# Classifying multiply connected wandering domains

Gustavo R. Ferreira  Centre de Recerca Matemàtica, Barcelona, Spain
<EMAIL_ADDRESS>and  Lasse Rempe  University of Liverpool, Liverpool, UK

###### Abstract.

We study the internal dynamics of multiply connected wandering domains of
meromorphic functions. We do so by considering the sequence of injectivity
radii along the orbit of a base point, together with the hyperbolic
distortions along the same orbit. The latter sequence had been previously used
in this context; the former introduces geometric information about the shape
of the wandering domains that interacts with the dynamic information given by
the hyperbolic distortions. Using this idea, we complete the description of
the internal dynamics of any wandering domain of a meromorphic function, and
also unify previous approaches to the question. We conclude that the internal
dynamics of a wandering domain, from the point of view of hyperbolic geometry,
can be classified into six different types. Five of these types are realised
by wandering domains of entire functions, while the sixth can only arise for
meromorphic functions: a locally but not globally eventually isometric
wandering domain. We construct a meromorphic function with such a domain,
demonstrating that this new phenomenon does in fact occur. Our results show
that, on a local level, the dynamics of multiply connected wandering domains
of meromorphic functions is similar to those of entire functions, although new
global phenomena can arise for non-entire functions.

###### 2020 Mathematics Subject Classification:

37F10 (primary); 30D05, 30F45 (secondary)

##  1\. Introduction

Let $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a meromorphic function,
where $\widehat{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ denotes the Riemann
sphere. The Fatou set of $f$, denoted $F(f)$, is defined as the set of points
$z\in\mathbb{C}$ such that, on some neighbourhood of $z$, the iterates
$(f^{n})_{n\in\mathbb{N}}$ are defined and form a normal family in the sense
of Montel. In other words, it is the set where the dynamics is not sensitive
to initial conditions – nearby points will have similar long-term behaviour.
The complement of the Fatou set, the Julia set $J(f)$, is the locus of
“chaotic” dynamics. Both sets are completely invariant under $f$, meaning that
$f^{-1}(F(f))\subset F(f)$ and $f(F(f))\subset F(f)$, and analogously for the
Julia set (with special care taken for the point at infinity). In particular,
if $U$ is a connected component of the Fatou set (henceforth a Fatou
component), then $f^{n}(U)$ is, for all $n\geq 0$, contained in some Fatou
component $U_{n}$ (clearly, $U_{0}=U$). If $U_{n}\neq U_{m}$ for all $n\neq
m$, we say that $U$ is a wandering domain.

The dynamical behaviour of wandering domains of meromorphic functions is an
area of intense research. Recent efforts [BEF+22, Fer22, Fer24] to classify
their internal dynamics have focused on the behaviour of the hyperbolic metric
– something intrinsic to the domains themselves. More precisely, let
$f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a meromorphic function with a
wandering domain $U$, and let $z$ and $w$ be distinct points in $U$. The
primary question of this line of research has been: what is the long-term
behaviour of the sequence

$\left(d_{U_{n}}\left(f^{n}(z),f^{n}(w)\right)\right)_{n\in\mathbb{N}},$ (1)

where $d_{\Omega}$ denotes the hyperbolic distance in the domain
$\Omega\subset\widehat{\mathbb{C}}$? By the Schwarz–Pick lemma (see Lemma
2.2), this sequence of non-negative numbers is non-increasing, and hence tends
to a limit $c(z,w)$. The next questions, then, are whether this limit is zero
or positive, and whether it is achieved. These were analysed by Benini et al.
[BEF+22] in the case when all the domains $U_{n}$ are simply connected, and by
the first author [Fer22, Fer24] for multiply connected wandering domains
falling in various cases. The purpose of this paper is to provide a framework
that both unifies the existing progress and enables us to tackle previously
unaddressed types of wandering domains.

Let $U$ be a wandering domain of a meromorphic function $f$, and let $z\in U$.
The sequence of hyperbolic distortions of $f$ along the orbit of $z$ is the
sequence $(\lambda_{n}(z))_{n=0}^{\infty}$, where

$\lambda_{n}(z)=\|Df\left(f^{n-1}(z)\right)\|_{U_{n-1}}^{U_{n}}.$

Here $\|Df\left(f^{n-1}(z)\right)\|_{U_{n-1}}^{U_{n}}$ denotes the hyperbolic
distortion of $f\colon U_{n-1}\to U_{n}$ at $f^{n-1}(z)$. This object was
first introduced in [BEF+22] in the context of simply connected wandering
domains where it plays an important role in the study of internal dynamics. It
will be important for us as well, and we start by showing that it behaves in
the same way at all points in the domain. This claim can be deduced from the
results of [BEF+22] and [Fer24], but we give here a proof of a stronger result
that does not rely on previous knowledge of internal dynamics of wandering
domains.

######  Proposition 1.1.

Let $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a meromorphic function with
a wandering domain $U$. Then, for any $z,w\in U$,

$e^{-2d_{U}(z,w)}(1-\lambda_{n}(w))\leq 1-\lambda_{n}(z)\leq
e^{2d_{U}(z,w)}(1-\lambda_{n}(w)).$

In particular, whether the series

$\sum_{n\geq 1}(1-\lambda_{n}(z))$

converges, diverges, or has a finite number of non-zero terms does not depend
on the choice of $z\in U$.

