of inner functions near the unit circle is provided, which is of
independent interest.

##  1 Introduction

Chaos is a key concept related to the complexity of a topological dynamical
system, which is in turn, and according to any definition of chaos, strongly
connected with the density of periodic points. In this paper, we investigate
density of periodic points on invariant subset of the phase space, namely the
boundary of stable regions like, for example basins of attraction.

Throughout this paper, we consider the discrete dynamical system generated by
a holomorphic map $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$, i.e. we study
the sequences of iterates $\left\\{f^{n}(z)\right\\}_{n}$, where
$z\in{\mathbb{C}}$. We assume $f$ is neither constant nor Möbius. If infinity
is an essential singularity of $f$, then we say that $f$ is a transcendental
meromorphic map; otherwise $f$ extends holomorphically to
$\widehat{\mathbb{C}}$ as a rational map. These dynamical systems arise
naturally, for example, from the popular Newton’s root-finding method applied
to entire functions. For general background on the iteration of meromorphic
maps, we refer to [Ber93].

In this situation, the complex plane, regarded as the phase space of the
dynamical system, is divided into two totally invariant sets: the Fatou set
$\mathcal{F}(f)$, the set of points $z\in\mathbb{C}$ such that
$\left\\{f^{n}\right\\}_{n\in\mathbb{N}}$ is well-defined and forms a normal
family in some neighbourhood of $z$; and the Julia set $\mathcal{J}(f)$, its
complement, where the dynamics is chaotic. In particular, periodic points are
dense in $\mathcal{J}(f)$. Indeed, for rational maps, this was already
established by Fatou and Julia in the beginning of the twentieth century.
Baker [Bak68, Thm. 1] proved that this holds for transcendental entire
functions as well, relying on a deep theorem of Ahlfors, a proof that was
later generalized to transcendental meromorphic functions [BKL91, Thm. 1]. The
goal of this paper is to study density of periodic points in appropriate
invariant subsets of the Julia set, which will be determined by means of the
Fatou set.

More precisely, the Fatou set is open and consists in general of infinitely
many components, which are called Fatou components. Due to the invariance of
the Fatou and Julia sets, Fatou components are either periodic, preperiodic or
wandering. Periodic Fatou components are classified into rotation domains
(Siegel disks or Herman rings), attracting and parabolic basins, and Baker
domains, being the latter exclusive of transcendental functions.

Observe that, if $U$ is a $p$-periodic Fatou component, then
$\left\\{f^{n}(\partial U)\right\\}_{n=0}^{p-1}$ is a closed invariant subset
of the Julia set. Hence, we ask the following question.

######  Question.

Let $U$ be a periodic Fatou component. Are periodic points dense in $\partial
U$?

Note that although periodic points are dense in the Julia set, a priori they
could accumulate on $\partial U$ only from the complement of $\overline{U}$,
without being in $\partial U$. For instance, if $U$ is a rotation domain with
locally connected boundary, then there are no periodic points in $\partial U$
at all. Nevertheless, F. Przytycki and A. Zdunik gave a positive answer to the
question for rational maps, excluding Fatou components which are rotation
domains.

######  Theorem.

(Przytycki-Zdunik, [PZ94]) Let
$f\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ be a rational map, and
let $U$ be an attracting or parabolic basin for $f$. Then, periodic points are
dense in $\partial U$.

In the seminal paper [PZ94], two different proofs are provided: one for simply
connected attracting basins, and a general one, which works in the non-simply
connected or parabolic situations. The latter relies on a technique, known as
geometric coding trees, which has been shown not to work well in the infinite
degree case, even for hyperbolic maps (see [BK07, p. 405]).

Their proof relies on three specific features of rational maps: $f$ having
finitely many singular values (points on which some branch of $f^{-1}$ is not
locally well-defined, see Sect. 2.6), $f$ extending analytically to the
boundary of $U$ (taken in $\widehat{\mathbb{C}}$), and $f$ having finite
degree. Note that these three assumptions are no longer satisfied for
transcendental meromorphic maps. Indeed, a transcendental map $f$ can have
infinitely many singular values, and it may have essential singularities on
the boundary of $U$, implying that $f|_{\partial U}$ is no longer analytic. In
addition, $f|_{U}$ may have infinite degree. Moreover, when dealing with
transcendental meromorphic functions, one encounters other new challenges,
namely a new type of Fatou components (Baker domains, on which iterates
accumulate on the essential singularity), and the presence of asymptotic
values.

The goal of this paper is to show that, by imposing some mild assumptions on
the postsingular set of $f$

$P(f)\coloneqq\overline{\bigcup\limits_{s\in SV(f)}\bigcup\limits_{n\geq
0}f^{n}(s)},$

where $SV(f)$ denotes the set of singular values of $f$, we are able to
overcome these challenges, and prove the density of periodic boundary points.
Indeed, we show the following. (Recall that, given a simply connected domain
$U$, $C\subset U$ is a crosscut if $C$ is a Jordan arc such that
$\overline{C}=C\cup\left\\{a,b\right\\}$, with $a,b\in\partial U$, $a\neq b$;
any of the two connected components of $U\smallsetminus C$ is a crosscut
neighbourhood.)

######  Theorem A.

(Periodic boundary points are dense) Let
$f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a meromorphic function, and let
$U$ be a periodic simply connected Fatou component for $f$.

Assume the following conditions are satisfied.

  1. (i)

$U$ is either an attracting basin, a parabolic basin, or a doubly parabolic
Baker domain, with $f|_{\partial U}$ recurrent.

  2. (ii)

There exists $x\in\partial U$ and $r>0$ such that $P(f)\cap D(x,r)=\emptyset$.

  3. (iii)

There exists a crosscut neighbourhood $N\subset U$ with $P(f)\cap
N=\emptyset$.

Then, periodic points are dense in $\partial U$.

Next, we shall outline the main steps on the proof of [PZ94], and explain how
the new difficulties that appear for transcendental maps are overcome, showing
at the same time the need for the hypotheses of the theorem. For simplicity,
we shall assume that the Fatou component $U$ is invariant.

In the case of an attracting basin of a rational function, Pesin’s theory can
be applied to prove that for $\omega_{U}$-almost every $x\in\partial U$, there
exist inverse branches which are locally well-defined and contracting with
respect to the Euclidean metric [PZ94, Lemma 1] (see also [PUZ91, Lemma 1],
and [PU10, Thm. 11.2.3]). Crucial ingredients in this proof are the ergodic
properties of $f|_{\partial U}$, studied in [Prz85, Prz86], together with
$f|_{\partial U}$ being analytic and the finitude of critical values. None of
the previous conditions is satisfied for a general transcendental meromorphic
map, so a priori Pesin’s theory cannot be applied in our situation. We solve
this by assuming that inverse branches are well-defined $\omega_{U}$-almost
everywhere (this is a straightforward consequence of (ii)), and we prove
contraction of inverse branches with respect to the hyperbolic metric in a
suitable domain.

Next, we extend the proof of [PZ94] to other Fatou components, apart from
attracting basins. Indeed, our proof relies only on the ergodic properties of
the map $f|_{\partial U}$, not on the precise type of Fatou component we are
considering. More precisely, we only ask $f|_{\partial U}$ to be ergodic and
recurrent with respect to the harmonic measure $\omega_{U}$, which implies
that $\omega_{U}$-almost every orbit in $\partial U$ is dense in $\partial U$.
Hence, all Fatou components for which the boundary map is ergodic and
recurrent may be considered, and these include attracting and parabolic
basins, rotation domains and certain types of Baker domains (for instance,
doubly parabolic Baker domains with singular values compactly contained in
$U$, Thm. 5.7). However, note that rotation domains never satisfy the
hypothesis of our theorem, since $P(f)$ is always dense in their boundary, and
(ii) is never fulfilled.

Finally, as it is common in constructions of this kind, the proof relies
strongly on the inferred dynamics in the unit disk $\mathbb{D}$ via the
Riemann map $\varphi\colon\mathbb{D}\to U$. More precisely, let $U$ be a
$p$-periodic (simply connected) attracting basin of a rational map $f$, and
consider a Riemann map $\varphi\colon\mathbb{D}\to U$. Then, the function

$g\colon\mathbb{D}\to\mathbb{D},\hskip 14.22636ptg\coloneqq\varphi^{-1}\circ
f^{p}\circ\varphi$

is the inner function associated to $(f^{p},U)$ by $\varphi$ (see Sect. 5.1).

A careful study of the associated inner function is required. In the case of a
rational attracting basin considered in [PZ94], $g$ is a finite Blaschke
product, which can be chosen to satisfy $g(0)=0$. We shall view $g$ as a
rational map $g\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$, extended by
Schwarz reflection. Then, its critical values (which are finitely many) are
compactly contained in $\mathbb{D}$ (and, by reflection, in
$\widehat{\mathbb{C}}\smallsetminus\overline{\mathbb{D}}$) and their orbits
converge uniformly to 0 (or to $\infty$), which are attracting fixed points.
Hence, inverse branches of $g$ are well-defined for all points in
$\partial\mathbb{D}$. Moreover, precise estimates on the behaviour of such
inverse branches are given in [PZ94, Lemma 2].

In contrast to this setting, in the general situation we consider, $g$ is no
longer a finite Blaschke product, and may not have an attracting fixed point
in $\mathbb{D}$. However, having singular values of $f|_{U}$ compactly
contained in $U$ allows us to control inverse branches for the associated
inner function $g$ at $\lambda$-almost every point in $\mathbb{D}$, even if
$g$ has infinite degree. Indeed, we consider the maximal meromorphic extension
of $g$:

$g\colon\widehat{\mathbb{C}}\smallsetminus E(g)\to\widehat{\mathbb{C}},$

where $E(g)\subset\partial\mathbb{D}$ denotes the set of singularities of $g$
(points at which $g$ cannot be extended analytically), and denote by
$g^{*}\colon\partial\mathbb{D}\to\partial\mathbb{D}$ its radial extension (see
Sect. 3). In this situation, we prove the following result concerning inner
functions (not necessarily associated to Fatou components), which is of
independent interest.

######  Theorem B.

(Inverse branches at boundary points) Let $g\colon\mathbb{D}\to\mathbb{D}$ be
an inner function, such that $g^{*}|_{\partial\mathbb{D}}$ is recurrent.
Assume that there exists a crosscut neighbourhood $N\subset\mathbb{D}$ with
$P(g)\cap N=\emptyset$. Then, for $\lambda$-almost every
$\xi\in\partial\mathbb{D}$, there exists $\rho_{0}\coloneqq\rho_{0}(\xi)>0$
such that all branches $G_{n}$ of $g^{-n}$ are well-defined in
$D(\xi,\rho_{0})$. In particular, the set $E(g)$ of singularities of $g$ has
$\lambda$-measure zero.

In addition, for all $0<\alpha<\frac{\pi}{2}$, there exists
$\rho_{1}<\rho_{0}$ such that, for all $n\geq 0$, all branches $G_{n}$ of
$g^{-n}$ are well-defined in $D(\xi,\rho_{1})$ and, if $\gamma$ denotes the
radial segment at $\xi$, then the curve $G_{n}(\gamma)$ tends to $G_{n}(\xi)$
non-tangetially with angle at most $\alpha$.

Note that $\alpha$ does not depend on $n$, nor on the chosen inverse branch.
Apart from giving a precise characterization of inverse branches, Theorem B
also describes measure-theoretically the set of singularities, improving the
results in [EFJS19, ERS20, JF23]. Compare also with the situation for one
component inner functions (a more restrictive class of inner functions)
described in [IU23, Part III].

At this point, we shall make some additional remarks, in order to clarify and
contextualize our results.

First, one should observe that the class of (transcendental) meromorphic
functions is not closed under composition: iterates $f^{n}$ of a
transcendental meromorphic function $f$ have, in general, countably many
analytic singularities, so they are no longer meromorphic functions of the
plane. Hence, we consider functions in class $\mathbb{K}$, the smallest class
of functions which includes transcendental meromorphic functions and which is
closed under composition. Formally, $f\in\mathbb{K}$ if there exists a compact
countable set $E(f)\subset\widehat{\mathbb{C}}$ such that

$f\colon\widehat{\mathbb{C}}\smallsetminus E(f)\to\widehat{\mathbb{C}}$

is meromorphic in $\widehat{\mathbb{C}}\smallsetminus E(f)$ but in no larger
set. The theory of Fatou and Julia of iteration of rational maps was extended
to class $\mathbb{K}$ by Bolsch, and Baker, Domínguez and Herring [Bol96,
Bol97, Bol99, BDH01, BDH04, Dom10, DOS22]. Although more sophisticated tools
are needed, the main features of iteration theory extend successfully to class
$\mathbb{K}$ (see Sect. 5). In particular, if $f\in\mathbb{K}$, then for any
$k\geq 1$, $f^{k}\in\mathbb{K}$ and $\mathcal{F}(f)=\mathcal{F}(f^{k})$. This
allows us to reduce the study of $k$-periodic Fatou components to the study of
the invariant ones, just replacing $f$ by $f^{k}$.

Second, Theorem A concerns periodic Fatou components which are simply
connected. It is well-known that Fatou components of meromorphic functions
(and for functions in class $\mathbb{K}$) are either simply connected, doubly
connected or infinitely connected. There are plenty of examples of functions
and classes of functions whose Fatou components are simply connected. For
instance, periodic Fatou components of entire maps are always simply connected
[Bak84]. Moreover, if $f$ is an entire function, then its Newton’s method

$N_{f}\colon\mathbb{C}\to\widehat{\mathbb{C}},\hskip
14.22636ptN_{f}(z)\coloneqq z-\dfrac{f(z)}{f^{\prime}(z)}$

is a meromorphic function, whose Fatou components are simply connected [FJT08,
FJT11, BFJK14, BFJK18].

Third, for a transcendental meromorphic function, it is not even known whether
there exists a periodic point in the boundary of every periodic Fatou
component. In particular, our result also answers this question for a wide
class of Fatou components.

Finally, Theorem A and Theorem B are presented in the introduction in a
simplified form. For the stronger version of Theorem A, see Theorem 6.1; while
for Theorem B in its complete form, look at Theorem 4.6.

######  Remark.

(Doubly connected Fatou components) It is well-known that periodic Fatou
components of functions in class $\mathbb{K}$ are either simply connected,
doubly connected or infinitely connected. However, in contrast with the
rational case, doubly connected Fatou components are not necessarily Herman
rings. Indeed, an explicit example of an invariant doubly connected attracting
basin of a holomorphic self-map of the punctured plane $\mathbb{C}^{*}$ is
provided in [Bol99, p. 545] (see also [Kee88, Sect. 3 and 6]). Moreover, in
[EMS21] an example of a doubly connected Baker domain of a holomorphic self-
map of the punctured plane $\mathbb{C}^{*}$ is constructed using approximation
theory. More examples are provided in [HWZ22].

It is our belief that doubly connected Fatou components can be studied in a
similar maner as simply connected ones. Indeed, in this case, there always
exists a universal covering $\pi\colon\mathbb{D}\to U$, which behaves locally
near $\partial\mathbb{D}$ as a conformal map. Hence, it seems plausible that
the same arguments apply to prove that periodic points are dense in the
boundary of such Fatou components, under the same conditions of Theorem A (or
its general version 6.1).

Structure of the paper. In Section 2, some preliminary definitions and results
are collected, which are used through the article. This includes distortion
estimates for univalent maps, abstract ergodic theory, boundary extension of
holomorphic maps, and harmonic measure. Section 3 is devoted to the iteration
of inner functions, including the classical Denjoy-Wolff Theorem, Cowen’s
classification of self-maps of the unit disk, and Doering-Mañé’s results on
the ergodic properties of inner functions. Next, in Section 4, we deal with
inverse branches of inner functions near the unit circle, proving Theorem B.
In Section 5 we include the Fatou and Julia theory of functions in class
$\mathbb{K}$, and the results needed for the statement and the proof of
Theorem 6.1. Finally, Section 6 is devoted to prove Theorem A using all the
tools previously developed.

Notation. Throughout this article, $\mathbb{C}$ and $\widehat{\mathbb{C}}$
denote the complex plane and the Riemann sphere, respectively. Given a domain
$U\subset\widehat{\mathbb{C}}$, we denote its boundary (in
$\widehat{\mathbb{C}}$) by $\widehat{\partial}U$. We save the notation of
${\partial}U$ to denote the boundary of $U$ with respect to the domain of
definition of the function we are considering (see Sec. 5).

If $U$ is a simply connected domain, $\omega_{U}$ stands for the class of
harmonic measures in $\widehat{\partial}U$, while the notation
$\omega_{U}(z_{0},\cdot)$ stands for the harmonic measure in
$\widehat{\partial}U$ with respect to $z_{0}\in U$ (see Sect. 2.5). We denote
by $\mathbb{D}$, the unit disk; by $\partial\mathbb{D}$, the unit circle; and
by $\lambda$, the Lebesgue measure on $\partial\mathbb{D}$, normalized so that
$\lambda(\partial\mathbb{D})=1$.

Acknowledgments. I am indebted to my supervisor, Núria Fagella, for all her
support and encouragement about this project. I also wish to thank Oleg Ivrii,
for interesting discussions and carefully reading de manuscript, Lasse Rempe,
and specially Anna Zdunik for interesting discussions and comments. Finally, I
want to express my gratitude to the Complex Dynamics Group of the University
of Barcelona for a series of lectures on ergodic theory, which inspired this
paper.

##  2 Preliminaries

In this section we gather the tools we use throughout the article, including
distortion estimates for univalent maps, abstract ergodic theory, boundary
extension of holomorphic maps, and harmonic measure. Although all the results
in this section seem to be well-known, we include the proof of those for which
we could not find a written reference.

###  2.1 Distortion estimates for univalent maps

We need the following results concerning the distortion for univalent maps.

######  Theorem 2.1.

(De Branges, [Bra85]) Let $\varphi\colon\mathbb{D}\to\mathbb{C}$ be univalent,
with $\varphi(0)=0$ and $\varphi^{\prime}(0)=1$. Then,

$\varphi(z)=z+\sum_{n\geq 2}a_{n}z^{n},$

with $\left|a_{n}\right|\leq n$, for $n\geq 2$.

######  Corollary 2.2.

(Distortion estimates for univalent maps) Let $\varphi\colon
D(z_{0},r_{0})\to\mathbb{C}$ be univalent, and let $r\in(0,r_{0})$. Then,
there exists $C\coloneqq C(r,r_{0})$, with $C(r,r_{0})\to 0$ as
$\frac{r}{r_{0}}\to 0$, such that, for all $z\in D(z_{0},r)$,

$\left|\varphi(z)-L(z)\right|\leq
C\left|\varphi^{\prime}(z_{0})\right|\left|z-z_{0}\right|,$

where $L$ stands for the liner map
$L(z)\coloneqq\varphi(z_{0})+\varphi^{\prime}(z_{0})(z-z_{0})$.

In particular, if $\varphi$ additionally satisfies $\varphi(0)=0$ and
$\varphi^{\prime}(0)=1$. Then, for all $r\in(0,1)$, there exists $C\coloneqq
C(r)$, with $C(r)\to 0$ as $r\to 0$, for all $z\in D(0,r)$,

$\left|\frac{\varphi(z)}{z}-1\right|\leq C.$

Note that, in both cases, $C$ does not depend on the univalent map considered.

###### Proof.

Let us start by proving the particular case of
$\varphi\colon\mathbb{D}\to\mathbb{C}$, satisfying $\varphi(0)=0$ and
$\varphi^{\prime}(0)=1$. Then, by Theorem 2.1, for all $z\in\mathbb{D}$, it
holds

$\dfrac{\varphi(z)}{z}=1+\sum_{n\geq 2}a_{n}z^{n-1},$

with $\left|a_{n}\right|\leq n$, for $n\geq 2$. Hence, for $r\in(0,1)$ and
$z\in D(0,r)$, it holds

$\left|\dfrac{\varphi(z)}{z}-1\right|=\left|\sum_{n\geq
2}a_{n}z^{n-1}\right|\leq\sum_{n\geq
2}\left|a_{n}\right|r^{n-1}\leq\sum_{n\geq 2}nr^{n-1}\eqqcolon C(r).$

Note that the last power series converges for $r<1$, and $C(r)\to 0$ as $r\to
0$, as desired.

Now, consider any univalent map $\varphi\colon\mathbb{D}\to\mathbb{C}$, and
let $\psi\colon\mathbb{D}\to\mathbb{C}$ be defined as

$\psi(w)\coloneqq\dfrac{\varphi(z_{0}+r_{0}w)-\varphi(z_{0})}{r_{0}\varphi^{\prime}(z_{0})}.$

Note that $\psi$ is univalent, and satisfies $\psi(0)=0$ and
$\psi^{\prime}(0)=1$. Let $r<r_{0}$ and $\rho\coloneqq\frac{r}{r_{0}}<1$.
Hence, there exists $C\coloneqq C(\rho)$, such that, for $w\in D(0,\rho)$,

$\left|\frac{\psi(w)}{w}-1\right|\leq C.$

Letting $z=z_{0}+r_{0}w$, we get that, for $z\in D(z_{0},r)$,

$\dfrac{\left|\varphi(z)-(\varphi(z_{0})+\varphi^{\prime}(z_{0})(z-z_{0}))\right|}{\left|z-z_{0}\right|\left|\varphi^{\prime}(z_{0})\right|}\leq
C,$

as desired. $\square$

###  2.2 Abstract ergodic theory

We recall some basic notions used in abstract ergodic theory (for more
details, see e.g. [Aar97, PU10, Haw21]).

######  Definition 2.3.

(Ergodic properties of measurable maps) Let $(X,\mathcal{A},\mu)$ be a measure
space, and let $T\colon X\to X$ be measurable. Then, $T$ is:

  * •

non-singular, if for every $A\in\mathcal{A}$, it holds $\mu(T^{-1}(A))=0$ if
and only if $\mu(A)=0$;

  * •

$\mu$\- preserving, if for every $A\in\mathcal{A}$, it holds
$\mu(T^{-1}(A))=\mu(A)$ (we also say that $\mu$ is $T$-invariant);

  * •

recurrent, if for every $A\in\mathcal{A}$ and $\mu$-almost every $x\in A$,
there exists a sequence $n_{k}\to\infty$ such that $T^{n_{k}}(x)\in A$;

  * •

ergodic, if $T$ is non-singular and for every $A\in\mathcal{A}$ with
$T^{-1}(A)=A$, it holds $\mu(A)=0$ or $\mu(X\smallsetminus A)=0$.

Clearly, invariance implies non-singularity. Moreover, the following holds
true.

######  Theorem 2.4.

(Poincaré Recurrence Theorem, [Haw21, Thm. 2.12]) Let $(X,\mathcal{A},\mu)$ be
a measure space, and let $T\colon X\to X$ be a measurable transformation.
Assume $\mu(X)<\infty$, and $T$ is $\mu$-preserving. Then, $T$ is recurrent
with respect to $\mu$.

######  Theorem 2.5.

(Almost every orbit is dense, [Aar97, Prop. 1.2.2]) Let $(X,\mathcal{A},\mu)$
be a measure space, and let $T\colon X\to X$ be non-singular. Then, the
following are equivalent.

  1. (a)

$T$ is ergodic and recurrent.

  2. (b)

For every $A\in\mathcal{A}$ with $\mu(A)>0$, we have that for $\mu$-almost
every $x\in X$, there exists a sequence $n_{k}\to\infty$ such that
$T^{n_{k}}(x)\in A$.

Note that, if the space $X$ is endowed with a topology whose open sets are
measurable and have positive measure, then statement (b) implies that
$\mu$-almost every orbit is dense in $X$.

In holomorphic dynamics, it is possible to replace the function by an iterate
of it, since the dynamics remain essentially the same. Thus, we are interested
in knowing which ergodic properties remain under taking iterates of the
function.

######  Lemma 2.6.

(Ergodic properties for $T^{k}$) Let $(X,\mathcal{A},\mu)$ be a measure space,
and let $T\colon X\to X$ be non-singular. Let $k$ be a positive integer. Then,

  1. (a)

$T$ is recurrent if and only if $T^{k}$ is recurrent.

  2. (b)

If $T^{k}$ is ergodic, so is $T$. The converse is not true in general.

###### Proof.

  1. (a)

It is clear that $T^{k}$ recurrent implies $T$ recurrent. We shall see the
converse. To do so, consider $A\in\mathcal{A}$ with $\mu(A)>0$. Since $T$ is
assumed to be recurrent, for $\mu$-almost every $x\in A$ there exists a
sequence $n_{j}\to\infty$ such that $T^{n_{j}}(x)\in A$. For such $x$,
consider the following subsequences

$\left\\{T^{kn}(x)\right\\}_{n},\left\\{T^{kn+1}(x)\right\\}_{n},\dots,\left\\{T^{2kn-1}(x)\right\\}_{n}.$

At least one of them, say $\left\\{T^{kn+l}(x)\right\\}_{n}$, contains
infinitely many $T^{n_{j}}(x)$’s. Choose $n$ and $k$ so that

$y\coloneqq T^{kn+l}(x)\in A.$

Then, $y$ is a point in $A$ whose orbit returns to $A$ infinitely many times
under $T^{k}$. We claim that such points have full measure in $A$. Assume
that, on the contrary, there exists $B\subset A$ with $\mu(B)>0$ such that,
for all $x\in B$, $T^{nk}(x)\in A$, only for finitely many $n$’s. Applying the
same procedure as before to $B$ we can find a point in $B$ whose orbit returns
to $B$, and hence to $A$, infinitely many times under $T^{k}$, which is a
contradiction. Hence, $T^{k}$ is recurrent.

  2. (b)

Let $A\in\mathcal{A}$ be such that $T^{-1}(A)=A$. Then, $T^{-k}(A)=A$, and,
since $T^{k}$ is assumed to be ergodic, either $\mu(A)=0$ or
$\mu(X\smallsetminus A)=0$. Thus, $T$ is ergodic.

To see that the converse is not true in general, consider the space
$\mathbb{Z}$ endowed with the counting measure $\mu$, i.e. given
$X\subset\mathbb{Z}$, $\mu(X)$ is the number of elements of $X$. Then, the
translation $T\coloneqq x\mapsto x+1$ is ergodic, since there are no proper
$T$-invariant subsets of $X$. However, $T^{2}=x\mapsto x+2$ is not ergodic,
since $2\mathbb{Z}$ is invariant, and $\mu(2\mathbb{Z})>0$ and
$\mu(\mathbb{Z}\smallsetminus 2\mathbb{Z})>0$.

$\square$

Mostly, we will use the measure space
$(\partial\mathbb{D},\mathcal{B}(\partial\mathbb{D}),\lambda)$, where
$\mathcal{B}(\partial\mathbb{D})$ denotes the Borel $\sigma$-algebra of
$\partial\mathbb{D}$, and $\lambda$, its normalized Lebesgue measure. We need
the concept of Lebesgue density.

######  Definition 2.7.

(Lebesgue density) Given a Borel set $A\in\mathcal{B}(\partial\mathbb{D})$,
the Lebesgue density of $A$ at $\xi\in\partial\mathbb{D}$ is defined as

$d_{\xi}(A)\coloneqq\lim\limits_{\rho\to 0}\dfrac{\lambda(A\cap
D(\xi,\rho))}{\lambda(D(\xi,\rho))}.$

A point $\xi\in\partial\mathbb{D}$ is called a Lebesgue density point for $A$
if $d_{\xi}(A)=1$.

Given any Borel set $A\in\mathcal{B}(\partial\mathbb{D})$, with
$\lambda(A)>0$, then $\lambda$-almost every point in $A$ is a Lebesgue density
point for $A$ [Rud87, p. 138].

###  2.3 Generalized radial arcs and Stolz angles

Throughout the article, the following concepts will be needed, which describe
ways one may approach a boundary point $\xi\in\partial\mathbb{D}$.

In the sequel, we denote the (Euclidean) disk of radius $\rho>0$ centered at
$\xi\in\partial\mathbb{D}$ by $D(\xi,\rho)$. We also consider the radial
segment at $\xi$ of length $\rho>0$,

$R_{\rho}(\xi)\coloneqq\left\\{r\xi\colon r\in(1-\rho,1)\right\\}.$

######  Definition 2.8.

(Landing set) Given a curve
$\gamma\colon\left[0,1\right)\to\widehat{\mathbb{C}}$, we consider its landing
set

$L(\gamma)\coloneqq\left\\{v\in\widehat{\mathbb{C}}\colon\textrm{ there exists
}\left\\{t_{n}\right\\}_{n}\subset\left[0,1\right),t_{n}\to 1\textrm{ such
that }\gamma(t_{n})\to v\right\\}.$

By definition, $L(\gamma)$ is a connected, compact subset of
$\widehat{\mathbb{C}}$. We say that $\gamma$ lands at
$v\in\widehat{\mathbb{C}}$ if $L(\gamma)=\left\\{v\right\\}$, or,
equivalently, if

$\lim\limits_{t\to 1^{-}}\gamma(t)=v.$

Given a domain $U\subset\widehat{\mathbb{C}}$, a point
$p\in\widehat{\partial}U$ is accessible from $U$ if there exists a curve
$\gamma\subset U$ landing at $p$.

######  Definition 2.9.

(Crosscut neighbourhoods and Stolz angles) Let $\xi\in\partial\mathbb{D}$.

  * •

A crosscut $C$ is an open Jordan arc $C\subset\mathbb{D}$ such that
$\overline{C}=C\cup\left\\{a,b\right\\}$, with $a,b\in\partial\mathbb{D}$. If
$a=b$, we say that $C$ is degenerate; otherwise it is non-degenerate.

  * •

A crosscut neighbourhood of $\xi\in\partial\mathbb{D}$ is an open set
$N\subset\mathbb{D}$ such that $\xi\in\partial N$, and $C\coloneqq\partial
N\cap\mathbb{D}$ is a non-degenerate crosscut. We usually write $N_{\xi}$ or
$N_{C}$, to stress the dependence on the point $\xi$ or on the crosscut $C$.
Note that for a crosscut neighbourhood $N$,
$\partial\mathbb{D}\cap\overline{N}$ is a non-trivial arc.

  * •

Given $\xi\in\partial\mathbb{D}$, a Stolz angle111Note that the usual
defintion of Stolz angle is
$\Delta=\left\\{z\in\mathbb{D}\colon\left|\textrm{Arg }\xi-\textrm{Arg
}(\xi-z)\right|<\alpha,\left|\xi-z\right|<\rho\right\\}.$ However, both
definitions are equivalent for our purposes, and the stated one is slightly
more convenient in our setting. at $\xi$ is a set of the form

$\Delta_{\alpha,\rho}=\left\\{z\in\mathbb{D}\colon\left|\textrm{Arg
}\xi-\textrm{Arg }(\xi-z)\right|<\alpha,\left|z\right|>1-\rho\right\\}.$

  * •

We say that $\gamma$ lands non-tangentially at $\xi\in\partial\mathbb{D}$ if
$\gamma$ lands at $\xi$, and there exists a Stolz angle $\Delta_{\alpha,\rho}$
at $\xi$ with $\gamma\subset\Delta_{\alpha,\rho}$.

$\xi$

(a) Radial segment

$\xi$

(b) (Euclidean) disk

$\xi$

(c) Degenerate crosscut

$\xi$

(d) (Non-degenerate) crosscut

$\xi$

(e) Crosscut neighbourhood

$\xi$

(f) Stolz angle
text

Figure 2.1: Different sets related to $\xi\in\partial\mathbb{D}$.

Some times it is more convenient to work in the upper half-plane $\mathbb{H}$
rather than in the unit disk $\mathbb{D}$. The previous concepts can be
defined analogously for points in $\partial\mathbb{H}$. In particular, the
specific formulas for both the radial segment and Stolz angles at a point
$x\in\mathbb{R}$ are

$R_{\rho}^{\mathbb{H}}(x)\coloneqq\left\\{z\in\mathbb{H}\colon{\textrm{Im
}}w<\rho,{\textrm{Re }}w=x\right\\};$
$\Delta^{\mathbb{H}}_{\alpha,\rho}(x)\coloneqq\left\\{z\in\mathbb{H}\colon{\textrm{Im
}}w<\rho,\dfrac{\left|\textrm{Re }w-x\right|}{{\textrm{Im
}}w}<\tan\alpha\right\\}.$

A more flexible notion of radial segment and Stolz angle will be needed for
our purposes.

######  Definition 2.10.

(Generalized radial arc and Stolz angle)  Let $p\in\overline{\mathbb{D}}$ and
let $\xi\in\partial\mathbb{D}$, $\xi\neq p$. Let $\rho>0$ and
$0<\alpha<\pi/2$.

  * •

If $p\in\mathbb{D}$, consider the Möbius transformation
$M\colon\mathbb{D}\to\mathbb{D}$, $M(z)=\dfrac{p-z}{1-\overline{p}z}$. Then,
the (generalized) radial segment $R_{\rho}(\xi,p)$ of length $\rho$ at $\xi$
is defined as the preimage under $M$ of the radial segment $R_{\rho}(M(\xi))$.
Analogously, the (generalized) Stolz angle $\Delta_{\alpha,\rho}(\xi,p)$ of
angle $\alpha$ and length $\rho$ is the preimage under $M$ of the Stolz angle
$\Delta_{\alpha,\rho}(M(\xi))$. That is,

$R_{\rho}(\xi,p)\coloneqq M^{-1}(R_{\rho}(M(\xi))),$
$\Delta_{\alpha,\rho}(\xi,p)\coloneqq M^{-1}(\Delta_{\alpha,\rho}(M(\xi))).$

  * •

If $p\in\partial\mathbb{D}$, consider the Möbius transformation
$M\colon\mathbb{D}\to\mathbb{H}$, $M(z)=i\dfrac{p+z}{p-z}$. Then, the
(generalized) radial segment and Stolz angle at $\xi$ are defined as the
preimages of the corresponding radial segment and Stolz angle at
$M(\xi)\in\mathbb{R}$. That is,

$R_{\rho}(\xi,p)\coloneqq M^{-1}(R_{\rho}^{\mathbb{H}}(M(\xi)))$
$\Delta_{\alpha,\rho}(\xi,p)\coloneqq
M^{-1}(\Delta_{\alpha,\rho}^{\mathbb{H}}(M(\xi))).$

See Figures 2.2 and 2.3.

Observe that $R_{\rho}(\xi)=R_{\rho}(\xi,0)$, and
$\Delta_{\alpha,\rho}(\xi)=\Delta_{\alpha,\rho}(\xi,0)$. Note also that
$R_{\rho}(\xi,p)$ is a curve landing non-tangentially at
$\xi\in\partial\mathbb{D}$, while $\Delta_{\alpha,\rho}(\xi,p)$ is an angular
neighbourhood of $\xi$, since Möbius transformations are conformal, and hence
angle-preserving.

$M$$p$$0$$\xi_{1}$$\xi_{2}$$M(\xi_{1})$$M(\xi_{2})$$\mathbb{D}$$\mathbb{D}$$\Delta_{\alpha,\rho}(\xi_{1},p)$$R_{\rho}(\xi_{2},p)$$\Delta_{\alpha,\rho}(M(\xi_{1}))$$R_{\rho}(M(\xi_{2}))$$\rho$$\alpha$

Figure 2.2: Radial arc and angular neighbourhood with respect to
$p\in\mathbb{D}$.

$M$$p$$\xi_{1}$$\Delta_{\alpha,\rho}(\xi_{1},p)$$R_{\rho}(\xi_{2},p)$$\Delta^{\mathbb{H}}_{\alpha,\rho}(M(\xi_{1}))$$R^{\mathbb{H}}_{\rho}(M(\xi_{2}))$$\xi_{2}$$M(\xi_{1})$$M(\xi_{2})$$\mathbb{D}$$\mathbb{H}$$\rho
i$$\alpha$

Figure 2.3: Radial arc and angular neighbourhood with respect to
$p\in\partial\mathbb{D}$.

###  2.4 Boundary behaviour of meromorphic maps in $\mathbb{D}$

In this section, we are interested in the boundary behaviour of meromorphic
maps $h\colon\mathbb{D}\to\widehat{\mathbb{C}}$. Since $h$ may not extend
continuously to $\partial\mathbb{D}$, the concepts of radial and angular limit
are a keystone on studying the boundary behavior of $h$.

######  Definition 2.11.

(Radial and angular limit) Let $h\colon\mathbb{D}\to\widehat{\mathbb{C}}$ be a
meromorphic map, and let $\xi\in\partial\mathbb{D}$. We say that $h$ has
radial limit at $\xi$ if the limit

$h^{*}(\xi)\coloneqq\lim\limits_{t\to 1^{-}}h(t\xi)$

exists. We say that $h$ has angular limit at $\xi$ if, for any Stolz angle
$\Delta$ at $\xi$, the limit

$\lim\limits_{z\to\xi,z\in\Delta}h(z)$

exists.

Note that, whenever we write $h^{*}(\xi)=v$ we are assuming implicitly that
the radial limit exists, and equals $v$. The map

$h^{*}\colon\partial\mathbb{D}\to\widehat{\mathbb{C}}$

is called the radial extension of $h$ (defined wherever the radial limit
exists).

For maps $h\colon\mathbb{D}\to\widehat{\mathbb{C}}$ omitting three values in
$\widehat{\mathbb{C}}$, the following theorem relates radial and angular
limits.