This means that every wandering domain exhibits one of three types of
behaviour:

######  Definition 1.2.

Let $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a meromorphic function with
a wandering domain $U$.

  1. (i)

If $\sum_{n\geq 1}\left(1-\lambda_{n}(z)\right)=+\infty$ for all $z\in U$, we
say that $U$ is _infinitesimally contracting_.

  2. (ii)

If $\sum_{n\geq 1}\left(1-\lambda_{n}(z)\right)<+\infty$ and
$\lambda_{n}(z)<1$ for infinitely many values of $n$ for every $z\in U$, we
say that $U$ is _infinitesimally semi-contracting_.

  3. (iii)

If there exists $N\in\mathbb{N}$ such that, for $n\geq N$, $\lambda_{n}(z)=1$
for every $z\in U$, we say that $U$ is _infinitesimally eventually isometric_.

Let us take a moment to discuss the significance of the series

$\sum_{n\geq 1}\left(1-\lambda_{n}(z)\right).$ (2)

For any $z\in U$ and $k\in\mathbb{N}$, the chain rule implies that

$\|Df^{k}(z)\|_{U}^{U_{k}}=\prod_{n=1}^{k}\|Df\left(f^{n-1}(z)\right)\|_{U_{n-1}}^{U_{n}}.$
(3)

Since the hyperbolic distortion is always in the interval $[0,1]$, the product
on the right-hand side necessarily converges to a non-negative number as
$k\to+\infty$. If $f^{n}(z)$ is not a critical point of $f$ for any $n\geq 0$,
then this limit is non-zero if and only if the sum $\sum_{n\geq
1}\left(1-\lambda_{n}(z)\right)$ is convergent. So the classification in
Definition 1.2 is really about the behaviour of the hyperbolic derivative
$\|Df^{k}(z)\|_{U}^{U_{k}}$ for all but a countable set of $z$: if $U$ is
infinitesimally contracting, this limit tends to $0$, if it is infinitesimally
eventually isometric, the sequence is eventually constant, and in the
infinitesimally semi-contracting case, it converges to, but is not eventually
equal to, a non-zero limit. Stating Proposition 1.1 and Definition 1.2 in
terms of the sum (2) allows us to state the definition without having to
exclude iterated preimages of critical points.

Using the terminology from Definition 1.2 we can restate a key result from
[BEF+22] as follows.

######  Theorem A ([BEF+22], Theorem A).

Let $f\colon\mathbb{C}\to\mathbb{C}$ be an entire function with a simply
connected wandering domain $U$. Define the set

$E\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{(z,w)\in U\times
U\colon\text{$f^{k}(z)=f^{k}(w)$ for some $k\in\mathbb{N}$}\\}.$

  1. (i)

If $U$ is infinitely contracting, then
$d_{U_{n}}\left(f^{n}(z),f^{n}(w)\right)\to 0$ for all $z$ and $w$ in $U$;

  2. (ii)

If $U$ is infinitesimally semi-contracting, then for all $(z,w)\in(U\times
U)\setminus E$, there is $c(z,w)>0$ such that
$d_{U_{n}}\left(f^{n}(z),f^{n}(w)\right)\to c(z,w)>0$ but
$d_{U_{n}}\left(f^{n}(z),f^{n}(w)\right)\neq c(z,w)$ for all $n$;

  3. (iii)

If $U$ is infinitesimally eventually isometric, then there exists
$N\in\mathbb{N}$ such that, for all $(z,w)\in(U\times U)\setminus E$, there is
$c(z,w)>0$ with $d_{U_{n}}\left(f^{n}(z),f^{n}(w)\right)=c(z,w)$ for $n\geq
N$.

Benini et al. called the behaviours described in Theorem Ai-iii contracting,
semi-contracting, and eventually isometric (respectively). When we wish to
emphasise the distinction with the notions from Definition 1.2, we shall refer
to this as _globally_ contracting, semi-contracting, or eventually isometric
behaviour. Theorem A also holds for simply connected wandering domains of
meromorphic functions, as long as one assumes that all iterates of the domain
are simply connected. Thus, from a hyperbolic-geometric perspective, their
work completely describes the internal dynamics of simply connected wandering
domains as: “if a simply connected wandering domain $U$ has only simply
connected iterates and is infinitesimally contracting (resp. semi-contracting,
eventually isometric), then it is globally contracting (resp. semi-
contracting, eventually isometric)”.

Because of the complex interplay between the mapping properties of the
function and the topology and geometry of the domains, the above maxim fails
for multiply connected wandering domains. Indeed, it was observed in [Fer22]
that such a wandering domain may also be  _bimodal_ – i.e., there are pairs
$(z,w)\notin E$ such that the sequence (1) converges to $0$, while for others
it tends nontrivially to a non-zero limit – or _trimodal_ – the sequence may
exhibit all three of the behaviours listed in Theorem A. In order to provide a
unified explanation for the internal dynamics of wandering domains, we
introduce the following definition.