######  Theorem 2.12.

(Lehto-Virtanen, [Pom92, Sect. 4.1]) Let
$h\colon\mathbb{D}\to\widehat{\mathbb{C}}$ be a meromorphic map omitting three
values in $\widehat{\mathbb{C}}$. Let $\gamma$ be a curve in $\mathbb{D}$
landing at $\xi\in\partial\mathbb{D}$. If $h(\gamma)$ lands at a point
$v\in\mathbb{C}$, then $h$ has angular limit at $\xi$ equal to $v$. In
particular, radial and angular limits are the same.

######  Remark 2.13.

(Limit on generalized radial arcs and Stolz angles) Note that, in particular,
the Lehto-Virtanen Theorem justifies that, for meromorphic maps omitting three
values, it is equivalent to take the limit along the radial segment, than
along any generalized radial arc. Likewise, the angular limit can be computed
along generalized Stolz angles.

The following theorems describe more precisely the boundary behaviour of
meromorphic maps $h\colon\mathbb{D}\to\widehat{\mathbb{C}}$ in terms of
measure.

######  Theorem 2.14.

(Radial extensions are measurable, [Pom92, Prop. 6.5]) Let
$h\colon\mathbb{D}\to\widehat{\mathbb{C}}$ be continuous. Then, the points
$\xi\in\partial\mathbb{D}$ where the radial limit $h^{*}$ exists form a Borel
set, and if $A\subset\widehat{\mathbb{C}}$ is a Borel set, then

$(h^{*})^{-1}(A)\coloneqq\left\\{\xi\in\partial\mathbb{D}\colon h^{*}(\xi)\in
A\right\\}\subset\partial\mathbb{D}$

is also a Borel set.

In the particular case where $h=\varphi\colon\mathbb{D}\to U$ is a Riemann
map, the following theorem, due to Fatou, Riesz and Riesz, ensures the
existence of radial limits almost everywhere.

######  Theorem 2.15.

(Existence of radial limits)  Let $\varphi\colon\mathbb{D}\to U$ be a Riemann
map. Then, for $\lambda$-almost every point $\xi\in\partial\mathbb{D}$, the
radial limit $\varphi^{*}(\xi)$ exists. Moreover, if we fix
$\xi\in\partial\mathbb{D}$ for which $\varphi^{*}(\xi)$ exists, then
$\varphi^{*}(\xi)\neq\varphi^{*}(\zeta)$, for $\lambda$-almost every point
$\zeta\in\partial\mathbb{D}$.

###  2.5 Harmonic measure

Let $U$ be a simply connected domain. Then, the Riemann map
$\varphi\colon\mathbb{D}\to U$ induces a measure in $\widehat{\partial}U$, the
harmonic measure, which is the appropriate one when dealing with the
boundaries of Fatou components. We define harmonic measure in
$\widehat{\partial}U$ in terms of the push-forward under a Riemann map of the
normalized measure on the unit circle $\partial\mathbb{D}$.

######  Definition 2.16.

(Harmonic measure) Let $U\subsetneq{\mathbb{C}}$ be a simply connected domain,
$z\in U$, and let $\varphi\colon\mathbb{D}\to U$ be a Riemann map, such that
$\varphi(0)=z\in U$. Let $(\partial\mathbb{D},\mathcal{B},\lambda)$ be the
measure space on $\partial\mathbb{D}$ defined by $\mathcal{B}$, the Borel
$\sigma$-algebra of $\partial\mathbb{D}$, and $\lambda$, its normalized
Lebesgue measure. Consider the measurable space
$(\widehat{\mathbb{C}},\mathcal{B}(\widehat{\mathbb{C}}))$, where
$\mathcal{B}(\widehat{\mathbb{C}})$ is the Borel $\sigma$-algebra of
$\widehat{\mathbb{C}}$. Then, given $A\in\mathcal{B}(\widehat{\mathbb{C}})$,
the harmonic measure at $z$ relative to $U$ of the set $A$ is defined as:

$\omega_{U}(z,A)\coloneqq\lambda((\varphi^{*})^{-1}(A)).$

Note that the harmonic measure $\omega_{U}(z,\cdot)$ is well-defined. Indeed,
by Theorem 2.14, the set

$(\varphi^{*})^{-1}(A)=\left\\{\xi\in\partial\mathbb{D}\colon\varphi^{*}(\xi)\in
A\right\\}$

is a Borel set of $\partial\mathbb{D}$, and hence measurable. We also note
that the definition of $\omega_{U}(z,\cdot)$ is independent of the choice of
$\varphi$, provided it satisfies $\varphi(0)=z$, since $\lambda$ is invariant
under rotation.

We refer to [GM05, Pom92] for equivalent definitions and further properties of
the harmonic measure. We only need the following simple facts.

######  Lemma 2.17.

(Sets of zero and full harmonic measure) Let $U\subsetneq{\mathbb{C}}$ be a
simply connected domain, and $B\in\mathcal{B}(\widehat{\mathbb{C}})$. If there
exists $z_{0}\in U$ such that $\omega_{U}(z_{0},B)=0$ (resp.
$\omega_{U}(z_{0},B)=1$), then $\omega_{U}(z,B)=0$ (resp. $\omega_{U}(z,B)=1$)
for all $z\in U$. In this case, we say that the set $B$ has zero (resp. full)
harmonic measure relative to $U$, and we write $\omega_{U}(B)=0$ (resp.
$\omega_{U}(B)=1$).

Finally, we are interested in the support of $\omega_{U}$. Recall that

$\textrm{supp
}\omega_{U}\coloneqq\left\\{x\in\widehat{\mathbb{C}}\colon\textrm{ for all
}r>0,\textrm{ }\omega_{U}(D(x,r))>0\right\\}.$

Note that it only depends on the sets of zero measure, hence it is well-
defined without specifying the base-point of the harmonic measure.

######  Lemma 2.18.

(Support of harmonic measure) Let $U\subsetneq{\mathbb{C}}$ be a simply
connected domain. Then,

$\textrm{\em supp }\omega_{U}=\widehat{\partial}U.$

This follows easily from considering an equivalent definition of harmonic
measure in terms of solutions to the Dirichlet problem, see e.g. [Con95, Chap.
21].

###  2.6 Regular and singular values for holomorphic maps

Throughout the paper, we will make an extensive use of the concepts of regular
and singular values. Although these definitions are quite standard in the
context of entire or meromorphic maps (i.e. with one single essential
singularity), we believe it is useful to give definitions in the rather
general context of functions of class $\mathbb{K}$ or inner functions.

We consider the following class of meromorphic functions, denoted by
$\mathbb{M}$, consisting of functions

$f\colon\widehat{\mathbb{C}}\smallsetminus
E(f)\longrightarrow\widehat{\mathbb{C}},$

where $\Omega(f)\coloneqq\widehat{\mathbb{C}}\smallsetminus E(f)$ is the
largest set where $f$ is meromorphic, and, for all $z\in E(f)$, the cluster
set $Cl(f,z)$ of $f$ at $z$ is $\widehat{\mathbb{C}}$, that is

$\displaystyle Cl(f,z)=\left\\{w\in\widehat{\mathbb{C}}\colon\textrm{ there
exists }\left\\{z_{n}\right\\}_{n}\subset\Omega(f),\,z_{n}\to z,\,f(z_{n})\to
w\right\\}=\widehat{\mathbb{C}}.$ (1)

If $E(f)=\emptyset$, then $f$ is rational and we make the further assumption
that $f$ is non-constant. Note that $\Omega(f)$ is open, and $E(f)$ has empty
interior. Indeed, if $z$ is an interior point for $E$, there does not exist
any sequence in $\Omega(f)$ converging to $z$, and hence $Cl(f,z)$ is empty, a
contradiction.

In the case that $f$ is an inner function (Sect. 3), $E(f)$ will be a closed
subset of $\partial\mathbb{D}$, while if $f\in\mathbb{K}$, $E(f)$ is a
countable subset of the Riemann sphere. In both cases, the assumption on the
cluster set is satisfied (see e.g. [BD99, BDH01]).

In this general setting, regular and singular values, and critical and
asymptotic values, are defined as follows. Note that appropriate charts have
to be used when dealing with $\infty$.

######  Definition 2.19.

(Regular and singular values) Given a value $v\in\widehat{\mathbb{C}}$, we say
that $v$ is a regular value for $f$ if there exists $r\coloneqq r(v)>0$ such
that all branches $F_{1}$ of $f^{-1}$ are well-defined (and, hence, conformal)
in $D(v,r)$. Otherwise we say that $v$ is a singular value for $f$.

The set of singular values of $f$ is denoted by $SV(f)$. Note that $SV(f)$ is
closed by definition, and it is the smallest set for which

$f\colon\widehat{\mathbb{C}}\smallsetminus(E(f)\cup
f^{-1}(SV(f)))\longrightarrow\widehat{\mathbb{C}}\smallsetminus SV(f)$

is a covering map.

######  Definition 2.20.

(Critical and asymptotic values) Given a value $v\in\widehat{\mathbb{C}}$, we
say that $v$ is a critical value if there exists $z\in\Omega$ such that
$f^{\prime}(z)=0$ and $f(z)=v$. We say that $z$ is a critical point.

We say that $v$ is an asymptotic value if there exists a curve
$\gamma\colon\left[0,1\right)\to\Omega$ such that $\gamma(t)\to\partial\Omega$
and $f(\gamma(t))\to v$, as $t\to 1$. We say that the curve $\gamma$ is an
asymptotic path.

The set of critical values of $f$ is denoted by $CV(f)$, while $AV(f)$ stands
for the set of asymptotic values.

Note that we do not assume, in general, that $\gamma(t)$ lands at a definite
point $\partial\Omega$ as $t\to 1$. However, this will be the case for both
inner functions (Lemma 4.1) and functions in class $\mathbb{K}$ (Sect. 5.1).
We say that $v$ is an asymptotic value corresponding to $x\in\partial\Omega$
if $v=\lim\limits_{t\to 1}f(\gamma(t))$, where $\gamma$ is a curve such that
$\gamma(t)\to x$ as $t\to 1$. Note that an asymptotic value may correspond to
more than one point in $\partial\Omega$.

The following lemma makes explicit the relation between regular and singular
values, and critical and asymptotic values, in the sense of Iversen [Ive14,
BE95]. Although its content is well-known for meromorphic functions and for
functions of finite type, we were unable to find a proof in the literature
which fits into the general setting we are considering. We follow the ideas of
[BE95].

######  Lemma 2.21.

(Characterization of singular values) Let $f\in\mathbb{M}$. Then,

$SV(f)=\overline{CV(f)\cup AV(f)}.$

###### Proof.

Let $v\in\widehat{\mathbb{C}}$. For every $r>0$, choose a component $U(r)$ of
$f^{-1}(D(v,r))$ in such a way that $r_{1}<r_{2}$ implies $U(r_{1})\subset
U(r_{2})$. Note that $f^{-1}(D(v,r))$ is non-empty for all
$v\in\widehat{\mathbb{C}}$ and $r>0$, because of (1) if $E(f)\neq\emptyset$
and trivially if $E(f)=\emptyset$.

Two possibilities can occur.

  * •

$\bigcap_{r>0}U(r)\neq\emptyset$. In such a case, there exists $z\in\Omega(f)$
such that $\bigcap_{r>0}U(r)=\left\\{z\right\\}$ and, hence, $f(z)=v$ (indeed,
note that if the previous intersection was larger than a point, this would
contradict the Open Mapping Principle). If $f^{\prime}(z)\neq 0$, then the
inverse branch $F_{1}$ of $f^{-1}$ sending $v$ to $z$ is well-defined and
conformal in $D(v,r_{0})$, for some $r_{0}>0$. If $f^{\prime}(z)=0$, $f$ acts
as a branched covering around $z$, and the corresponding inverse branch of
$f^{-1}$ is not well-defined around $v$. Note that, this latter case occurs if
and only if $z$ is a critical point and $v$ is a critical value.

  * •

$\bigcap_{r>0}U(r)=\emptyset$. We show that this case correponds with the case
of $v$ being an asymptotic value. First, assume $\bigcap_{r>0}U(r)=\emptyset$,
and we shall construct an asymptotic path $\gamma$ for $v$. Let $r_{k}$ be a
sequence of positive real numbers tending to 0, and let $z_{k}\in U(r_{k})$.
Let $\gamma_{k}\subset U(r_{k})$ be a curve connecting $z_{k}$ to $z_{k+1}$.
Then, $\gamma=\cup_{k}\gamma_{k}$ satisfies $\gamma(t)\to\partial\Omega$, and
hence is an asymptotic path for $v$.

On the other hand, if $v$ is an asymptotic value, let $\gamma$ be an
asymptotic path. Then, define $U(r)$ to be the connected component of
$f^{-1}(D(v,r))$ containing the tail of $\gamma$. Then,
$\bigcap_{r>0}U(r)=\emptyset$, as desired.

Then, it is clear that $v$ is a regular value (as in Def. 2.19) if and only if
$v$ is not a critical, nor an asymptotic value, nor an accumulation thereof.
$\square$

##  3 Iteration of inner functions

Consider a holomorphic map $g\colon\mathbb{D}\to\mathbb{D}$. Since $g$ is
bounded, the radial extension $g^{*}$ exists $\lambda$-almost everywhere (Thm.
2.15). We are interested in the case where $g^{*}$ preserves
$\partial\mathbb{D}$ $\lambda$-almost everywhere.

######  Definition 3.1.

(Inner function) A holomorphic self-map of the unit disk
$g\colon\mathbb{D}\to\mathbb{D}$ is an inner function if its radial extension
$g^{*}$ satisfies $g^{*}(\xi)\in\partial\mathbb{D}$, for $\lambda$-almost
every point $\xi\in\partial\mathbb{D}$.

In general, inner functions present a highly discontinuous behaviour in
$\partial\mathbb{D}$.

######  Definition 3.2.

(Singularity) Let $g$ be an inner function. A point $\xi\in\partial\mathbb{D}$
is called a singularity of $g$ if $g$ cannot be continued analytically to any
neighbourhood of $\xi$. Denote the set of singularities of $g$ by $E(g)$.

Throughout the paper, given any inner function $g$, we consider it continued
to $\widehat{\mathbb{C}}\smallsetminus\overline{\mathbb{D}}$ by the reflection
principle, and to $\partial\mathbb{D}\smallsetminus E(g)$ by analytic
continuation. In other words, $g$ is considered as its maximal meromorphic
extension

$g\colon\widehat{\mathbb{C}}\smallsetminus E(g)\to\widehat{\mathbb{C}}.$

If an inner function has finite degree, then it is a finite Blaschke product.
In this case, $g$ has no singularities, and it extends to the Riemann sphere
as a rational map. On the other hand, infinite degree inner functions must
have at least one singularity. The following lemma characterizes the
singularities of an inner function.

######  Lemma 3.3.

(Characterization of singularities, [Gar07, Thm. II.6.6]) Let
$g\colon\mathbb{D}\to\mathbb{D}$ be an inner function. Then, $\xi\in E(g)$ if
and only if, for any crosscut neighbourhood $N_{\xi}$ of $\xi$,

$\overline{g(N_{\xi})}=\overline{\mathbb{D}}.$

###  3.1 Iteration of holomorphic self-maps of the unit disk

The asymptotic behaviour of the iterates of a holomorphic self-map of the unit
disk is essentially described by the Denjoy-Wolff theorem. Note that the
results in this section are valid for any holomorphic self-map of
$\mathbb{D}$, not necessarily an inner function.

######  Theorem 3.4.

(Denjoy-Wolff, [Mil06, Thm. 5.2]) Let $g\colon\mathbb{D}\to\mathbb{D}$ be
holomorphic, which is not the identity nor an elliptic Möbius transformation.
Then, there exists a point $p\in\overline{\mathbb{D}}$, the Denjoy-Wolff point
of $g$, such that for all $z\in\mathbb{D}$, $g^{n}(z)\to p$.

Hence, holomorphic self-maps of $\mathbb{D}$ are classified into two types:
the elliptic ones, for which $p\in\mathbb{D}$, and the non-elliptic ones, with
$p\in\partial\mathbb{D}$. In the first case, the Schwarz lemma describes the
dynamics precisely.

######  Theorem 3.5.

(Schwarz lemma, [Mil06, Lemma 1.2]) Let $g\colon\mathbb{D}\to\mathbb{D}$ be
holomorphic, with $g(0)=0$. Then, for all $z\in\mathbb{D}$,
$\left|g(z)\right|\leq\left|z\right|$, and $\left|g^{\prime}(0)\right|\leq 1$.

An analogous result was obtained by Wolff for non-elliptic self-maps of
$\mathbb{D}$.

######  Theorem 3.6.

(Wolff lemma, [Wol26]) Let $g\colon\mathbb{D}\to\mathbb{D}$ be holomorphic,
with Denjoy-Wolff point $p\in\partial\mathbb{D}$. Let $D\subset\mathbb{D}$ be
an open disk tangent to $\partial\mathbb{D}$ at $p$. Then, $g(D)\subset D$. In
particular, $g^{*}(p)=p$.

Another equivalent way of stating Wolff lemma is that, for any holomorphic
function $h\colon\mathbb{H}\to\mathbb{H}$ with Denjoy-Wolff point $\infty$ and
any upper half-plane $H$, $h(H)\subset H$ (see also [Bar08, Lemma 2.33]).

Note that, in the elliptic case, $g$ is holomorphic in a neighbourhood of the
Denjoy-Wolff point $p\in\mathbb{D}$, which is fixed and it is either
attracting (if $\left|g^{\prime}(p)\right|\in(0,1)$) or superattracting (if
$g^{\prime}(p)=0$). In the former case, $g$ is conjugate to
$z\mapsto\left|g^{\prime}(p)\right|z$ in a neighourhood of $p$ (by Koenigs
Theorem, see e.g. [Mil06, Chap. 8]). In the latter case, the dynamics are
conjugate to those of $z\mapsto z^{d}$, where $d$ stands for the local degree
of $g$ at $p$ (by Böttcher Theorem, see e.g. [Mil06, Chap. 9]).

An analogous result for the non-elliptic case is given by the following result
of Cowen, which leads to a classification of non-elliptic self-maps of
$\mathbb{D}$ in terms of the dynamics near the Denjoy-Wolff point.

######  Definition 3.7.

(Absorbing domains and fundamental sets) Let $U$ be a domain in $\mathbb{C}$
and let $f\colon U\to U$ be a holomorphic map. A domain $V\subset U$ is said
to be an absorbing domain for $f$ in $U$ if $f(V)\subset V$ and for every
compact set $K\subset U$ there exists $n\geq 0$ such that $f^{n}(K)\subset V$.
If, additionally, $V$ is simply connected and $f|_{V}$ is univalent, $V$ is
said to be a fundamental set for $f$ in $U$.

######  Theorem 3.8.

(Cowen’s classification of self-maps of $\mathbb{D}$, [Cow81]) Let $g$ be a
holomorphic self-map of $\mathbb{D}$ with Denjoy-Wolff point
$p\in\partial\mathbb{D}$. Then, there exists a fundamental set $V$ for $g$ in
$\mathbb{D}$.

Moreover, given a fundamental set $V$, there exists a domain $\Omega$ equal to
$\mathbb{C}$ or $\mathbb{H}=\left\\{\textrm{\em Im }z>0\right\\}$, a
holomorphic map $\psi\colon\mathbb{D}\to\Omega$, and a Möbius transformation
$T\colon\Omega\to\Omega$, such that:

  1. (a)

$\psi(V)$ is a fundamental set for $T$ in $\Omega$,

  2. (b)

$\psi\circ g=T\circ\psi$ in $\mathbb{D}$,

  3. (c)

$\psi$ is univalent in $V$.

Moreover, $T$ and $\Omega$ depend only on the map $g$, not on the fundamental
set $V$. In fact (up to a conjugacy of $T$ by a Möbius transformation
preserving $\Omega$), one of the following cases holds:

  * •

$\Omega=\mathbb{C}$, $T=\textrm{\em id}_{\mathbb{C}}+1$ (doubly parabolic
type),

  * •

$\Omega=\mathbb{H}$, $T=\lambda\textrm{\em id}_{\mathbb{H}}$, for some
$\lambda>1$ (hyperbolic type),

  * •

$\Omega=\mathbb{H}$, $T=\textrm{\em id}_{\mathbb{H}}\pm 1$ (simply parabolic
type).

Finally, note that if $g$ is a self-map of $\mathbb{D}$, so is $g^{k}$, for
all $k\geq 1$. The type in Cowen’s classification is preserved by taking
iterates.

######  Lemma 3.9.

(Cowen’s classification for $g^{k}$) Let $g\colon\mathbb{D}\to\mathbb{D}$
holomorphic, and let $k$ be a positive integer. Then, $g$ is elliptic (resp.
doubly parabolic, hyperbolic, simply parabolic) if and only if so is $g^{k}$.

###### Proof.

It is clear that $g$ is of elliptic type if and only if so is $g^{k}$. Now,
assume that $p\in\partial\mathbb{D}$ is the Denjoy-Wolff point of $g$, and
choose a fundamental set $V$ for $g$ in $\mathbb{D}$. Then, $V$ is a
fundamental set for $g^{k}$ in $\mathbb{D}$. It follows that $g|_{V}$ is
conformally conjugate to $T_{1}\colon\Omega_{1}\to\Omega_{1}$, and
$g^{k}|_{V}$ is conformally conjugate to $T_{2}\colon\Omega_{2}\to\Omega_{2}$.
Hence, $T_{1}\colon\Omega_{1}\to\Omega_{1}$ and
$T_{2}\colon\Omega_{2}\to\Omega_{2}$ are conformally conjugate. Since $T$ and
$\Omega$ are unique up to conformal conjugacy, and do not depend on the choice
of the fundamental set, it follows that $g$ and $g^{k}$ are of the same type
in Cowen’s classification. $\square$

###  3.2 Ergodic properties of the radial extension
$g^{*}\colon\partial\mathbb{D}\to\partial\mathbb{D}$

Let $g\colon\mathbb{D}\to\mathbb{D}$ be an inner function. We consider the
dynamical system induced by its the radial extension

$g^{*}\colon\partial\mathbb{D}\to\partial\mathbb{D}.$

Recall that if $g$ is an inner function, so is $g^{k}$ [BD99, Lemma 4], so the
equality

$(g^{n})^{*}(\xi)=(g^{*})^{n}(\xi)$

holds $\lambda$-almost everywhere. Moreover, the radial extension $g^{*}$ is
measurable (Thm. 2.14), and hence analyzable from the point of view of ergodic
theory. The following is a recollection of ergodic properties of $g^{*}$, with
precise references.

######  Theorem 3.10.

(Ergodic properties of $g^{*}$) Let $g\colon\mathbb{D}\to\mathbb{D}$ be an
inner function with Denjoy-Wolff point $p\in\overline{\mathbb{D}}$. The
following holds.

  1. (a)

$g^{*}$ is non-singular. In particular, for $\lambda$-almost every
$\xi\in\partial\mathbb{D}$, its infinite orbit under $g^{*}$,
$\left\\{(g^{n})^{*}(\xi)\right\\}_{n}$, is well-defined.

  2. (b)

$g^{*}$ is ergodic if and only if $g$ is elliptic or doubly parabolic.

  3. (c)

If $g^{*}$ is recurrent, then it is ergodic. In this case, for every
$A\in\mathcal{B}(\mathbb{D})$ with $\lambda(A)>0$, we have that for
$\lambda$-almost every $\xi\in\partial\mathbb{D}$, there exists a sequence
$n_{k}\to\infty$ such that $(g^{n_{k}})^{*}(\xi)\in A$. In particular,
$\lambda$-almost every $\xi\in\partial\mathbb{D}$,
$\left\\{(g^{n})^{*}(\xi)\right\\}_{n}$ is dense in $\partial\mathbb{D}$.

  4. (d)

If $g$ is an elliptic inner function, then $g^{*}$ is recurrent.

  5. (e)

The radial extension of a doubly parabolic inner function is not recurrent in
general. However, if $g$ is doubly parabolic and the Denjoy-Wolff point $p$ is
not a singularity for $g$, then $g^{*}$ is recurrent. Moreover, if $g$ is
doubly parabolic and there exists $z\in\mathbb{D}$ and $r>1$ such that

$\textrm{\em
dist}_{\mathbb{D}}(g^{n+1}(z),g^{n}(z))\leq\frac{1}{n}+O\left(\frac{1}{n^{r}}\right),$

as $n\to\infty$, then $g^{*}$ is recurrent.

  6. (f)

Let $k$ be a positive integer. Then, $g^{k}$ is an inner function. Moreover,
$g^{*}$ is ergodic (resp. recurrent) if and only if $(g^{k})^{*}$ is ergodic
(resp. recurrent).

###### Proof.

  1. (a)

The proof that $g^{*}$ is non-singular can be found in [Aar97, Prop. 6.1.1].
We claim that this already implies that, for $\lambda$-almost every
$\xi\in\partial\mathbb{D}$, its infinite orbit
$\left\\{(g^{n})^{*}(\xi)\right\\}_{n}$ is well-defined. We shall prove it by
induction. First, it is clear that the set

$\left\\{\xi\in\partial\mathbb{D}\colon g^{*}(\xi)\textrm{ is well-
defined}\right\\}$

has full measure, and, since $g^{*}$ is non-singular,
$g^{*}(\partial\mathbb{D})=1$. Now, assume that the set

$\left\\{\xi\in\partial\mathbb{D}\colon\left\\{(g^{n})^{*}(\xi)\right\\}_{n=0}^{k-1}\textrm{
is well-defined}\right\\}$

has full measure, and $\lambda((g^{k-1})^{*}(\partial\mathbb{D}))=1$. Then,
the set

$(g^{k-1})^{*}(\partial\mathbb{D})\cap\left\\{\xi\in\partial\mathbb{D}\colon(g^{n})^{*}(\xi)\textrm{
is well-defined}\right\\}$

has also full measure, proving that the orbit
$\left\\{g^{*}(\xi)\right\\}_{n=0}^{k}$ is well-defined for $\lambda$-almost
every $\xi\in\partial\mathbb{D}$, as desired.

  2. (b)

It follows from combining [DM91, Thm. G] with [Bon97, Thm. 1.4].

  3. (c)

See [DM91, Thm. E, F], as well as [Aar97, Thm. 6.1.7].

  4. (d)

The first statement is [DM91, Corol. 1] (see also [Aar97, Thm. 6.1.8]). The
second statement follows from applying Theorem 2.5, and applying that open
sets in $\partial\mathbb{D}$ have positive measure. See also [BEFRS22, Sect.
8.3].

  5. (e)

An example of a doubly parabolic inner function whose boundary map is not
recurrent is given in [BFJK19, Example 1.3]. Conditions which imply recurrence
are found in [BFJK19, Thm. B] and [BFJK19, Thm. E], respectively.

  6. (f)

Finally, the proof that $g^{k}$ is an inner function can be found in [BD99,
Lemma 4]. Since ergodicity depends only on the type in Cowen’s classification
3.8, which is invariant under talking iterates (Lemma 3.9), $g^{*}$ is ergodic
if and only if $(g^{k})^{*}$ is ergodic. Recurrence is always preserved under
taking iterates (see Lemma 2.6).

$\square$

##  4 Inverse branches for inner functions at boundary points. Proof of
Theorem B

Let $g\colon\mathbb{D}\to\mathbb{D}$ be an inner function, considered as its
maximal meromorphic extension

$g\colon\widehat{\mathbb{C}}\smallsetminus E(g)\to\widehat{\mathbb{C}},$

as before. We shall consider regular and singular values for $g$ as introduced
in Section 2.6. We observe that for any inner function $g$ and any $z\in
E(g)$, the cluster set (as defined in Section 2.6) is $\widehat{\mathbb{C}}$.
Indeed, if $z\in E(g)$, then Lemma 3.3 and Schwarz reflection imply that
$Cl(g,z)=\widehat{\mathbb{C}}$ (compare also with [BD99]). Thus,
$g\in\mathbb{M}$, and all the previous description of singular values applies.

Note that, since $\mathbb{D}$ and
$\widehat{\mathbb{C}}\smallsetminus\overline{\mathbb{D}}$ are totally
invariant, there cannot be critical points nor critical values in
$\partial\mathbb{D}$. Moreover, by symmetry, if $v\in\mathbb{D}$ is regular
(resp. singular), then so is $1/\overline{v}$. Hence, we consider

$SV(g,\overline{\mathbb{D}})=\left\\{v\in\overline{\mathbb{D}}\colon v\textrm{
is singular}\right\\}.$

We start by proving that asymptotic paths actually land at points in $E(g)$.

######  Lemma 4.1.

(Asymptotic paths land) Let $v\in\overline{\mathbb{D}}$ be an asymptotic value
for $g$, and let
$\gamma\colon\left[0,1\right)\to\widehat{\mathbb{C}}\smallsetminus E(g)$ be an
asymptotic path for $v$. Then, there exists a singularity $\xi\in
E(g)\subset\partial\mathbb{D}$ such that $\gamma(t)\to\xi$, as $t\to 1$.

###### Proof.

Assume, on the contrary, that the landing set $L(\gamma)$ of the asymptotic
path $\gamma$ is a continuum in $E(g)$. Then, $L(\gamma)$ is a closed non-
degenerate interval in the unit circle.

On the one hand, for $\lambda$-almost every point $\xi$ in $L(\gamma)$, the
radial limit $g^{*}(\xi)$ exists. Let us denote the radial segment at $\xi$ by
$R_{\xi}$.

On the other hand, without loss of generality, we can assume
$\gamma\colon\left[0,1\right)\to\overline{\mathbb{D}}\smallsetminus E(g)$.
Then, since $L(\gamma)\subset E(g)$ and
$\gamma\subset\overline{\mathbb{D}}\smallsetminus E(g)$, for every point
$\xi\in L(\gamma)$ (except at the endpoints), there exists a sequence
$\left\\{\xi_{n}\right\\}\subset\gamma\cap R_{\xi}$ with $\xi_{n}\to\xi$.
Then, $g(\xi_{n})\to v$, implying that the radial limit $g^{*}(\xi)$ equals
$v$. This contradicts the fact that radial limits are different almost
everywhere (Thm. 2.15). $\square$

Next we prove that singular values in $\partial\mathbb{D}$ correspond to
accumulation points of singular values in $\mathbb{D}$.

######  Proposition 4.2.

(Singular values in $\partial\mathbb{D}$) Let $g\colon\mathbb{D}\to\mathbb{D}$
be an inner function, and let $\xi\in\partial\mathbb{D}$. The following are
equivalent.

  1. (a)

There exists a crosscut $C$, with crosscut neighbourhood $N_{C}$ and
$\xi\in\partial N_{C}$ such that $SV(g)\cap N_{C}=\emptyset$.

  2. (b)

$v$ is regular, i.e. there exists $\rho\coloneqq\rho(\xi)>0$ such that all
inverse branches $G_{1}$ of $g$ are well-defined in $D(\xi,\rho)$.

###### Proof.

The implication (b)$\Rightarrow$(a) is trivial. Let us prove
(a)$\Rightarrow$(b). Without loss of generality, we can assume that there are
no singular values in $\partial\mathbb{D}\cap\overline{N_{C}}$. Moreover, note
that (a) implies that all inverse branches $G_{1}$ are well-defined (and
conformal) in $N_{C}$. The assumption that there are no singular values in
$\partial\mathbb{D}\cap\overline{N_{C}}$ implies that $G_{1}$ is holomorphic
in $\overline{N_{C}}\cap\mathbb{D}$. We shall show that $G_{1}$ can be
extended across $\partial\mathbb{D}$ by Schwarz reflection (see Fig. 4.1).

To this end, let $\varphi\colon\mathbb{D}\to N_{C}$ be a Riemann map. Note
that it extends homeomorphically to $\partial N_{C}$; we shall denote this
extension again by $\varphi$. Then,

$G_{1}\circ\varphi\colon\mathbb{D}\to G_{1}(N_{C})$

is a Riemann map for the simply connected domain $G_{1}(N_{C})$, where $G_{1}$
is any branch of $g^{-1}$.

Consider the radial extension

$G^{*}_{1}\colon\partial N_{C}\to\partial G_{1}(N_{C}),$

defined as

$\ G_{1}^{*}(x)=(G_{1}\circ\varphi)^{*}(\varphi^{-1}(x)),$

for $x\in\partial N_{C}$. Note that, since $G_{1}|_{N_{C}}$ is bounded, the
radial extension is well-defined almost everywhere in $\partial N_{C}$. Since
we assume that $G_{1}$ is holomorphic in $\overline{N_{C}}\cap\mathbb{D}$, it
follows that $G_{1}(\overline{N_{C}}\cap\mathbb{D})\subset\mathbb{D}$. Indeed,
assume $z\in\overline{N_{C}}\cap\mathbb{D}$ and
$G_{1}(z)\in\partial\mathbb{D}$. Since $G_{1}$ is conformal, it would map
points in $\mathbb{D}$ to points in
$\widehat{\mathbb{C}}\smallsetminus\overline{\mathbb{D}}$, a contradiction.

Moreover, modifying slightly the crosscut if needed, we can assume that, for
the two endpoints $\left\\{\xi_{1},\xi_{2}\right\\}=\partial
N_{C}\cap\partial\mathbb{D}$, the limit

$G_{1}^{*}(\xi_{i})=\lim\limits_{z\to\xi_{i},z\in
N_{C}\cap\mathbb{D}}G_{1}(z)$

exists, for $i=1,2$. This is possible since the radial extension is well-
defined almost everywhere. Note that, since $g$ is assumed to be an inner
function, $G_{1}^{*}(\xi_{i})\in\partial\mathbb{D}$ (otherwise, there would
exist a point in $\mathbb{D}$ mapped by $g$ to $\partial\mathbb{D}$, a
contradiction). Hence, $G_{1}(\partial N_{C}\cap\mathbb{D})$ is a crosscut in
$\mathbb{D}$; we shall denote this crosscut by $C^{\prime}$. On the other
hand, note that, for $x\in\partial N_{C}\cap\partial\mathbb{D}$, it holds

$Cl(G_{1},x)\coloneqq\left\\{w\in\overline{\mathbb{D}}\colon\textrm{ there
exists }\left\\{x_{n}\right\\}_{n}\subset N_{C}\textrm{ with }x_{n}\to
x\textrm{ and }G_{1}(x_{n})\to w\right\\}\subset\partial\mathbb{D},$

since $g(\mathbb{D})\subset\mathbb{D}$. Hence,

$\partial G_{1}(N_{C})\subset C^{\prime}\cup\partial\mathbb{D}.$

Therefore, $\partial G_{1}(N_{C})$ is locally connected (and, in fact, a
Jordan curve), so $G_{1}$ extends homeomorphically to $\partial N_{C}$, and
$G_{1}(\partial N_{C}\cap\partial\mathbb{D})\subset\partial\mathbb{D}$.

Thus, by Schwarz reflection, we can extend holomorphically $G_{1}$ to

$N_{C}\cup(\partial
N_{C}\cap\partial\mathbb{D})\cup\left\\{z\in\widehat{\mathbb{C}}\colon\frac{1}{\overline{z}}\in
N_{C}\right\\},$

(see Fig. 4.1). In particular, there exists $\rho\coloneqq\rho(\xi)>0$ such
that all inverse branches of $g$ are well-defined in $D(\xi,\rho)$, as
desired.

$G_{1}$$\rho$$G_{1}(\xi)$$\mathbb{D}$$\xi$

Figure 4.1: Whenever an inverse branch $G_{1}$ is well-defined in a crosscut
neighbourhood, it can be extended across the unit circle by Schwarz
reflection.

$\square$

It follows from Proposition 4.2 that a value $v\in\partial\mathbb{D}$ is
singular for $g$ if and only if it is accumulated by singular values in
$\mathbb{D}$, i.e. $v\in\overline{SV(g)\cap\mathbb{D}}$. Clearly, for finite
Blaschke products, all values $v\in\partial\mathbb{D}$ are regular, and the
same is true if $SV(g)\cap\mathbb{D}$ is compactly contained in $\mathbb{D}$.
Moreover,

######  Corollary 4.3.

(Non-singular Denjoy-Wolff point) Let $g\colon\mathbb{D}\to\mathbb{D}$ be an
inner function with Denjoy-Wolff point $p\in\partial\mathbb{D}$. If
$p\notin\overline{SV(g)}$, then $p$ is not a singularity for $g$.