######  Definition 1.3.

Let $f$ be a meromorphic function with a wandering domain $U$. For $z\in U$,
let $\delta_{n}(z)$ denote the injectivity radius of $U_{n}$ at $f^{n}(z)$
(see Definition 2.5). We say that $U$ is essentially thin if there exists
$z_{0}\in U$ and a sequence $(n_{k})_{k\in\mathbb{N}}$ such that
$\delta_{n_{k}}(z_{0})\to 0$ as $k\to+\infty$. A wandering domain that is not
essentially thin is said to be essentially thick.

Let us reassure ourselves that this definition does not depend on the choice
of $z_{0}\in U$.

######  Proposition 1.4.

Let $f$ be a meromorphic function with a wandering domain $U$, and let
$z_{0}\in U$. If $(n_{k})_{k\geq 1}$ is a subsequence such that
$\delta_{n_{k}}(z_{0})\to 0$, then $\delta_{n_{k}}(z)\to 0$ for all $z\in U$.

Propositions 1.1 and 1.4 show that the qualitative behaviour of the sequences

$\left(\lambda_{n}(z)\right)_{n\in\mathbb{N}}$ and
$\left(\delta_{n}(z)\right)_{n\in\mathbb{N}}$

does not depend on our choice of $z\in U$. This behaviour gives us almost
complete information about the internal dynamics of the wandering domain, in
terms of the behaviour of the sequence (1). This is summarised in the
following theorem, which combines our main results with previous work. We say
that a wandering domain $U$ has _eventual connectivity_ $n$, where $1\leq
n\leq\infty$, if $U_{n}$ has connectivity $n$ for all sufficiently large $n$.
Note that, if $U$ has eventual connectivity $1$, then it is essentially thick
by definition; in this case, we also say that $U$ is _eventually simply
connected_.

Let us also say that $U$ is “locally eventually isometric” if condition iii of
Theorem A holds for every compact subset $K\subset U$. That is, there exists
$N=N(K)\in\mathbb{N}$ such that the sequence (1) is constant for $n\geq N$,
when $(z,w)\in K\times K\setminus E$. (It was shown in [Fer24, Theorem 1.1]
that it is equivalent to require only that the condition holds for _some_ $K$
with non-empty interior.)

######  Theorem 1.5.

Let $U$ be a wandering domain of a transcendental meromorphic function. Then
one of the following holds.

  1. (1)

$U$ is globally contracting. $U$ has eventual connectivity $1$, $\infty$, or
none.

  2. (2)

$U$ is essentially thick and globally semi-contracting. $U$ has eventual
connectivity $1$, $\infty$, or none.

  3. (3)

$U$ is essentially thin, infinitesimally semi-contracting and bimodal. $U$ has
eventual connectivity $\infty$.

  4. (4)

$U$ is essentially thick and globally eventually isometric. $U$ has finite or
infinite eventual connectivity.

  5. (5)

$U$ is essentially thick and locally, but not globally, eventually isometric.
$U$ has eventual connectivity $\infty$.

  6. (6)

$U$ is essentially thin, infinitesimally eventually isometric and trimodal.
$U$ has eventual connectivity $2$.

With the exception of 5, all possibilities listed above may be realised using
known constructions. If $f$ is entire, then $\delta_{n}(z_{0})\to 0$ when $U$
is multiply connected; in particular, 5 cannot occur for entire functions.
(All other cases do arise, subject to the restriction that all $U_{n}$ are
simply connected in cases 1, 2 and 4.)

It is natural to ask whether 5 can be realised in the meromorphic case. Our
next theorem shows that this is indeed the case.

######  Theorem 1.6.

There exists a transcendental meromorphic function $g$ with a wandering domain
that is locally eventually isometric but not globally eventually isometric.

### Singular foliations in essentially thin domains

For multiply connected wandering domains of entire functions, there exists an
absorbing sequence of round annuli $A_{n}\subset U_{n}$ such that any orbit in
$U$ is contained in $A_{n}$ for all sufficiently large $n$ [BRS13]. Moreover,
$\operatorname{mod}A_{n}\to+\infty$ as $n\to+\infty$. (Recall that the
_modulus_ of a topological annulus $A$ is the unique value of $M>0$ such that
$A$ is biholomorphic to $\\{z\colon 1<|z|<e^{2\pi M}\\}$, see [Hub06,
Proposition 3.2.1].) Furthermore, there exist dynamically defined singular
foliations that determine the long-term behaviour of the sequence (1). (See
[Fer22, pp. 1899–1900], and [Fer22, p. 1902] for the definition of a singular
foliation.) The same turns out to be true for the bi- and trimodal cases in
Theorem 1.5.

######  Definition 1.7.

Let $U$ be a wandering domain of the meromorphic function $f$.

  * •

We say that $U$ admits a contracting (singular) foliation if there exists a
singular foliation $\mathcal{C}$ of $U$ such that
$d_{U_{n}}\left(f^{n}(z),f^{n}(w)\right)\to 0$ for every $z$ and $w$ on the
same leaf of $\mathcal{C}$.