###### Proof.

If $p\notin\overline{SV(g)}$, then there exists a crosscut neighbourhood
$N_{p}$ such that $p\in\partial N_{p}$ and $SV(g)\cap N_{p}=\emptyset$. Since
the Denjoy-Wolff point is radially fixed (Thm. 3.6), there exists a curve
$\gamma\subset N_{p}$ landing at $p$, such that $g(\gamma)\subset N_{p}$ also
lands at $p$. Consider $G_{1}$ the inverse branch of $g^{-1}$ defined in
$N_{p}$ such that $G_{1}(g(\gamma))=\gamma$. By Proposition 4.2, $G_{1}$
extends conformally to $D(p,\rho)$ for some $\rho>0$, and $G_{1}(p)=p$. Then,
$D_{1}\coloneqq G_{1}(D(p,\rho))$ is a negihbourhood of $p$, and

$g\colon D_{1}\to D(p,\rho)$

conformally. Therefore, by Lemma 3.3, $p$ is not a singularity for $g$, as
desired. $\square$

######  Remark 4.4.

(At most one asymptotic value per singularity) By the Lehto-Virtanen Theorem
2.12, given a singularity $\xi\in E(g)$, there exists at most one asymptotic
value $v\in\overline{\mathbb{D}}$ corresponding to $\xi$. Indeed, if $v$ is an
asymptotic value corresponding to the singularity $\xi$, there exists a curve
landing at $\xi$ whose image lands at $v$. By Lehto-Virtanen Theorem,
$g^{*}(\xi)=v$. Since radial limits, if they exist, are unique, there cannot
be more asymptotic values corresponding to $\xi$. Hence,

$\\#\textrm{Sing}(g)\geq\\#AV(g)\cap\overline{\mathbb{D}}.$

This differs from when a meromorphic function
$f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ is considered, where the essential
singularity (infinity) can have infinitely many asymptotic values
corresponding to it.

Next, we analyse the distortion induced by inverse branches near
$\partial\mathbb{D}$, and how we can control the preimages of radial limits in
terms of Stolz angles.

######  Proposition 4.5.

(Control of radial limits in terms of Stolz angles) Let
$g\colon\mathbb{D}\to\mathbb{D}$ be an inner function with Denjoy-Wolff point
$p\in\overline{\mathbb{D}}$. Let $\xi\in\partial\mathbb{D}$, $\xi\neq p$.
Assume there exists $\rho_{0}>0$ such that $D(\xi,\rho_{0})\cap SV(g)\neq 0$.
Then, for all $0<\alpha<\frac{\pi}{2}$, there exists
$\rho_{1}\coloneqq\rho_{1}(\alpha,\rho_{0})<\rho_{0}$ such that all branches
$G_{1}$ of $g^{-1}$ are well-defined in $D(\xi,\rho_{1})$ and, for all
$\rho<\rho_{1}$,

$G_{1}(R_{\rho}(\xi,p))\subset\Delta_{\alpha,\rho}(G_{1}(\xi),p),$

where $R_{\rho}(\cdot,p)$ and $\Delta_{\alpha,\rho}(\cdot,p)$ stand for the
generalized radial segment and Stolz angle with respect to $p$ (Def. 2.10).

$G_{1}$$R$$G_{1}(\xi)$$\mathbb{D}$$p=0$$\xi$

Figure 4.2: Whenever an inverse branch $G_{1}$ is well-defined at a boundary
point $\xi\in\partial\mathbb{D}$, $G_{1}$ sends radial segments into angular
neighbourhoods of a given openning. In the figure, $p=0\in\mathbb{D}$.

Note that $\rho_{1}$ depends only on $\rho_{0}$ and $\alpha$, but not on the
point $\xi\in\partial\mathbb{D}$, nor on the inverse branch $G_{1}$.

###### Proof.

Note that, since $D(\xi,\rho_{0})\cap SV(g)\neq\emptyset$, all branches
$G_{1}$ of $g^{-1}$ are well-defined in $D(\xi,\rho_{0})$. We shall
distinguish two cases.

  * •

Assume first that $g$ is elliptic, so $p\in\mathbb{D}$. According to
Definition 2.10, it is enough to consider $g\colon\mathbb{D}\to\mathbb{D}$
with $g(0)=0$ and prove

$G_{1}(R_{\rho}(\xi,0))\subset\Delta_{\alpha,\rho}(G_{1}(\xi),0).$

By Schwarz lemma 3.5, $\left|G_{1}(z)\right|\geq\left|z\right|$ for $z\in
D(\xi,\rho_{0})\cap\mathbb{D}$. It is left to see that, for $z\in
R_{\rho}(\xi,0)$,

$\left|\textrm{Arg }G_{1}(\xi)-\textrm{Arg
}(G_{1}(\xi)-G_{1}(z))\right|<\alpha.$

To do so, consider the linear map

$L_{1}(z)\coloneqq G_{1}(\xi)+G^{\prime}_{1}(\xi)(z-\xi).$

Note that
$\left|L_{1}(z)-G_{1}(\xi)\right|=\left|G^{\prime}_{1}(\xi)\right|\left|z-\xi\right|$.
Moreover, by Corollary 2.2, there exists $\rho_{1}<\rho_{0}$ and a constant
$C(\rho_{1})>0$ such that

$\left|G_{1}(z)-L_{1}(z)\right|\leq
C(\rho_{1})\left|z-\xi\right|\left|G^{\prime}_{1}(\xi)\right|,$

for all $z\in D(\xi,\rho_{1})$. That is, the point $G_{1}(z)$ belongs to the
disk of center $L_{1}(z)$ and radius
$C(\rho_{1})\left|z-\xi\right|\left|G^{\prime}_{1}(\xi)\right|$ (see Fig.
4.3).

Since $C(\rho_{1})\to 0$ as $\rho_{1}\to 0$, we have
$\frac{C(\rho_{1})}{1-C(\rho_{1})}\to 0$ as $\rho_{1}\to 0$. Without loss of
generality, we assume $C(\rho_{1})$ satisfies

$\frac{C(\rho_{1})}{1-C(\rho_{1})}<\tan\alpha.$

Let

$\beta\coloneqq\left|\textrm{Arg }G_{1}(\xi)-\textrm{Arg
}(G_{1}(\xi)-G_{1}(z))\right|.$

We claim that, if $z\in R_{\rho}(\xi,0)$, then $\textrm{Arg
}L_{1}(z)=\textrm{Arg }(G_{1}(\xi))$. Indeed, $L_{1}$ is the affine map
associated to $G_{1}$, which is an inverse branch of $g$. The map $G_{1}$ is
conformal in $D(\xi,\rho_{0})$, and hence angle preserving, and
$G_{1}(D(\xi,\rho_{0})\cap\partial\mathbb{D})\subset\partial\mathbb{D}$. From
this, it follows that, if $\textrm{Arg }z=\textrm{Arg }\xi$, then $\textrm{Arg
}L_{1}(z)=\textrm{Arg }(G_{1}(\xi))$, i.e. if $z$ lies on the radial segment
at $\xi$, then $L_{1}(z)$ lies on the radial segment at $G_{1}(\xi)$.

Then, noting that $G_{1}(z)$ belongs to the disk of center $L_{1}(z)$ and
radius $C(\rho_{1})\left|z-\xi\right|\left|G^{\prime}_{1}(\xi)\right|$, it
follows

$\tan\beta\leq\dfrac{C(\rho_{1})\left|G^{\prime}_{1}(\xi)\right|\left|z-\xi\right|}{(1-C(\rho_{1}))\left|G^{\prime}_{1}(\xi)\right|\left|z-\xi\right|}=\frac{C(\rho_{1})}{1-C(\rho_{1})}\leq\tan\alpha,$

as desired. See also Figure 4.3.

$\xi$$G_{1}(\xi)$$0$$z$$L_{1}(z)$$G_{1}(z)$$\partial\mathbb{D}$

(a)

$G_{1}(\xi)$$\beta$$L_{1}(z)$$C(\rho_{1})\left|G_{1}^{\prime}(\xi)\right|\left|z-\xi\right|$$(1-C(\rho_{1}))\left|G_{1}^{\prime}(\xi)\right|\left|z-\xi\right|$$G_{1}(z)$

(b)

Figure 4.3: Image (a) shows how the inverse branch $G_{1}$ acts near $\xi$,
and (b) provides more detail on it. Indeed, the point $G_{1}(z)$ lies on the
disk
$D(L_{1}(z),C(\rho_{1})\left|G_{1}^{\prime}(\xi)\right|\left|z-\xi\right|)$.
The angle $\beta$ captures the opening of the vector $G_{1}(z)-G_{1}(\xi)$
with respect to the vector $L_{1}(z)-G_{1}(\xi)$. Hence, $\tan\beta$ is
bounded above by the maximal distance between $G_{1}(z)$ and $L_{1}(z)$
divided by the minimal distance between $G_{1}(\xi)$ and $G_{1}(z)$.

  * •

Assume $g$ is non-elliptic, so $p\in\partial\mathbb{D}$. Note that
$G_{1}(\xi)\neq p$, since $p$ is the Denjoy-Wolff point and, hence, it is
radially fixed (Thm. 3.6).

Now, consider $h\colon\mathbb{H}\to\mathbb{H}$, $h\coloneqq M\circ g\circ
M^{-1}$, where $M\colon\mathbb{D}\to\mathbb{H}$, $M(z)=i\dfrac{p+z}{p-z}$.
Then, there exists $\widetilde{\rho}_{0}$ such that
$D(M(\xi),\widetilde{\rho}_{0})\cap SV(h)=\emptyset$, and consider $H_{1}$ the
branch of $h^{-1}$ corresponding to $G_{1}$, well-defined in
$D(M(\xi),\widetilde{\rho}_{0})$. It is enough to prove that there exists
$\widetilde{\rho}_{1}<\widetilde{\rho}_{0}$ such that, for all
$\rho<\widetilde{\rho}_{1}$,

$H_{1}(R^{\mathbb{H}}_{\rho}(M(\xi)))\subset\Delta^{\mathbb{H}}_{\alpha,\rho}(H_{1}(M(\xi))).$

First note that, by Wolff lemma 3.6, if $\textrm{Im }w<\rho$, then $\textrm{Im
}H_{1}(w)<\rho$. Now, consider the linear map

$L_{1}(w)\coloneqq H_{1}(M(\xi))+H^{\prime}_{1}(M(\xi))(w-M(\xi)).$

Note that
$\left|L_{1}(w)-H_{1}(\xi)\right|=\left|H^{\prime}_{1}(M(\xi))\right|\left|w-M(\xi)\right|$.
Moreover, by Corollary 2.2, there exists
$\widetilde{\rho}_{1}<\widetilde{\rho}_{0}$ and a constant
$C(\widetilde{\rho}_{1})>0$ such that

$\left|H_{1}(w)-L_{1}(w)\right|\leq
C(\widetilde{\rho}_{1})\left|w-M(\xi)\right|\left|H^{\prime}_{1}(M(\xi))\right|,$

for all $w\in D(\xi,\widetilde{\rho}_{1})$. We assume, without loss of
generality,

$\dfrac{C(\widetilde{\rho}_{1})}{1-C(\widetilde{\rho}_{1})}<\tan\alpha.$

Since $h(\mathbb{H})\subset\mathbb{H}$, and $H_{1}$ is a branch of $h^{-1}$,
conformal where defined, then if $w\in R_{\rho}^{\mathbb{H}}(M(\xi))$, then
$\textrm{Re }L_{1}(w)=H_{1}(M(\xi))$, i.e. if $w$ lies on the radial segment
at $M(\xi)$, then $L_{1}(w)$ lies on the radial segment at $L_{1}(M(\xi))$.

We claim that, for $w\in R_{\rho}^{\mathbb{H}}(M(\xi))$, it holds

$\dfrac{\left|\textrm{Re }H_{1}(w)-H_{1}(M(\xi))\right|}{\textrm{Im
}H_{1}(w)}<\tan\alpha.$

Indeed,

$\left|\textrm{Re }H_{1}(w)-H_{1}(M(\xi))\right|=\left|\textrm{Re
}H_{1}(w)-\textrm{Re }L_{1}(w)\right|\leq
C(\widetilde{\rho}_{1})\left|w-M(\xi)\right|\left|H^{\prime}_{1}(M(\xi))\right|,$
$\textrm{Im }H_{1}(w)=\textrm{Im
}(H_{1}(w)-H_{1}(M(\xi)))\geq(1-C(\widetilde{\rho}_{1}))\left|H_{1}^{\prime}(M(\xi))\right|\left|w-M(\xi)\right|,$

as desired. See also Figure 4.4.

$w$$M(\xi)$$H_{1}(M(\xi))$$L_{1}(w)$$H_{1}(w)$$C(\widetilde{\rho_{1}})\left|H_{1}^{\prime}(M(\xi))\right|\left|w-M(\xi)\right|$$(1-C(\widetilde{\rho_{1}}))\left|H_{1}^{\prime}(M(\xi))\right|\left|w-M(\xi)\right|$

Figure 4.4: The point $H_{1}(w)$ lies on the disk
$D(L_{1}(w),C(\widetilde{\rho}_{1})\left|H_{1}^{\prime}(\xi)\right|\left|z-\xi\right|)$,
and this gives the estimates on its real and imaginary part.

$\square$

We are now ready to prove Theorem B. In fact, we show a stronger statement
(Thm. 4.6), from which Theorem B is an immediate corollary.

Consider the postsingular set

$P(g)\coloneqq\overline{\bigcup\limits_{v\in SV(g)}\bigcup\limits_{n\geq
0}g^{n}(v)}.$

With our accurate study of the singular values, the following theorem is quite
straight forward.

######  Theorem 4.6.

(Inverse branches at boundary points) Let $g\colon\mathbb{D}\to\mathbb{D}$ be
an inner function, such that $g^{*}|_{\partial\mathbb{D}}$ is recurrent.
Assume there exists $\zeta\in\partial\mathbb{D}$ and a crosscut neighbourhood
$N_{\zeta}$ of $\zeta$ such that $P(g)\cap N_{\zeta}=\emptyset$. Then, for
$\lambda$-almost every $\xi\in\partial\mathbb{D}$, there exists
$\rho_{0}\coloneqq\rho_{0}(\xi)>0$ such that all branches $G_{n}$ of $g^{-n}$
are well-defined in $D(\xi,\rho_{0})$. In particular, the set $E(g)$ of
singularities of $g$ has $\lambda$-measure zero.

In addition, for all $0<\alpha<\frac{\pi}{2}$, there exists
$\rho_{1}<\rho_{0}$ such that, for all $n\geq 0$, all branches $G_{n}$ of
$g^{-n}$ are well-defined in $D(\xi,\rho_{1})$ and, for all $\rho<\rho_{1}$,

$G_{n}(R_{\rho}(\xi),p)\subset\Delta_{\alpha,\rho}(G_{n}(\xi),p),$

where $R_{\rho}(\cdot,p)$ and $\Delta_{\alpha,\rho}(\cdot,p)$ stand for the
radial segment and the Stolz angle with respect to $p$ (Def. 2.10).

###### Proof.

By Proposition 4.2, the existence of a crosscut neighbourhood $N_{\zeta}$ of
$\zeta\in\partial\mathbb{D}$ such that $P(g)\cap N_{\zeta}=\emptyset$, implies
the existence of $\rho_{\zeta}>0$ such that all branches $G_{n}$ of $g^{-n}$
are well-defined in $D(\zeta,\rho_{\zeta})$.

We have to see that, for $\lambda$-almost every $\xi\in\partial\mathbb{D}$,
there exists $\rho_{\xi}>0$ such that all branches $G_{n}$ of $g^{-n}$ are
well-defined in $D(\xi,\rho_{\xi})$. Since we are assuming $g^{*}$ to be
recurrent, for $\lambda$-almost every $\xi\in\partial\mathbb{D}$,
$\left\\{(g^{n})^{*}(\xi)\right\\}_{n}$ is dense in $\partial\mathbb{D}$ (Thm.
3.10(c)). Therefore, there exists $n_{0}\coloneqq n_{0}(\xi)$ such that
$(g^{n_{0}})^{*}(\xi)\in D(\zeta,\rho_{\zeta})$. This already implies the
existence of $\rho_{\xi}>0$ such that all inverse branches $G_{n}$ of $g^{-n}$
are well-defined in $D(\xi,\rho_{\xi})$.

Next we prove that $\lambda(E(g))=0$. To do so, let

$K\coloneqq\left\\{\xi\in\partial\mathbb{D}\colon\exists\rho>0\textrm{ such
that all branches }G_{n}\textrm{ of }g^{-n}\textrm{ are well-defined in
}D(\xi,\rho)\right\\}.$

Note that points in $g^{-1}(K)$ do not belong to $E(g)$. Indeed, if $\zeta\in
g^{-1}(K)$, then there exists a neighbourhood of $\zeta$ which is mapped
conformally to $D(g(\zeta),\rho)$, and hence $\zeta$ cannot be a singularity.
Therefore, it is enough to prove that $\lambda(g^{-1}(K))=1$. This follows
from the fact that $\lambda(K)=1$ and that $g^{*}$ is non-singular (Thm.
3.10(a)).

Finally, the control of the image of radial segments by inverse branches in
terms of angular neighbourhoods follows from Proposition 4.5. Indeed, note
that the estimates obtained therein do not depend on the inverse branches
considered, but only on the radius of the disk where they are defined.
$\square$

###### Proof of Theorem B.

It follows straightforward from applying Theorem 4.6. $\square$

######  Remark 4.7.

Observe that a sufficient condition so that hypotheses of Theorem 4.6 are
satisfied is that singular values are compactly contained in $\mathbb{D}$.
Indeed, it is enough to show that, if singular values of $g|_{\mathbb{D}}$ are
compactly contained in $\mathbb{D}$, then there exists
$\zeta\in\partial\mathbb{D}$ and a crosscut neighbourhood $N_{\zeta}$ of
$\zeta$ such that $P(g)\cap N_{\zeta}=\emptyset$.

Assume first that $g$ is elliptic, with Denjoy-Wolff point $p\in\mathbb{D}$.
After conjugating by a Möbius transformation, we assume $p=0$. Now, consider
an Euclidean disk $D(0,r)$, with $r\in(0,1)$ big enough so that $SV(g)\subset
D(0,r)$. By Schwarz Lemma 3.5, $D(0,r)$ is forward invariant under $g$, so
$P(g)\subset D(0,r)$. This implies that, for all $\zeta\in\partial\mathbb{D}$,
we can find a crosscut neighbourhood of $\zeta$ disjoint from the postsingular
set.

If $g$ is doubly parabolic, the procedure is analogous, with the difference
that we should work with a holodisk thangent to the Denjoy-Wolff point instead
of an Euclidean disk, and we apply Wolff Lemma 3.6 instead of Schwarz Lemma.
Note that we can find a crosscut neighbourhood disjoint from the postsingular
set for any $\zeta\in\partial\mathbb{D}$, except for the Denjoy-Wolff point.

##  5 Invariant Fatou components of functions in class $\mathbb{K}$

As mentioned in the introduction, consider $f\in\mathbb{K}$, i.e.

$f\colon\widehat{\mathbb{C}}\smallsetminus E(f)\to\widehat{\mathbb{C}},$

where $\Omega(f)\coloneqq\widehat{\mathbb{C}}\smallsetminus E(f)$ is the
largest set where $f$ is meromorphic and $E(f)$ is the set of singularities of
$f$, which is assumed to be closed and countable. Note that $\Omega(f)$ is
open.

Notation. Having fixed a function $f\in\mathbb{K}$, we denote $\Omega(f)$ and
$E(f)$ simply by $\Omega$ and $E$, respectively. Given a domain
$U\subset\Omega$, we denote by $\partial U$ the boundary of $U$ in $\Omega$,
and we keep the notation $\widehat{\partial}U$ for the boundary with respect
to $\widehat{\mathbb{C}}$.

The dynamics of such functions was studied in [Bol96, Bol97, Bol99, BDH01,
BDH04, Dom10, DOS22]. The Fatou set $\mathcal{F}(f)$ is defined as the largest
open set in which $\left\\{f^{n}\right\\}_{n}$ is well defined and normal, and
the Julia set $\mathcal{J}(f)$, as its complement in $\widehat{\mathbb{C}}$.
The standard theory of Fatou and Julia for rational or entire functions
extends successfully to this more general setting. We need the following
properties.

######  Theorem 5.1.

(Properties of Fatou and Julia sets, [BDH01, Thm. A]) Let $f\in\mathbb{K}$.
Then,

  1. (a)

$\mathcal{F}(f)$ is completely invariant in the sense that
$z\in\mathcal{F}(f)$ if and only if $f(z)\in\mathcal{F}(f)$;

  2. (b)

for every positive integer $k$, $f^{k}\in\mathbb{K}$,
$\mathcal{F}(f^{k})=\mathcal{F}(f)$ and $\mathcal{J}(f^{k})=\mathcal{J}(f)$;

  3. (c)

$\mathcal{J}(f)$ is perfect;

  4. (d)

repelling periodic points are dense in $\mathcal{J}(f)$.

By (a), Fatou components (i.e. connected components of $\mathcal{F}(f)$) are
mapped among themselves, and hence classified into periodic, preperiodic or
wandering. By (b), the study of periodic Fatou components reduces to the
invariant ones, i.e. those for which $f(U)\subset U$. Those Fatou components
are classified into attracting basins, parabolic basins, Siegel disks, Herman
rings and Baker domains [BDH01, Thm. C]. A Baker domain is, by definition, a
periodic Fatou component $U$ of period $k\geq 1$ for which there exists
$z_{0}\in\widehat{\partial}U$ such that $f^{nk}(z)\to z_{0}$, for all $z\in U$
as $n\to\infty$, but $f^{k}$ is not meromorphic at $z_{0}$. In such case,
$z_{0}$ is accessible from $U$ [BDH01, 658].

######  Theorem 5.2.

(Connectivity of Fatou components, [Bol99]) Let $f\in\mathbb{K}$, and let $U$
be a periodic Fatou component of $f$. Then, the connectivity of $U$ is 1, 2,
or $\infty$.

In this paper, we focus on simply connected periodic Fatou components, which
we assume to be invariant. We analyse them by means of the Riemann map
$\varphi\colon\mathbb{D}\to U$.

###  5.1 Inner function associated to an invariant Fatou component

Let $f\in\mathbb{K}$, and let $U$ be an invariant Fatou component for $f$,
which we assume to be simply connected. Consider $\varphi\colon\mathbb{D}\to
U$ to be a Riemann map. Then, $f\colon U\to U$ is conjugate by $\varphi$ to a
holomorphic map $g\colon\mathbb{D}\to\mathbb{D}$, such that the diagram

${U}$${U}$${\mathbb{D}}$${\mathbb{D}}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi}$

commutes.

For entire functions, where the unique essential singularity lies at $\infty$,
it is well-known that $g$ is an inner function (see e.g. [EFJS19, Sect. 2.3],
or [ERS20, Prop. 1.1]). Actually, the same holds for functions in class
$\mathbb{K}$ (see Prop. 5.6), and we say that $g$ is an inner function
associated to $(f,U)$. Note that two inner functions associated to the same
$(f,U)$ are conformally conjugate and hence have the same dynamical behaviour.

As usual, singular values (defined in Sect. 2.6) play a distinguished role in
the dynamics. It follows from [Bol97, Thm. 1.2] that
$Cl(f,z)=\widehat{\mathbb{C}}$, for every $z\in E(f)$ (see also [BDH01, p.
651]). Thus, $\mathbb{K}\subset\mathbb{M}$, and the discussion in Section 2.6
applies.

Note that, as in the case of inner functions, asymptotic paths land. This
comes from the fact that $\partial\Omega=E(f)$ is countable, and hence if the
landing set $L(\gamma)$ of a curve $\gamma\subset\Omega$ consists of more than
two points $\partial\Omega$, then $L(\gamma)\cap\Omega\neq\emptyset$.

The dynamics of $f|_{U}$ is controlled by the following subset of singular
values.

######  Definition 5.3.

(Relevant singular values) Let $f\in\mathbb{K}$, and let $U$ be an invariant
Fatou component for $f$. We define:

  * •

Relevant critical values for $f|_{U}$,

$CV(f,U)\coloneqq\left\\{v\in U\colon v=f(z),f^{\prime}(z)=0\right\\}.$

  * •

Relevant asymptotic values for $f|_{U}$,

$AV(f,{U})\coloneqq\left\\{v\in{U}\cap AV(f)\colon\textrm{ there exists an
asymptotic path }\gamma\subset U\textrm{ for }v\right\\}.$

  * •

Relevant singular values for $f|_{U}$,
$SV(f,{U})\coloneqq\overline{CV(f,{U})\cup AV(f,{U})}\cap U$.

  * •

Relevant postsingular set for $f|_{U}$,
$P(f,{U})\coloneqq\overline{\bigcup\limits_{v\in SV(f,{U})}\textrm{
}\bigcup\limits_{n\geq 0}f^{n}(v)}\cap U$.

Note that there may be other singular values in $U$, if another Fatou
component is mapped into $U$ non-conformally. However, these singular values
do not play a role in the internal dynamics of $(f,U)$, and the inner function
does not see them. Note that

$SV(g,\mathbb{D})=\varphi^{-1}(SV(f,U)),$

and

$P(g,\mathbb{D})=\varphi^{-1}(P(f,U)).$

It is well-known that $SV(f,U)\neq\emptyset$ whenever $U$ is an attracting or
parabolic basin, or a doubly parabolic Baker domain. If $SV(f,U)$ is compactly
contained in $U$, the behaviour of the associated inner function $g$ is well
understood, in the following sense.

######  Proposition 5.4.

(Singular values compactly contained) Let $f\in\mathbb{K}$, let $U$ be an
invariant simply connected Fatou component for $f$, and $g$ its associated
inner function. Assume $SV(f,U)$ is compactly contained in $U$. Then, the
following hold.

  1. (a)

If $U$ is an attracting basin, then, for all $\xi\in\partial\mathbb{D}$, there
exists a crosscut neighbourhood $N_{\xi}$ of $\xi$, such that $N_{\xi}\cap
P(g,\mathbb{D})=\emptyset$.

  2. (b)

If $U$ is either a parabolic basin or a Baker domain, then the Denjoy-Wolff
point $p\in\partial\mathbb{D}$ of $g$ is not a singularity for $g$. Moreover,
for all $\xi\in\partial\mathbb{D}$, $\xi\neq p$, there exists a crosscut
neighbourhood $N_{\xi}$ of $\xi\in\partial\mathbb{D}$, such that $N_{\xi}\cap
P(g,\mathbb{D})=\emptyset$.

###### Proof.

  1. (a)

Let $z_{0}\in U$ be the attracting fixed point, and consider $g$ to be the
inner function associated by a Riemann map $\varphi$, with $\varphi(0)=z_{0}$.
Then, there exists $r\in(0,1)$ big enough so that $SV(g,\mathbb{D})\subset
D(0,r)$.

By Schwarz lemma 3.5, $D(0,r)$ is forward invariant, so
$P(g,\mathbb{D})\subset D(0,r)$, and (a) follows trivially.

  2. (b)

By Corollary 4.3, it is enough to find a crosscut neighbourhood $N_{p}$ of $p$
such that $N_{p}\cap SV(g,\mathbb{D})=\emptyset$, and this is immediate from
the hypothesis.

The second statement follows for applying the same argument as in (a), using a
tangent disk at the Denjoy-Wolff point and Wolff Lemma 3.6.

$\square$

###  5.2 Ergodic properties of the boundary map $f\colon\partial U\to\partial
U$

Let $\varphi\colon\mathbb{D}\to U$ be the Riemann map, as in Section 5.1.
Then, the radial extension

$\varphi^{*}\colon\partial\mathbb{D}\to\widehat{\partial}U$

is well-defined $\lambda$-almost everywhere, and $\widehat{\partial}U$ admits
a harmonic measure $\omega_{U}$, which stands for the push-forward of the
normalized Lebesgue measure $\lambda$ of $\partial\mathbb{D}$ (see Sect. 2.5).
The ergodic properties of $f|_{\partial U}$ will be derived from the ergodic
properties of $g^{*}$, where $g$ is the inner function associated to $(f,U)$.

We start by proving that $g$ is actually an inner function. To that end,
consider the following subsets of $\partial\mathbb{D}$.

$\Theta_{E}\coloneqq\left\\{\xi\in\partial\mathbb{D}\colon\varphi^{*}(\xi)\in
E(f)\right\\}$
$\Theta_{\Omega}\coloneqq\left\\{\xi\in\partial\mathbb{D}\colon\varphi^{*}(\xi)\in\Omega(f)\right\\}$

Note that, since $E(f)$ is countable, $\lambda(\Theta_{E})=0$, so
$\lambda(\Theta_{\Omega})=1$. Moreover, the conjugacy
$f\circ\varphi=\varphi\circ g$ extends for the radial extensions wherever it
makes sense, as it is shown in the following lemma.

######  Lemma 5.5.

(Radial limits commute) Let $\xi\in\Theta_{\Omega}$, then $g^{*}(\xi)$ and
$\varphi^{*}(g^{*}(\xi))$ are well-defined, and

$f(\varphi^{*}(\xi))=\varphi^{*}(g^{*}(\xi)).$

###### Proof.

Let $r_{\xi}(t)=t\xi$, with $t\in\left[0,1\right)$. By assumption,
$\varphi(r_{\xi}(t))\to\varphi^{*}(\xi)\eqqcolon w\in\partial U$, as $t\to
1^{-}$ (recall that $\partial U$ denotes the boundary of $U$ taken in
$\Omega$). Since $f$ is continuous at $w$ and $f\circ\varphi=\varphi\circ g$,
for all $0<t<1$,

$\varphi(g(r_{\xi}(t)))=f(\varphi(r_{\xi}(t)))\to f(w)\in\widehat{\partial}U,$

as $t\to 1^{-}$. We claim that this already implies that $\gamma(t)\coloneqq
g(r_{\xi}(t))$ lands at some $\zeta\in\partial\mathbb{D}$. Indeed, consider

$L_{\gamma,1}\coloneqq\left\\{z\in\overline{\mathbb{D}}\colon\textrm{ there
exists }t_{n}\to 1^{-}\textrm{ such that }\gamma(t_{n})\to z\right\\},$

which is a non-empty, compact, connected set contained in
$\partial\mathbb{D}$, since points in $\mathbb{D}$ are mapped to $U$ by
$\varphi$. If $L_{\gamma,1}$ is a non-degenerate arc $I$, with $\lambda(I)>0$,
for $\lambda$-almost every $\zeta\in I$, $\varphi^{*}(\zeta)=f(w)$, which is a
contradiction with Fatou’s Theorem 2.15. Hence,
$L_{\gamma,1}=\zeta\in\partial\mathbb{D}$.

Finally, the Lehto-Virtanen Theorem 2.12 implies that $g^{*}(\xi)=\zeta$ and
$\varphi^{*}(g^{*}(\xi))=f(w)$, as desired. $\square$

######  Proposition 5.6.

($g$ is inner) Let $f\in\mathbb{K}$, and let $U$ be an invariant simply
connected Fatou component of $f$. Then, the associated map
$g\colon\mathbb{D}\to\mathbb{D}$ is an inner function.

###### Proof.

Since $g$ is a self-map of the unit disk, its radial extension $g^{*}$ is
well-defined $\lambda$-almost everywhere. We have to see that
$\left|g^{*}(\xi)\right|=1$, for $\lambda$-almost every
$\xi\in\partial\mathbb{D}$.

Assume, on the contrary, that there exists $A\subset\partial\mathbb{D}$ with
$\lambda(A)>0$ and $\left|g^{*}(\xi)\right|<1$, for all $\xi\in A$. We can
assume, without loss of generality, that $\varphi^{*}(\xi)\in\partial
U\subset\mathcal{J}(f)$, for all $\xi\in A$. By Lemma 5.5, for all $\xi\in A$,
$\varphi^{*}(\xi)\in\mathcal{J}(f)\smallsetminus E(f)$, and

$f(\varphi^{*}(\xi))=\varphi^{*}(g^{*}(\xi))\in U\subset\mathcal{F}(f)$

which is a contradiction with the total invariance of the Fatou set. $\square$

We note that, since $\omega_{U}(E(f))=0$, for every Borel set
$B\subset\widehat{\mathbb{C}}$ and $z\in U$, we have

$\omega_{U}(z,B)=\omega_{U}(z,B\cap\Omega(f)).$

With these tools at hand, and those developed in the previous sections, we can
now prove ergodic properties like ergodicity and recurrence for the boundary
map of Fatou components of maps in class $\mathbb{K}$, generalizing the
results of Doering and Mañé [DM91] for rational maps.

######  Theorem 5.7.

(Ergodic properties of the boundary map) Let $f\in\mathbb{K}$, and let $U$ be
an invariant simply connected Fatou component for $f$. Let $g$ be an inner
function associated to $(f,U)$. Then, the following are satisfied.

  1. (i)

If $U$ is either an attracting basin, a parabolic basin, or a Siegel disk,
then $g^{*}|_{\partial\mathbb{D}}$ is ergodic and recurrent with respect to
the Lebesgue measure $\lambda$.

  2. (ii)

If $U$ is a doubly parabolic Baker domain, $g^{*}|_{\partial\mathbb{D}}$ is
ergodic with respect to $\lambda$. In addition, assume one of the following
conditions is satisfied.

    1. (a)

$f|_{U}$ has finite degree.

    2. (b)

Relevant singular values $SV(f,U)$ are compactly contained in $U$.

    3. (c)

The Denjoy-Wolff point of $g$ is not a singularity for $g$.

    4. (d)

There exists $z\in U$ and $r>1$ such that

$\textrm{\em
dist}_{U}(f^{n+1}(z),f^{n}(z))\leq\frac{1}{n}+O\left(\frac{1}{n^{r}}\right),$

as $n\to\infty$, where $\textrm{\em dist}_{U}$ denotes the hyperbolic distance
in $U$.

Then, $g^{*}|_{\partial\mathbb{D}}$ is recurrent with respect to $\lambda$.

  3. (iii)

If $g^{*}|_{\partial\mathbb{D}}$ is ergodic (resp. recurrent) with respect to
$\lambda$, so is $f|_{{\partial}U}$ with respect to $\omega_{U}$.

If $g^{*}|_{\partial\mathbb{D}}$ is recurrent with respect to $\lambda$, then
for $\omega_{U}$-almost every point $x\in\partial U$,
$\left\\{f^{n}(x)\right\\}_{n}$ is dense in $\partial U$.

  4. (iv)

Let $k$ be a positive integer. Then, the inner function associated to $(f,U)$
has the same ergodic properties than the inner function associated to
$(f^{k},U)$.

###### Proof.

  1. (i)

The associated inner function to these Fatou components is either elliptic or
doubly parabolic, so $g^{*}$ is ergodic (Thm. 3.10(b)).

Recurrence for the inner function associated to a Siegel disk or an attracting
basin follows from the fact that these inner functions are always elliptic
(Thm. 3.10(d)). Recurrence for parabolic basins follows from [DM91, Thm. 6.1]
(although it is stated for rational maps, the proof only uses the local
behaviour around the parabolic fixed point, so it is valid for
$f\in\mathbb{K}$).

  2. (ii)

Ergodicity of doubly parabolic Fatou components follows from Theorem 3.10(b).
Recurrence (under the stated conditions) follows from Theorem 3.10(e). Indeed,
note that, if $f|_{U}$ has finite degree, then relevant singular values
$SV(f,U)$ are compactly contained in $U$. By Proposition 5.4, in this case,
the Denjoy-Wolff point $p\in\partial\mathbb{D}$ is not a singularity for $g$.
Hence, by Theorem 3.10(e), the boundary map $g^{*}|_{\partial\mathbb{D}}$ is
recurrent with respect to $\lambda$.

To see that condition (d) implies recurrence, note that

$\textrm{dist}_{U}(f^{n+1}(z),f^{n}(z))=\textrm{
dist}_{\mathbb{D}}(g^{n+1}(\varphi^{-1}(z)),g^{n}(\varphi^{-1}(z))),$

and apply again Theorem 3.10(e).

  3. (iii)

We start with ergodicity. Let $A\subset\partial U$ be measurable. If
$A=f^{-1}(A)$, then

$(\varphi^{*})^{-1}(A)=(\varphi^{*})^{-1}(f^{-1}(A))=(g^{*})^{-1}((\varphi^{*})^{-1}(A)).$

If $g^{*}$ is ergodic, then $\lambda((\varphi^{*})^{-1}(A))=0$ or
$\lambda((\varphi^{*})^{-1}(A))=1$. Then, $\omega_{U}(A)=0$ or
$\omega_{U}(A)=1$, and $f|_{\partial U}$ is ergodic.

For the recurrence, assume $A\subset\partial U$ is measurable, and consider
$(\varphi^{*})^{-1}(A)$. Then, for $\lambda$-almost every
$\xi\in(\varphi^{*})^{-1}(A)$, there exists $n_{k}\to\infty$ such that
$(g^{n_{k}})^{*}(\xi)\in(\varphi^{*})^{-1}(A)$. Since $A\subset\partial
U\subset\Omega$, Lemma 5.5 applies, and

$\varphi^{*}((g^{n_{k}})^{*}(\xi))=f^{n_{k}}(\varphi^{*}(\xi))\in A,$

proving recurrence for $f|_{\partial U}$. Since $\textrm{supp
}\omega_{U}=\widehat{\partial}U$, it follows from Theorem 2.5 that
$\omega_{U}$-almost every orbit is dense.

  4. (iv)

It follows from Theorem 3.10(f).

$\square$

##  6 Density of periodic boundary points. Proof of Theorem A

With the tools developed in the previous sections we are now able to prove a
more general version of Theorem A, namely Theorem 6.1.

######  Theorem 6.1.

(Periodic boundary points are dense) Let $f\in\mathbb{K}$, and let $U$ be an
invariant simply connected Fatou component for $f$. Let
$\varphi\colon\mathbb{D}\to U$ be a Riemann map, and let
$g\colon\mathbb{D}\to\mathbb{D}$ be the inner function associated to $(f,U)$
by $\varphi$.
Assume the following conditions are satisfied.