  * •

We say that $U$ admits an eventually isometric (singular) foliation if there
exists a singular foliation $\mathcal{L}$ of $U$ such that, for all large $n$,
$d_{U_{n}}\left(f^{n}(z),f^{n}(w)\right)=c(z,w)>0$ for every $z$ and $w$ on
the same leaf of $U$.

With this notion, we can make cases 6 and 3 of Theorem 1.5 more precise as
follows.

######  Theorem 1.8.

Let $f$ be a meromorphic function, and let $U$ be an essentially thin
infinitesimally eventually isometric wandering domain of $f$. Then $U$ has
eventual connectivity $2$ and $\delta_{n}(z)\to 0$ for all $z\in U$; in
particular, $\operatorname{mod}U_{n}\to 0$ as $n\to\infty$. Furthermore, $U$
admits both contracting and eventually isometric singular foliations.

######  Theorem 1.9.

Let $f$ be a meromorphic function with an essentially thin infinitesimally
semi-contracting wandering domain $U$. Then $\delta_{n}(z)\to 0$ for all $z\in
U$, and there is a sequence of topological annuli $A_{n}\subset U_{n}$ such
that $f(A_{n})\subset A_{n+1}$ for all $n$,
$\operatorname{mod}A_{n}\to+\infty$ as $n\to+\infty$, and for any compact
subset $K\subset U$, there exists $N\in\mathbb{N}$ such that $f^{n}(K)\subset
A_{n}$. Furthermore, $U$ admits a contracting singular foliation.

Recall that any modulus of sufficiently large modulus contains a round annulus
of not much smaller modulus. (This was first proved by Teichmüller; see
[Ahl73, Equations (4-21) and (4-23)]; a sharp version is given by Avkhadiev
and Wirths in [AW09, Theorem 3.17].) With this in mind, Theorems 1.8 and 1.9
generalise results of Zheng [Zhe06] and Bergweiler, Rippon, and Stallard
[BRS13] on the existence of “large” round annuli in multiply connected
wandering domains of meromorphic functions with finitely many poles. In
[Zhe06] and [BRS13], the wandering domains themselves are large in the
Euclidean sense, and so are the relevant subannuli. For meromorphic maps, the
wandering domains may be small, in which case they contain subannuli that are
small in the Euclidean sense, but with large modulus. See also [Zhe17] for
more results on the size of annuli contained in multiply connected wandering
domains.

The idea of the proofs of Theorems 1.8 and 1.9 can be described as follows.
Suppose that $U$ is essentially thin and not infinitesimally contracting. By
considering the universal coverings of the domains $U_{n}$, we show (Lemma
4.1) that the sequence $\delta_{n}(z)$ is eventually non-increasing; in
particular, $\delta_{n}(z)\to 0$. By principles of hyperbolic geometry, this
means that $f^{n}(z)$ is indeed contained in a sub-annulus of $U_{n}$ of large
modulus. We must then show that each of these annuli can be chosen
independently of the point $z$, and such that each annulus maps inside the
next one (Theorem 4.2). This argument is partly inspired by an argument for
the case of entire functions, due to Shishikura [Shi21]. Finally, we conclude
that the covering groups of the domains $U_{n}$ converge geometrically to a
continuous one-parameter group; the orbits of this group (and their
transversals, in the infinitesimally eventually isometric case) give rise to
the desired singular foliations.

Although we have stated and proved our results only for wandering domains of
meromorphic functions, they in fact hold in the more general setting of a
sequence of holomorphic maps between hyperbolic Riemann surfaces with no
punctures, with minor modifications to the proofs. More specifically, uses of
the argument principle should be replaced by homotopy-based arguments and
homomorphisms between fundamental groups. In these replacements, “winding
numbers” become powers of a primitive closed geodesic in the relevant homotopy
class.

The structure of this paper is as follows. In Section 2, we collect some
concepts and results on approximation theory and hyperbolic geometry to make
use of during our proofs. Section 3 is dedicated to proving Propositions 1.1
and 1.4. We then prove Theorems 1.8 and 1.9 in Section 4, and deduce Theorem
1.5 in Section 5. Finally, in Section 6, we discuss how all possibilities in
Theorem 1.5 can arise. In particular, we prove Theorem 1.6.

## Acknowledgements

The first author acknowledges financial support from the Spanish State
Research Agency through the María de Maeztu Program for Centers and Units of
Excellence in R&D (CEX2020-001084-M).

##  2\. Preliminaries

The proof of Theorem 1.6 is based on approximation theory. More specifically,
we have the following technique by Evdoridou, Martí-Pete, and Rempe [EMPR] for
constructing meromorphic functions:

######  Theorem 2.1.

Let $(K_{n})_{n\geq 0}\subset\mathbb{C}$ be a sequence of pairwise disjoint
compact sets such that $K_{n}\to\infty$ as $n\to+\infty$. Let
$(\varphi_{n})_{n\geq 1}$ be a sequence of functions with $\varphi_{n}$
meromorphic in a neighbourhood of $K_{n-1}$, and assume that
$\varphi_{n}^{-1}(K_{n})=K_{n-1}$. Let $(\epsilon_{n})_{n\geq 0}$ be a
sequence of positive numbers. Then, there exists a meromorphic function
$f\colon\mathbb{C}\to\widehat{\mathbb{C}}$, compact sets $\widetilde{K}_{n}$,
and homeomorphisms $\theta_{n}\colon K_{n}\to\widetilde{K}_{n}$ such that, for
all $n\geq 0$:

  1. (a)

$f\circ\theta_{n}=\theta_{n+1}\circ\varphi_{n+1}$ on $K_{n}$;

  2. (b)

$\partial\widetilde{K}_{n}\subset J(f)$;

  3. (c)

$\theta_{n}$ is conformal on $\mathrm{int}(K_{n})$; and

  4. (d)

$|\theta_{n}(z)-z|\leq\epsilon_{n}$ for $z\in K_{n}$.

For the other results in this paper, we will need techniques from hyperbolic
geometry. Most of these are standard, and we refer the reader to [BM06] for an
overview of the notation and concepts. We start with the Schwarz–Pick lemma:

######  Lemma 2.2.

Let $f\colon U\to V$ be a holomorphic map between hyperbolic domains. Then,

$\frac{\rho_{V}\left(f(z)\right)|f^{\prime}(z)|}{\rho_{U}(z)}=\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}\|Df(z)\|_{U}^{V}\leq
1\text{ for all $z\in U$},$

with equality if and only if $f\colon U\to V$ is an unbranched covering map.

In order to extract dynamical information from the hyperbolic distortion of
$f$ along an orbit, we have the following lifting property (see [Fer24, Lemmas
2.3 and 3.2]):

######  Lemma 2.3.

Let $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a meromorphic function with
a wandering domain $U$, and let $z_{0}\in U$. Given uniformising maps
$\varphi_{n}\colon\mathbb{D}\to U_{n}$ such that
$\varphi_{n}(0)=f^{n}(z_{0})$, there exists a unique sequence of non-constant
holomorphic functions $g_{n}\colon\mathbb{D}\to\mathbb{D}$ such that, for all
$n\geq 0$:

  1. (i)

$g_{n}$ fixes the origin;

  2. (ii)

$f|_{U_{n}}$ lifts to $g_{n}$, i.e. $\varphi_{n+1}\circ
g_{n}=f\circ\varphi_{n}$;

  3. (iii)

$|g_{n}^{\prime}(0)|=\|Df\left(f^{n}(z_{0})\right)\|_{U_{n}}^{U_{n+1}}$.

Next, we must consider how the geometry of $U_{n}$ will affect the dynamics of
$f^{n}\colon U\to U_{n}$. First, we have a tool to help us find large annuli
in hyperbolic Riemann surfaces: the collar theorem.

######  Theorem 2.4.

Let $S$ be a hyperbolic Riemann surface, and let $\gamma$ be a simple closed
geodesic of length $\ell$. Then, the neighbourhood

$A_{\gamma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{z\in
S\colon\mathrm{dist}_{S}(z,\gamma)<\eta(\ell)\\},$

where

$\eta(\ell)=\frac{1}{2}\log\frac{\cosh(\ell/2)+1}{\cosh(\ell/2)-1},$

is isometric to the $\eta(\ell)$-neighbourhood of the unique closed geodesic
in an annulus of modulus $\pi/\ell$. In particular, $A$ is a topological
annulus. Furthermore, if $\sigma\neq\gamma$ is another simple closed geodesic,
then $A_{\gamma}\cap A_{\sigma}=\emptyset$.

Next, we must be able to show that the orbit of a point enters theese large
annuli. A key tool in this endeavour is the injectivity radius of a hyperbolic
surface at a point; a precise definition is as follows.

######  Definition 2.5.

Let $S$ be a hyperbolic Riemann surface, and let $p\in S$. The injectivity
radius of $S$ at $p$ is

$\operatorname{inj}(p)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\inf\\{\ell_{S}(\gamma)/2\colon\text{$\gamma\subset
S$ is a non-trivial loop through $p$}\\}.$

An equivalent way of stating Definition 2.5 (and one that better explains the
name) is that $\operatorname{inj}(p)$ is the supremum of $r>0$ such that a
universal covering $\varphi\colon\mathbb{D}\to S$, normalised so that
$\varphi(0)=p$, is injective in the hyperbolic ball $D_{\mathbb{D}}(0,r)$ (cf.
[Thu97, Section 4.5]). In particular, $D_{S}(p,r)$ is isometric to the
hyperbolic ball $D_{\mathbb{D}}(0,r)$ for any $r<\operatorname{inj}(p)$. With
this terminology, we have another major result of hyperbolic geometry, the
thick-thin decomposition, to help us place orbits inside large annuli (see,
for instance, [CEG06, Theorem I.2.2.2], or [Thu97, Theorem 4.5.6]):

######  Theorem 2.6.

There exists an absolute constant $M>0$ (the Margulis constant) such that, for
any hyperbolic surface $S$, any point $p\in S$ with $\operatorname{inj}(p)<M$
belongs to either a collar of a simple closed geodesic or to a neighbourhood
of a cusp homeomorphic to $\mathbb{D}^{*}$.