  1. (a)

$g^{*}|_{\partial\mathbb{D}}$ is recurrent with respect to $\lambda$.

  2. (b)

There exists $x_{0}\in\partial U$ and $r_{0}\coloneqq r_{0}(x_{0})>0$ such
that, for all $n\geq 0$, if $D_{n}$ is a connected component of
$f^{-n}(D(x_{0},r_{0}))$ such that $D_{n}\cap U\neq\emptyset$, then
$f^{n}|_{D_{n}}$ is conformal.

  3. (c)

There exists a crosscut $C\subset\mathbb{D}$ and a crosscut neighbourhood
$N_{C}$ with $N_{C}\cap P(g)=\emptyset$.

Then, accessible periodic points are dense in $\partial U$.

Note that in Theorem 6.1 we assume $U$ to be invariant, i.e. $f(U)\subset U$.
This is not restrictive: indeed, if $U$ was a $p$-periodic Fatou component
with $p\geq 2$, i.e. $f^{p}(U)\subset U$, we shall replace $f$ by $f^{p}$.
Note that $f^{p}\in\mathbb{K}$, since class $\mathbb{K}$ is closed under
composition.

The proof of Theorem 6.1 is postponed until the end of the section but we show
now how to deduce Theorem A from it.

###### Proof of Theorem A.

It is enough to show that, if the hypotesis (i)-(iii) in Theorem A are
satisfied, then (a)-(c) in Theorem 6.1 also hold. First, if hypotesis (i) and
(iii) in Theorem A are satisfied, then (a) holds, applying Theorem 5.7.
Second, it is clear that (ii) implies (b). Finally, (iii) is equivalent to
(c), by means of the Riemann map. $\square$

Finally, we shall state some other conditions under which the hypothesis of
Theorem 6.1 are satisfied, and hence periodic points are dense in the boundary
of the Fatou component. Note that the following conditions allow for the
possibility of having infinitely many singular values accumulating at
infinity.

First let $U$ be an attracting or a parabolic basin. In this case, the
boundary map is always recurrent. Hence, it is enough to ask that there exists
a domain $V\subset U$, such that

$P(f)\cap\overline{U}\subset\overline{V}$

and $\omega_{U}(\overline{V})=0$ (see Fig. 6.1).

$U$$V$$\varphi$$\mathbb{D}$

Figure 6.1: Schematic representation of a Fatou component satisfying the
previous condition of the postsingular set. The Fatou component of the right
is a Baker domain of $z+e^{-z}$. For this particular example, the domain $V$
could have been taken simpler. However, we wanted to illustrate how $V$ looks
like in general. It follows that, for the associated inner function, there
exist crosscut neighbourhoods without postsingular values, and hence
hypothesis (c) is satisfied.

Let us check that such a Fatou component satisfies the hypothesis of Theorem
6.1. It is clear that this already implies that for $\omega_{U}$-almost every
$x\in\partial U$, there exists $r\coloneqq r(x)>0$ such that $D(x,r)\cap
P(f)=\emptyset$, and hence (b) is satisfied. To see (c), let
$\varphi\colon\mathbb{D}\to U$ be a Riemann map, and note that
$\varphi^{-1}(\overline{V}\cap U)$ is a closed set in $\mathbb{D}$. By
hypothesis, $\lambda(\overline{\varphi^{-1}({V}\cap U)})=0$, implying (c).
Compare with the Fatou components considered in [JF23].

In the case of a doubly parabolic Baker domain, we need to ask additionally
that $f|_{\partial U}$ is recurrent with respect to $\omega_{U}$. This can be
ensured with any of the conditions given in Theorem 5.7(e).

###  6.1 Proof of Theorem 6.1

We shall split the proof into several steps.

  1. 1.

Inverse branches well-defined $\omega_{U}$-almost everywhere.

First note that hypothesis (b) implies that inverse branches of $f^{n}$
interplaying with $\partial U$ are well-defined in $D(x_{0},r_{0})$, being
$r_{0}$ uniform for all $n\geq 0$ and all inverse branches. Let us start by
proving that this actually holds for $\omega_{U}$-almost every $x\in\partial
U$, as an easy consequence of (b) together with the ergodic properties of
$f|_{\partial U}$.

######  Claim.

For $\omega_{U}$-almost every $x\in\partial U$, there exists $r\coloneqq
r(x)>0$ such that, for all $n\geq 0$, $f^{-n}(D(x,r))\cap U$ is non-empty,
and, if $D_{n}$ is a connected component of $f^{-n}(D(x,r))$ such that
$D_{n}\cap U\neq\emptyset$, then $f^{n}|_{D_{n}}$ is conformal.

We shall refer to the inverse branches considered above as relevant inverse
branches of $f^{n}$ at $x\in\partial U$. When we refer to a particular inverse
branch, we write $F_{n,y,x}$ meaning that $F_{n}$ is an inverse branch of
$f^{n}$ sending $y$ to $x$. When the points $x$ and $y$ are clear from the
context, we shall write only $F_{n}$ to lighten the notation. Since we are
interested in the study of $f|_{\partial U}$, relevant inverse branches are
the only ones that play a role in our construction.

###### Proof of the claim.

The first statement of the claim is deduced from the conjugacy between
$f|_{U}$ and the inner function $g$, and the fact that inner functions
associated to Fatou components of functions in class omit at most two values
(see e.g. [Bol99, Thm. 1]).

For the second assertion, let $x_{0}$ and $r_{0}$ be the ones given by
hypothesis (b). Since $\omega_{U}(D(x_{0},r_{0}))>0$, by hypothesis (a) and
Theorem 5.7, the orbit of $\omega_{U}$-almost every point visits infinitely
many times $D(x_{0},r{(x_{0})})$. Hence, $\omega_{U}$-almost every
$x\in\widehat{\partial}U$ there exists $n_{0}\coloneqq n_{0}(x)$ such that
$f^{n_{0}}(x)\in D(x_{0},r{(x_{0})})$.

Fix $x\in\widehat{\partial}U$ with this property. Then, there exists
$r\coloneqq r(x)>0$ such that $f^{n_{0}}(D(x,r))\subset D(x_{0},r_{0})$. By
hypothesis (b), $f^{n_{0}}|_{D(x,r)}$ is conformal.

Let $D_{m}$ be a connected component of $f^{-m}(D(x,r))$ such that $D_{m}\cap
U\neq\emptyset$. Then, $D_{m}$ is contained in a connected component of
$f^{-m-n_{0}}(D(x_{0},r_{0}))$, so there exists $x_{m}\in D_{m}$ and a
relevant inverse branch of $f^{m+n_{0}}$

$F_{m+n_{0},x_{0},x_{m}}\colon f^{n_{0}}(D(x,r))\subset
D(x_{0},r_{0})\longrightarrow D_{m}$

which is well-defined, and hence conformal. Then,

$F_{m,x,x_{m}}=F_{m+n_{0},x_{m},x_{0}}\circ f^{m}\colon D(x,r)\longrightarrow
D_{m}$

is a well-defined inverse branch of $f^{m}$. In particular, $f^{m}|_{D_{m}}$
is conformal, proving the claim. $\square$

  2. 2.

Construction of an expansive metric around $\partial U$.

######  Lemma.

There exists a hyperbolic open set $W\subset\mathbb{C}$ and a measurable set
$X\subset W$ for which the following are satisfied.

    1. (2.1)

The set $X$ is contained in ${\partial}U$, and it has full
$\omega_{U}$-measure.

    2. (2.2)

For all $x\in X$ and $n\geq 0$, there exists $r_{W}\coloneqq r_{W}(x)>0$ such
that the hyperbolic disk $D_{W}(x,r_{W})$ is simply connected and compactly
contained in $W$, with all relevant branches $F_{n}$ of $f^{-n}$ at $x$ well-
defined in $D_{W}(x,r_{W})$. Moreover, $F_{n}(D_{W}(x,r_{W}))\subset W$.

    3. (2.3)

Relevant inverse branches do not increase the hyperbolic distance $\textrm{\em
dist}_{W}$ between points, i.e. for any $x\in W$ and $F_{1}$ relevant branch
of $f^{-1}$ at $x$, if $z,w\in D_{W}(x,r_{W})$,

$\textrm{\em dist}_{W}(F_{1}(z),F_{1}(w))\leq\textrm{\em dist}_{W}(z,w).$

###### Proof.

For all $x\in\partial U$, let $r_{n}(x)\in\left[0,+\infty\right)$ be the
radius of the maximal Euclidean disk $D(x,r_{n}(x))$ for which all relevant
branches of $f^{-n}$ at $x$ are well-defined. We assume at least one such
inverse branch exists, otherwise set $r_{n}(x)=0$ (by the previous claim, this
situation only happens on a set of zero $\omega_{U}$-measure). Clearly,
$r_{n}(x)\geq r_{n+1}(x)$, so $\left\\{r_{n}(x)\right\\}_{n}$ is a convergent
sequence for all $x\in\partial U$. Consider

$X\coloneqq\left\\{x\in\partial U\colon r_{n}(x)\to r(x)>0\right\\}.$

By the claim in the first step, $\omega_{U}(X)=1$. Let

$W\coloneqq\bigcup\limits_{x\in X}\bigcup\limits_{n\geq
0}\left\\{F_{n}(D(x,r(x)))\colon F_{n}\textrm{ is relevant at }x\right\\}.$

Note that $W$ is open, and $X\subset W$. Hence, 2(2.1) holds.

Taking $r(x)>0$ smaller if needed, we can assume $W$ omits at least two
points, so it is hyperbolic and admits a hyperbolic metric on it.

Note that $W$ may be disconnected. In this case, the hyperbolic density is
defined on each connected component separatedly. Indeed, each connected
component $W_{1}$ of $W$ is a hyperbolic domain, and hence admits a hyperbolic
density $\rho_{W_{1}}$. Given $z\in W$, we define

$\rho_{W}(z)\coloneqq\rho_{W_{1}}(z),$

where $W_{1}$ stands for the connected component of $W$ with $z\in W_{1}$.
Given $z,w\in W$, the hyperbolic distance is defined as
$\textrm{dist}_{W}(z,w)=\textrm{dist}_{W_{1}}(z,w)$, if $z$ and $w$ lie in the
same connected component $W_{1}$ of $W$; and $\textrm{dist}_{W}(z,w)=\infty$,
otherwise.

By construction, it holds that, for every $x\in X$, all relevant branches of
$f^{-n}$ are well-defined in $D(x,r(x))\subset W$. Since the Euclidean and the
hyperbolic metrics are locally equivalent, there exists $r_{W}\coloneqq
r_{W}(x)$ such that $\overline{D_{W}(x,r_{W})}\subset{D(x,r)}$. Hence,
$D_{W}(x,r_{W})$ is simply connected and compactly contained in $W$, and all
relevant inverse branches $F_{n}$ are well-defined in $D_{W}(x,r_{W})$, and
$F_{n}(D_{W}(x,r_{W}))\subset F_{n}(D(x,r))\subset W$. Thus, 2(2.2) holds.

Finally, we claim that $f$ does not increase the hyperbolic distance between
points. Consider

$W^{\prime}\coloneqq\bigcup\limits_{x\in X}\bigcup\limits_{n\geq
1}\left\\{F_{n}(D(x,r(x)))\colon F_{n}\textrm{ is relevant at
}x\right\\}\subset W.$

Consider the hyperbolic density $\rho_{W^{\prime}}$ in $W^{\prime}$, defined
component by component if needed. Note that each connected component $W_{1}$
of $W^{\prime}$ is mapped onto a connected component $W_{2}$ of $W$ as a
holomorphic covering. Hence, if $x\in W_{1}\subset W^{\prime}$ and $f(x)\in
W_{2}\subset W$, then

$\rho_{W^{\prime}}(x)=\rho_{W_{1}}(x)=\rho_{W_{2}}(f(x))\cdot\left|f^{\prime}(x)\right|=\rho_{W}(f(x))\cdot\left|f^{\prime}(x)\right|.$

Since each connected component of $W^{\prime}$ is contained in a connected
component of $W$, we have $\rho_{W}\leq\rho_{W^{\prime}}$ and, in particular,
if $x\in W^{\prime}$,

$\rho_{W}(x)\leq\rho_{W}(f(x))\cdot\left|f^{\prime}(x)\right|.$

Now, let $x\in W$ and let $F_{1}$ be a relevant branch of $f^{-1}$ at $x$.
Since $F_{1}$ is well-defined in $D_{W}(x,r_{W})$, it holds

$\rho_{W}(F_{1}(z))\left|F^{\prime}_{1}(z)\right|\leq\rho_{W}(z),$

for all $z\in D_{W}(x,r_{W})$. Next, take $z,w\in D_{W}(x,r_{W})$. Note that
$z,w\in W_{1}$, and $F_{1}(z),F_{1}(w)\in W_{2}$, for two connected components
$W_{1},W_{2}$ of $W$. Moreover, since hyperbolic disks are hyperbolically
convex (i.e. a geodesic joining two points in the disk is contained in the
disk), we can take $\gamma\subset D_{W}(x,r_{W})$ geodesic between $z$ and
$w$. Then,

$\textrm{dist}_{W}(F_{1}(z),F_{1}(w))=\textrm{dist}_{W_{2}}(F_{1}(z),F_{1}(w))\leq\int_{F_{1}(\gamma)}\rho_{W}(t)dt=$
$=\int_{\gamma}\rho_{W}(F_{1}(t))\left|F_{1}^{\prime}(t)\right|dt\leq\int_{\gamma}\rho_{W}(t)dt=\textrm{dist}_{W_{1}}(z,w)=\textrm{dist}_{W}(z,w).$

proving the claim. $\square$

  3. 3.

Control of radial limits in terms of Stolz angles.

Let us fix $\alpha\in(0,\frac{\pi}{2})$, and let $p\in\overline{\mathbb{D}}$
be the Denjoy-Wolff point of the associated inner function $g$. It follows
from Theorem 4.6 that, $\lambda$-almost every $\xi\in\partial\mathbb{D}$,
there exists $\rho\coloneqq\rho(\xi)>0$ such that:

    1. (3.1)

for all $n\geq 0$, every branch $G_{n}$ of $g^{-n}$ is well-defined in
$D(\xi,\rho)$;

    2. (3.2)

there exists $\rho_{1}\coloneqq\rho_{1}(\xi)$ such that, for all $n\geq 0$,

$G_{n}(R_{\rho_{1}},p)\subset\Delta_{\alpha,\rho_{1}}(G_{n}(\xi),p),$

where $R_{\rho}(\cdot,p)$ and $\Delta_{\alpha,\rho}(\cdot,p)$ stand for the
radial segment and the Stolz angle with respect to $p$ (Def. 2.10).

In the sequel, to lighten the notation we denote the radial segments and the
Stolz angles just by $R_{\rho}$ and $\Delta_{\rho}$, respectively. However,
one should keep in mind that they are radial segments and Stolz angles with
respect to the Denjoy-Wolff point, and that the opening $\alpha$ of the Stolz
angles is fixed throughtout the proof.

  4. 4.

Choice of a set $K_{\varepsilon}\subset\partial\mathbb{D}$, where bounds on
$\varphi$ and $G_{n}$ are uniform.

######  Lemma.

Fix $\varepsilon>0$. There exists a measurable set
$K_{\varepsilon}\subset\partial\mathbb{D}$ with $\lambda(K_{\varepsilon})\geq
1-\varepsilon$ such that the following holds.

    1. (4.1)

For all $\xi\in K_{\varepsilon}$, $\varphi^{*}(\xi)$ exists and
$\varphi^{*}(\xi)\in X$. Moreover, $(g^{n})^{*}(\xi)\in\Theta_{\Omega}$, for
all $n\geq 0$.

    2. (4.2)

There exists $r_{\varepsilon}>0$ such that for all $\xi\in K_{\varepsilon}$
and $n\geq 0$, all relevant inverse branches of $f^{n}$ are well-defined in
$D_{W}(\varphi^{*}(\xi),r_{\varepsilon})$.

    3. (4.3)

There exist $\rho_{\varepsilon}>0$ such that

      1. i.

For every $\xi\in K_{\varepsilon}$ and $n\geq 0$, every branch $G_{n}$ of
$g^{-n}$ is well-defined in $D(\xi,\rho_{\epsilon})$.

      2. ii.

For every $\xi\in K_{\varepsilon}$,

$G_{n}(R_{\rho_{\varepsilon}}(\xi))\subset\Delta_{\rho_{\varepsilon}}(G_{n}(\xi)).$

      3. iii.

For every $\xi\in K_{\varepsilon}$, if $z\in\Delta_{\rho_{\varepsilon}}(\xi)$,
then $\varphi(z)\in W$ and

$\textrm{\em
dist}_{W}(\varphi(z),\varphi^{*}(\xi))<\frac{r_{\varepsilon}}{3}.$

    4. (4.4)

There are no isolated points in $K_{\varepsilon}$. In fact, for every $\xi\in
K_{\varepsilon}$, there exists a subsequence $\left\\{n_{k}\coloneqq
n_{k}(\xi)\right\\}_{k}$, $n_{k}\to\infty$, such that $(g^{n_{k}})^{*}(\xi)\in
K_{\varepsilon}$ and $(g^{n_{k}})^{*}(\xi)\to\xi$.

    5. (4.5)

For every $\xi\in K_{\varepsilon}$, the orbit of $\varphi^{*}(\xi)$ under $f$
is dense in $\partial U$.

###### Proof.

Consider $K\coloneqq(\varphi^{*})^{-1}(X)\subset\partial\mathbb{D}$. Observe
that $\lambda(K)=1$. There is no loss of generality on assuming that
$(g^{n})^{*}(\xi)\in\Theta_{\Omega}$, for all $n\geq 0$, since this holds
$\lambda$-almost everywhere. Hence, all points in $K$ satisfy 4(4.1).

Next, by 2(2.2), for all $\xi\in K$, there exists $r\coloneqq r(\xi)>0$, such
that all relevant inverse branches $F_{n}$ are well-defined in
$D_{W}(\varphi^{*}(\xi),r(\xi))$. Hence, we can write $K$ as the countable
union of the nested measurable sets

$K_{m}\coloneqq\left\\{\xi\in K\colon\textrm{ all relevant }F_{n}\textrm{ are
well-defined in }D_{W}(\varphi^{*}(\xi),1/m)\right\\}\subset K_{m+1}.$

Choose $m_{0}$ such that $\lambda(K_{m_{0}})\geq 1-\varepsilon/3$, which
satisfies 4(4.2) with $r_{\varepsilon}=1/m_{0}$.