Finally, in the proof of Theorem 1.9, we will use the notion of geometric
convergence and some of its properties. In a way, it generalises the Hausdorff
metric to closed subsets of topological spaces, and it can be – for our
purposes – defined as follows***This is not the usual definition, but an
equivalent characterisation; see [BP92, Proposition E.1.2]..

######  Definition 2.7.

Let $X$ be a locally compact metrizable topological space, and let
$\mathcal{C}(X)$ denote the set of all closed subsets of $X$. We say that a
sequence $(C_{n})_{n\in\mathbb{N}}\subset\mathcal{C}(X)$ of subsets converges
geometrically to $C\in\mathcal{C}(X)$ if and only if:

  1. (i)

Any $x\in X$ for which there exists a subsequence $(n_{k})_{k\in\mathbb{N}}$
and points $x_{k}\in C_{n_{k}}$ such that $x_{k}\to x$ as $k\to+\infty$
satisfies $x\in C$; and

  2. (ii)

For any $x\in C$, there exists a sequence $(x_{n})_{n\in\mathbb{N}}$ with
$x_{n}\in C_{n}$ such that $x_{n}\to x$.

To us, the most interesting application is to set $X=\mathcal{A}(\mathbb{D})$,
the Lie group of conformal automorphisms of $\mathbb{D}$. In this case, we
have:

######  Lemma 2.8.

The subset $\mathcal{S}(\mathbb{D})\subset\mathcal{A}(\mathbb{D})$ of all
closed subgroups of $\mathcal{A}(\mathbb{D})$ is compact in the geometric
topology.

##  3\. Uniformity at the infinitesimal level

In this section, we prove Propositions 1.1 and 1.4. We start with Proposition
1.1; its proof uses the following formula for the hyperbolic distance between
points in the unit disc (see [BM06, p. 13]).

######  Lemma 3.1.

Let $z,w\in\mathbb{D}$ be such that $0\leq z\leq w<1$. Then,

$d_{\mathbb{D}}(z,w)=\log\left(\frac{1+w}{1-w}\cdot\frac{1-z}{1+z}\right).$

Using Lemma 3.1, we obtain the following bounds for the hyperbolic distortion:

######  Lemma 3.2.

Let $g\colon X\to Y$ be a holomorphic map between hyperbolic Riemann surfaces.
For $p\in X$, let
$\lambda(p)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\|Dg(p)\|_{X}^{Y}$.
Then, for any $z,w\in X$,

$e^{-2d_{X}(z,w)}(1-\lambda(w))\leq 1-\lambda(z)\leq
e^{2d_{X}(z,w)}(1-\lambda(w)).$

###### Proof.

First, if $\lambda(z)=1$, then by the Schwarz–Pick lemma $g$ is a covering
map, and so $\lambda(w)=1$ for every $w$ in $X$. Thus, in this case, the
inequality is trivial.

Assume, for now, that $\lambda(z)\leq\lambda(w)<1$. Applying Lemma 3.1 gives
us

$\frac{1+\lambda(w)}{1-\lambda(w)}\cdot\frac{1-\lambda(z)}{1+\lambda(z)}=e^{d_{\mathbb{D}}(\lambda(z),\lambda(w))}.$

Therefore,

$1-\lambda(z)=e^{d_{\mathbb{D}}(\lambda(z),\lambda(w))}(1-\lambda(w))\frac{1+\lambda(z)}{1+\lambda(w)}\leq
e^{d_{\mathbb{D}}(\lambda(z),\lambda(w))}(1-\lambda(w)),$

where the inequality follows from the fact that
$0\leq\lambda(z)\leq\lambda(w)<1$.

All that is left is to bound $d_{\mathbb{D}}(\lambda(z),\lambda(w))$ in terms
of $d_{X}(z,w)$. This is accomplished through Beardon and Minda’s Schwarz–Pick
lemma for hyperbolic distortion [BM06, Theorem 11.2], which tells us that
$d_{\mathbb{D}}(\lambda(z),\lambda(w))\leq 2d_{X}(z,w)$, yielding

$1-\lambda(z)\leq e^{2d_{X}(z,w)}(1-\lambda(w)).$

Reversing the roles of $z$ and $w$ in the proof gives the lower bound, and the
proof is complete. ∎

Finally, Proposition 1.1 follows easily from Lemma 3.2.

###### Proof of Proposition 1.1.

Let $z,w\in U$, and let $n\in\mathbb{N}$. By Lemma 3.2,

$e^{-2d_{U_{n-1}}(f^{n-1}(z),f^{n-1}(w))}(1-\lambda_{n}(w))\leq
1-\lambda_{n}(z)\leq
e^{2d_{U_{n-1}}(f^{n-1}(z),f^{n-1}(w))}(1-\lambda_{n}(w)),$

while by Lemma 2.2 $d_{U_{n-1}}(f^{n-1}(z),f^{n-1}(w))\leq d_{U}(z,w)$.
Proposition 1.1 follows with $C=e^{2_{U}(z,w)}$. ∎

Next, we prove Proposition 1.4. Its proof uses the following observation about
injectivity radii, which will also be useful later:

######  Lemma 3.3.

Let $X$ be a hyperbolic Riemann surface, let $p\in X$, and let
$\pi\colon\mathbb{D}\to X$ be a universal covering map such that $\pi(0)=p$.
Denote by $\tilde{p}^{*}$ a point such that

$|\tilde{p}^{*}|=\inf\\{|z|\colon z\in\mathbb{D}^{*},\pi(z)=p\\}.$

Then,

$\operatorname{inj}(p)=\frac{1}{2}d_{\mathbb{D}}(0,\tilde{p}^{*}).$

###### Proof.

Joining $0$ to $\tilde{p}^{*}$ by a geodesic arc $\gamma\in\mathbb{D}$, it
becomes clear that

$\operatorname{inj}(p)\leq\frac{1}{2}\ell_{X}(\pi(\gamma))=\frac{1}{2}\ell_{\mathbb{D}}(\gamma)=\frac{1}{2}d_{\mathbb{D}}(0,\tilde{p}^{*}).$

For the reverse inequality, assume that there exists some non-contractible
closed curve $\gamma\in X$ passing through $p$ such that
$2\operatorname{inj}(p)\leq\ell_{X}(\gamma)<d_{\mathbb{D}}(0,\tilde{p}^{*})$.
Then, we take its lift $\tilde{\gamma}\subset\mathbb{D}$ starting at $0$,
which must (by the unique path lifting property of $\pi$) terminate at some
$\tilde{p}\in\pi^{-1}(p)$ distinct from zero. This point also satisfies

$d_{\mathbb{D}}(0,\tilde{p})\leq\ell_{\mathbb{D}}(\tilde{\gamma})=\ell_{X}(\gamma)<d_{\mathbb{D}}(0,\tilde{p}^{*}).$

However, $\tilde{p}^{*}$ was defined as a smallest non-zero point on the fibre
of $p$, giving us a contradiction. The proof is complete. ∎

Without further ado:

###### Proof of Proposition 1.4.

Let $f$, $z_{0}$, and $U_{n}$, $n\geq 0$, be as in the statement. For $n\geq
0$, let $\varphi_{n}\colon\mathbb{D}\to U_{n}$ denote uniformising maps,
normalised so that $\varphi_{n}(0)=f^{n}(z_{0})$. Let $z_{n}^{*}\in\mathbb{D}$
be a point on the fibre of $f^{n}(z_{0})$ closest to the origin, i.e. so that

$|\tilde{z}_{n}^{*}|=\inf\\{|z|\colon
z\in\mathbb{D}^{*},\varphi_{n}(z)=z_{0}\\}.$

For each $n\geq 0$, we find a deck transformation
$\gamma_{n}\in\mathcal{A}(\mathbb{D})$, where $\mathcal{A}(\mathbb{D})$
denotes the group of conformal automorphisms of $\mathbb{D}$, so that
$\varphi_{n}(0)=\tilde{z}_{n}$. By Lemma 3.3, the hypothesis
$\operatorname{inj}(f^{n_{k}}(z_{0}))\to 0$ implies that $\tilde{z}_{n_{k}}\to
0$ as $k\to+\infty$, and we claim that this implies that the sequence
$(\gamma_{n_{k}})_{k\in\mathbb{N}}$ converges locally uniformly to the
identity as $k\to+\infty$. First, since the autormorphisms of $\mathbb{D}$ can
be parameterised by their image and derivative at zero, we see (since
$\tilde{z}_{n_{k}}=\gamma_{n_{k}}(0)\to 0$) that the sequence
$(\gamma_{n_{k}})_{k\in\mathbb{N}}$ is precompact in
$\mathcal{A}(\mathbb{D})$, and that any of its limit functions is an
automorphism of $\mathbb{D}$ fixing the origin. Since no $\gamma_{n_{k}}$ has
fixed points in $\mathbb{D}$, it follows from Hurwitz’s theorem that the only
possible limit function is the identity, and so that $\gamma_{n_{k}}(z)\to z$
locally uniformly as $k\to+\infty$.

Finally, let $w\in U$. Then, for $n\geq 0$, there exists
$\tilde{w}_{n}\in\varphi^{-1}(w_{n})$ such that

$d_{\mathbb{D}}(0,\tilde{w}_{n})=d_{U_{n}}(f^{n}(w),f^{n}(z_{0}))\leq
d_{U}(w,z_{0}),$

where the inequality comes from the Schwarz–Pick lemma. Since, by definition
of the injectivity radius,

$\operatorname{inj}(f^{n_{k}}(z))\leq\frac{1}{2}d_{\mathbb{D}}\left(\tilde{w}_{n_{k}},\gamma_{n_{k}}(\tilde{w})\right),$

the conclusion follows by the fact that $\gamma_{n_{k}}$ is converging to the
identity locally uniformly as $k\to+\infty$. ∎

##  4\. Essential thinness, large annuli, and contracting foliations

In this section, we prove Theorems 1.8 and 1.9. The two cases display many
similarities, and so we can combine parts of the proofs. We start with some
consequences of being essentially thin.