Now, by 3 (3.1) and 3 (3.2), we can assume that, for all $\xi\in K_{m_{0}}$,
there exists $\rho\coloneqq\rho(\xi)>0$ such that $G_{n}$ is well-defined in
$D(\xi,\rho)$ and $G_{n}(R_{\rho})\subset\Delta_{\rho}(G_{n}(\xi))$. Hence,
$K_{m_{0}}$ can be written as the countable union of the nested measurable
sets

$K_{m_{0}}^{k}\coloneqq\left\\{\xi\in K_{m_{0}}\colon G_{n}\textrm{ well-
defined in }D(\xi,1/k)\textrm{ and
}G_{n}(R_{1/k})\subset\Delta_{\frac{1}{k}}(G_{n}(\xi))\right\\}.$

Choose $k_{0}$ such that $\lambda(K^{k_{0}}_{m_{0}})\geq 1-\varepsilon/2$.
Finally, note that the angular limit of $\varphi$ exists at every $\xi\in
K^{k_{0}}_{m_{0}}$, that is, for all $\xi\in K^{k_{0}}_{m_{0}}$ there exists
$\rho_{1}(\xi)<\rho(\xi)$ such that, for all $z\in\Delta_{\rho_{1}}$,

$\textrm{dist}_{W}(\varphi(z),\varphi^{*}(\xi))<\frac{r_{\varepsilon}}{3}.$

Hence, proceeding as before, we find $K_{\varepsilon}\subset
K^{k_{0}}_{m_{0}}$, with $\lambda(K_{\varepsilon})\geq 1-\varepsilon$ and
satisfying 4(4.3) for some $\rho_{\varepsilon}>0$, uniform for $\xi\in
K_{\varepsilon}$.

Since $\lambda$-almost every point in $K_{\varepsilon}$ is a Lebesgue density
point (Def. 2.7), there is no loss of generality on assuming that every
$\xi\in K_{\varepsilon}$ is a Lebesgue density point. In particular, there are
no isolated points in $K_{\varepsilon}$.

Moreover, since $g^{*}|_{\partial\mathbb{D}}$ is recurrent, every measurable
set $E\subset X$ with $\lambda(E)>0$, we have that for $\lambda$-almost every
point $\xi\in K_{\varepsilon}$, there exists a subsequence
$\left\\{n_{k}\right\\}_{k}$, $n_{k}\to\infty$, with $(g^{n_{k}})^{*}(\xi)\in
E$.

Now, take a countable sequence $\left\\{\xi_{n}\right\\}_{n}\subset
K_{\varepsilon}$, such that $\left\\{\xi_{n}\right\\}_{n}$ is dense in
$K_{\varepsilon}$ and each $\xi_{n}$ is a Lebesgue density point for
$K_{\varepsilon}$. Consider $E_{j,n}\coloneqq D(\xi_{n},1/j)\cap
K_{\varepsilon}$, for $j,n\geq 1$. Since each $\xi_{n}$ is a Lebesgue density
point for $K_{\varepsilon}$, $\lambda(E_{j,n})>0$, for all $j,n\geq 1$.

Applying the previous property to the sequence
$\left\\{E_{j,n}\right\\}_{j,n}$, we have that, for $\lambda$-almost every
$\xi\in K_{\varepsilon}$, there exists a subsequence
$\left\\{n_{k}\right\\}_{k}$, $n_{k}\to\infty$, with $(g^{n_{k}})^{*}(\xi)\in
E_{k,k}$. Hence, there exists a subsequence $\left\\{n_{k}\right\\}_{k}$,
$n_{k}\to\infty$, with $(g^{n_{k}})^{*}(\xi)\in K_{\varepsilon}$,
$(g^{n_{k}})^{*}(\xi)\to\xi$, proving 4(4.4).

Finally, since points in $\partial U$ with dense orbit have full harmonic
measure, we can assume that $K_{\varepsilon}$ is chosen so that 4(4.5) holds.
$\square$

  5. 5.

Construction of a periodic point in $D_{W}(\varphi^{*}(\xi),r)$, for all
$\xi\in K_{\varepsilon}$ and $r\in(0,r_{\varepsilon})$.

Set $\xi\in K_{\varepsilon}$ and $r\in(0,r_{\varepsilon})$. The goal in this
section is to find a periodic point in $D_{W}(\varphi^{*}(\xi),r)\cap\partial
U$.

Write $\xi_{n}\coloneqq(g^{n})^{*}(\xi)$, for all $n\geq 0$. By 4(4.1),
$\varphi^{*}(\xi_{n})$ exists and belongs to $\Omega$, for all $n\geq 0$. By
4(4.4) and 4(4.2), $\xi_{n}\in K_{\varepsilon}$ infinitely often, and all
relevant inverse branches are well-defined in
$D_{W}(\varphi^{*}(\xi_{n}),r_{\varepsilon})$. In particular, for all $0\leq
m<n$, there is a relevant inverse branch $F_{n-m}$ of $f^{n-m}$ in
$D(\varphi^{*}(\xi_{n}),r_{\varepsilon})$. Hence, for all $n\geq 0$, $f$ maps
conformally a neighbourhood of $\varphi^{*}(\xi_{n})$ onto a neighbourhood of
$\varphi^{*}(\xi_{n+1})$. This implies that, although we are considering
$f\in\mathbb{K}$, in practice, everywhere where we consider $f$, it is
holomorphic (and, in fact, conformal).

Now, consider $D_{W}(\varphi^{*}(\xi_{0}),r_{\varepsilon})$, and let $W_{1}$
be the connected component of $W$ such that $\varphi^{*}(\xi_{0})\in W_{1}$.
Note that, by 4(4.2), all relevant branches of $f^{-n}$ at
$\varphi^{*}(\xi_{0})$ are well-defined in
$D_{W}(\varphi^{*}(\xi_{0}),r_{\varepsilon})$. In particular, there exists
$n_{0}\geq 1$ and a relevant inverse branch of $f^{n_{0}}$, say $F_{n_{0}}$,
such that $F_{n_{0}}(\varphi^{*}(\xi_{0}))\in W_{1}$ (recall that preimages of
any point are dense in the Julia set, with at most two exceptions). Consider
$D_{n_{0}}\coloneqq F_{n_{0}}(D_{W}(\varphi^{*}(\xi_{0}),r_{\varepsilon}))$.
Therefore,

$F_{n_{0}}\colon D_{W}(\varphi^{*}(\xi_{0}),r_{\varepsilon})\longrightarrow
D_{n_{0}}$

conformally.

######  Claim.

There exists $k\in(0,1)$ such that, for all $z,w\in
D_{W}(\varphi^{*}(\xi_{0}),r_{\varepsilon})$,

$\textrm{\em dist}_{W}(F_{n_{0}}(z),F_{n_{0}}(w))\leq k\cdot\textrm{\em
dist}_{W}(z,w).$

###### Proof.

Let $W_{1}$ be the connected component of $W$ in which $\varphi^{*}(\xi_{0})$
lies. Consider, as in the proof of 2(2.3),

$W^{\prime}\coloneqq\bigcup\limits_{x\in X\cap W_{1}}\bigcup\limits_{n\geq
1}\left\\{F_{n}(D(x,r(x)))\colon F_{n}\textrm{ is relevant at
}x\right\\}\subset W.$

Let $W^{\prime}_{1}$ be the connected component of $W^{\prime}$ that contains
$F_{n_{0}}(\varphi^{*}(\xi_{0}))$. Then, $W^{\prime}_{1}\subset W_{1}$, and

$f^{n_{0}}\colon W^{\prime}_{1}\to W_{1}$

is a holomorphic covering. Then, for $x\in W^{\prime}_{1}\subset W_{1}$, it
holds

$\rho_{W_{1}^{\prime}}(x)|=\rho_{W_{1}}(f^{n_{0}}(x))\cdot\left|(f^{n_{0}})^{\prime}(x)\right|.$

Note that the inclusion $W^{\prime}_{1}\subset W_{1}$ is strict (otherwise
$f^{n_{0}}(W_{1})=W_{1}$, and this is impossible since $W_{1}$ contains points
of $\mathcal{J}(f)$). Hence, $\rho_{W_{1}}<\rho_{W^{\prime}_{1}}$, so

$\rho_{W_{1}}(x)|<\rho_{W_{1}}(f^{n_{0}}(x))\cdot\left|(f^{n_{0}})^{\prime}(x)\right|.$

Moreover, since $D_{W}(\varphi^{*}(\xi_{0}),r_{W})$ is compactly contained in
$W$, there exists $k\in(0,1)$ such that, for all $x\in
D_{W}(\varphi^{*}(\xi_{0}),r_{W})$,

$\rho_{W_{1}}(F_{n_{0}}(x))\cdot\left|F_{n_{0}}^{\prime}(x)\right|\leq
k\cdot\rho_{W_{1}}(x).$

Finally, for $z,w\in D_{W}(\varphi^{*}(\xi_{0}),r_{W})$, take $\gamma\subset
D_{W}(\varphi^{*}(\xi_{0}),r_{W})$ geodesic between $z$ and $w$, and

$\textrm{dist}_{W}(F_{n_{0}}(z),F_{n_{0}}(w))=\textrm{dist}_{W_{1}}(F_{n_{0}}(z),F_{n_{0}}(w))\leq\int_{F_{n_{0}}(\gamma)}\rho_{W}(t)dt=$
$=\int_{\gamma}\rho_{W}(F_{n_{0}}(t))\left|F_{n_{0}}^{\prime}(t)\right|dt\leq
k\int_{\gamma}\rho_{W}(t)dt=k\textrm{ dist}_{W_{1}}(z,w)=k\textrm{
dist}_{W}(z,w),$

proving the claim. $\square$

Now, we claim that we can find $N\geq 1$ satisfying the following properties.

    1. (5.1)

If $N_{0}\coloneqq\\#\left\\{n\leq N\colon\varphi^{*}(\xi_{n})\in
D_{n_{0}}\right\\}$, then $k^{N_{0}}<\dfrac{r}{3r_{\varepsilon}}$.

    2. (5.2)

$\xi_{N}\coloneqq(g^{N})^{*}(\xi_{0})\in K_{\varepsilon}$

    3. (5.3)

There exists $t_{N}\in(0,1)$ such that $t_{N}\xi_{N}\in
R_{\rho_{\varepsilon}}(\xi_{N})\cap\Delta_{\rho_{\varepsilon}}(\xi_{0})$.

Indeed, by 4(4.5), the orbit of $\varphi^{*}(\xi_{0})$ is dense in $\partial
U$. In particular, it visits $D_{n_{0}}$ infinitely many times. Hence, there
exists $N^{\prime}$ so that 5 (5.1) is satisfied for $N^{\prime}$. By 4(4.4),
there exists a subsequence $\left\\{n_{k}\right\\}_{k}$, $n_{k}\to\infty$,
such that $\xi_{n_{k}}\in K_{\varepsilon}$ and $\xi_{n_{k}}\to\xi_{0}$. Thus,
we can find $N\geq N^{\prime}$ for which conditions 5 (5.2) and 5 (5.3) are
also satisfied. Note that the geometric condition in 5 (5.3) is satisfied as
long $\xi_{N}$ is close enough to $\xi_{0}$, since the radius
$\rho_{\varepsilon}$ and the angle $\alpha$ are uniform (see Fig. 6.2 for a
geometric intuition).

$\xi_{N}$$t_{N}\xi_{N}$$G_{N}(t_{N}\xi_{N})$$R(\xi_{N})$$\Delta_{\rho_{\varepsilon}}(\xi_{0})$$\partial\mathbb{D}$$\xi_{0}$

Figure 6.2: The choice of the point $t_{N}\xi_{N}\in
R_{\rho_{\varepsilon}}(\xi_{N})\cap\Delta_{\rho_{\varepsilon}}(\xi_{0})$.

######  Claim.

There exists a relevant inverse branch $F_{N}$ of $f^{N}$ at
$\varphi^{*}(\xi_{N})$ defined in
$D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon})$, which satisfies
$F_{N}(\varphi^{*}(\xi_{N}))=\varphi^{*}(\xi_{0})$ and

$F_{N}(D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon}))\subset
D_{W}\left(\varphi^{*}(\xi_{0}),\frac{r}{3}\right)\subset
D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon}).$

We note that, in particular,
$F_{N}(D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon}))\subset
D_{W}\left(\varphi^{*}(\xi_{0}),r\right)$.

###### Proof.

First note that $\xi_{N}\in K_{\varepsilon}$ 5 (5.2), so all relevant inverse
branches are well-defined in $D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon})$
4(4.2). Since $\xi_{N}=(g^{N})^{*}(\xi_{0})$, by Lemma 5.5, we have
$f^{N}(\varphi^{*}(\xi_{0}))=\varphi^{*}(\xi_{N})$. Hence, there exists a
relevant inverse branch $F_{N}$ of $f^{N}$ at $\varphi^{*}(\xi_{N})$ defined
in $D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon})$, which satisfies
$F_{N}(\varphi^{*}(\xi_{N}))=\varphi^{*}(\xi_{0})$.

Note that $F_{N}$ is the composition of different inverse branches of $f$, and
each of them does not increase the hyperbolic distance $\textrm{dist}_{W}$
between points 2(2.3). Moreover, $\left\\{f^{n}(\xi_{0})\right\\}_{n=0}^{N}$
visits $D_{n_{0}}$ at least $N_{0}$ times 5 (5.1). This means that applying
$F_{N}$ corresponds to applying the inverse $F_{n_{0}}$, which acts as a
contraction by $k$, at least $N_{0}$ times. Thus, we have

$F_{N}(D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon}))\subset
D_{W}\left(\varphi^{*}(\xi_{0}),k^{N_{0}}r_{\varepsilon}\right)\subset
D_{W}\left(\varphi^{*}(\xi_{0}),\frac{r}{3}\right).$

To see the remaining inclusion, note that, by 4(4.3) and 5 (5.1), we have

$\varphi(t_{N}\xi_{N})\in
D_{W}\left(\varphi^{*}(\xi_{0}),\frac{r_{\varepsilon}}{3}\right)\cap
D_{W}\left(\varphi^{*}(\xi_{N}),\frac{r_{\varepsilon}}{3}\right).$

Hence, applying the triangle inequality,

$\textrm{dist}_{W}(\varphi^{*}(\xi_{0}),\varphi^{*}(\xi_{N}))\leq\textrm{dist}_{W}(\varphi^{*}(\xi_{0}),\varphi(t_{N}\xi_{N}))+\textrm{dist}_{W}(\varphi(t_{N}\xi_{N}),\varphi^{*}(\xi_{N}))<\frac{2r_{\varepsilon}}{3},$

implying the desired inclusion. $\square$

######  Claim.

The map

$F_{N}\colon D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon})\longrightarrow
D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon})$

has an attracting fixed point in $D_{W}(\varphi^{*}(\xi_{0}),r)$, which is
accessible from $U$. Hence, $f$ has a repelling $N$-periodic point in
$D_{W}\left(\varphi^{*}(\xi_{0}),r\right)\cap\partial U$.

###### Proof.

Since $F_{N}(D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon}))\subset
D_{W}\left(\varphi^{*}(\xi_{0}),r\right)$, by the Denjoy-Wolff Theorem,
$F_{n}$ as a fixed point $p\in D_{W}(\varphi^{*}(\xi_{0}),r)$, which attracts
all points in $D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon})$ under the
iteration of $F_{n}$. Hence it is repelling under $f^{n}$ and thus belongs to
$\mathcal{J}(f)$.

It is left to see that $p$ is accessible from $U$. To do so, first note that,
by 4(4.3), there exists a branch $G_{N}$ of $g^{-N}$ such that $G_{N}$ is
well-defined in $D(\xi_{N},\rho_{\varepsilon})$ and $G_{N}(\xi_{N})=\xi_{0}$.
It holds that $\varphi\circ G_{N}=F_{N}\circ\varphi$ in
$\Delta_{\rho_{\varepsilon}}(\xi_{N})$. Moreover,
$G_{N}(R_{\rho_{\varepsilon}}(\xi_{N}))\subset\Delta_{\rho_{\varepsilon}}(\xi_{0})$.
In particular, $G_{N}(t_{N}\xi_{N})\in\Delta_{\rho_{\varepsilon}}(\xi_{0})$.

Since $t_{N}\xi_{N}\in\Delta_{\rho_{\varepsilon}}(\xi_{0})$, we can find a
curve $\gamma\subset\Delta_{\rho_{\varepsilon}}(\xi_{0})$ joining
$t_{N}\xi_{N}$ and $G_{N}(t_{N}\xi_{N})$. Then, $\varphi(\gamma)\subset
D_{W}(\varphi^{*}(\xi_{N}),r_{\varepsilon})$ joins $\varphi(t_{N}\xi_{N})$
with $F_{N}(\varphi(t_{N}\xi_{N}))$. Define

$\Gamma\coloneqq\bigcup\limits_{m\geq 0}F^{m}_{N}(\gamma).$

Then, $\Gamma\subset\partial U$ lands at $p$, proving the claim. $\square$

$\xi_{N}$$t_{N}\xi_{N}$$G_{N}(t_{N}\xi_{N})$$G_{N}$$F_{N}$$\gamma$$\varphi$$\partial\mathbb{D}$$\xi_{0}$$\varphi(\gamma)$$\varphi^{*}(\xi_{N})$$\varphi^{*}(\xi_{0})$$\varepsilon$

Figure 6.3: The construction of the curve $\gamma$ in $\mathbb{D}$, and its
image $\varphi(\gamma)$ in the dynamical plane.

  6. 6.

Periodic points are dense in $\partial U$.

Finally, to see that the previous construction leads to density of periodic
points in $\partial U$, one should take into account that $\textrm{supp
}\omega_{U}=\widehat{\partial}U$ (Lemma 2.18). Hence, for all $x\in\partial U$
and $\delta>0$, it holds $\omega_{U}(D(x,\delta))>0$. Take
$\varepsilon\coloneqq\omega_{U}(D(x,\delta))/2$, and consider
$K_{\varepsilon}$ as before. Note that, by the choice of $\varepsilon$, we
have $\omega_{U}(D(x,\delta)\cap\varphi^{*}(K_{\varepsilon}))>0$.

Let $\xi\in K_{\varepsilon}$ be such that $\varphi^{*}(\xi)\in D(x,\delta)$,
and let $r\leq r_{\varepsilon}$. In the previous step, we proved the existence
of a periodic point in $D_{W}(\varphi^{*}(\xi),r)$. Taking $r$ small enough,
since $\varphi^{*}(\xi)\in D(x,\delta)$, we can ensure that the periodic point
is in $D(x,\delta)$.

This ends the proof of Theorem 6.1.

Conflicts of interest. The author states that there is no conflict of
interest.

Data availability. Data sharing not applicable to this article as no datasets
were generated or analysed during the current study.

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