######  Lemma 4.1.

Let $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ be a meromorphic function with
an essentially thin wandering domain $U$, and let $z_{0}\in U$. Assume that
$U$ is infinitesimally semi-contracting or infinitesimally eventually
isometric. Then:

  1. (1)

For all large $n\in\mathbb{N}$,
$\operatorname{inj}(f^{n+1}(z_{0}))\leq\operatorname{inj}(f^{n}(z_{0}))$;

  2. (2)

$\operatorname{inj}(f^{n}(z))\to 0$ for any $z\in U$;

  3. (3)

For large enough $N\in\mathbb{N}$, there exist a non-trivial loop
$\gamma\subset U_{N}$ through $f^{N}(z_{0})$ and a sequence of points
$p_{n}\in\mathbb{C}\setminus U_{n}$, $n>N$, such that
$\operatorname{ind}(f^{n}(\gamma),p_{n})\to+\infty$.

###### Proof.

Let $(n_{k})_{k\in\mathbb{N}}$ be a sequence such that
$\operatorname{inj}(f^{n_{k}}(z_{0}))\to 0$; notice that this sequence exists
by the hypothesis that $U$ is eventually thin. As before, we start by taking
uniformising maps $\varphi_{n}\colon\mathbb{D}\to U_{n}$ normalised so that
$\varphi_{n}(0)=f^{n}(z_{0})$ and so that the points
$\tilde{z}_{n}\in\mathbb{D}^{*}$ chosen so that

$|\tilde{z}_{n}|=\inf\\{|z|\colon
z\in\mathbb{D}^{*},\varphi_{n}(z)=f^{n}(z_{0})\\}$

lie on the positive real axis for all $n\in\mathbb{N}$. Again by Lemma 3.3,
$\operatorname{inj}(f^{n_{k}}(z_{0}))\to 0$ now implies that $\tilde{z}_{k}\to
0$ as $k\to+\infty$. Then, we apply Lemma 2.3, obtaining a sequence
$g_{n}\colon\mathbb{D}\to\mathbb{D}$ of functions satisfying
$\varphi_{n+1}\circ g_{n}=f\circ\varphi_{n}$ for $n\geq 0$.

With that in mind, let us prove 1. Since $U$ is infinitesimally semi-
contracting (or eventually isometric), it follows that any limit function of
the lifts $(g_{n})_{n\in\mathbb{N}}$ is a rotation about the origin. Combining
this with the fact that $|\tilde{z}_{n_{k}}|\to 0$, we can choose $k$ large
enough that $g_{n}$ is injective in the disc $D(0,|\tilde{z}_{n_{k}}|)$ for
all $n\geq k$, and so in particular
$g_{n_{k}}(\tilde{z}_{n_{k}})=\tilde{w}_{k}\neq 0$. By the lift condition
$\varphi_{n+1}\circ g_{n}=f\circ\varphi_{n}$, it follows that
$\varphi_{n+1}(\tilde{w}_{k})=\varphi_{n+1}(0)=f^{n_{k}+1}(z_{0})$, and thus
$\operatorname{inj}(f^{n_{k}+1}(z_{0}))\leq
d_{\mathbb{D}}(0,\tilde{w}_{k})/2$. By Schwarz’s lemma,
$|\tilde{w}_{k}|\leq|\tilde{z}_{n_{k}}|$, and hence
$\operatorname{inj}(f^{n_{k}+1}(z_{0}))\leq\operatorname{inj}(f^{n_{k}}(z_{0}))$
(again by Lemma 3.3). 1 follows by induction, since $g_{n_{k}+1}$ is also
injective in the disc $D(0,|\tilde{z}_{n_{k}}|)\supseteq
D(0,|\tilde{w}_{k}|)$. 2 is now a simple consequence of 1 and Proposition 1.4.

We move on to 3. If we once again take $N$ large enough that $g_{n}$, $n\geq
N$, is injective in the disc $D(0,|\tilde{z}_{N}|)$, then for all $n>N$ the
function
$G_{N}^{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{N}$ maps $\tilde{z}_{N}$ onto some point $G_{N}^{n}(\tilde{z}_{N})\neq 0$.
The curve $\gamma=\varphi_{N}([0,\tilde{z}_{N}])\subset U_{N}$ is a non-
trivial loop through $f^{N}(z_{0})$, and by the lift condition so is any
iterate $f^{n}(\gamma)=\varphi_{n+N}(G_{N}^{n}([0,\tilde{z}_{N}])$. Hence, we
can take $p_{n}$, $n>N$, to be any point in $\mathbb{C}\setminus U_{n}$
surrounded by $f^{n}(\gamma)$; it remains to show that
$\operatorname{ind}(f^{n}(\gamma),p_{n})\to+\infty$. This, however, follows
from the following observation: since $U$ is infinitesimally semi-contracting
(or eventually isometric), $G_{N}^{n}$ converges locally uniformly to some
non-constant holomorphic function $G_{N}$ (for any fixed $N$), and furthermore
$G_{N}(\tilde{z}_{N})\neq 0$. The condition that
$\operatorname{inj}(f^{n}(z_{0}))\to 0$ now implies that, as $n$ increases,
there appear infinitely many points $\tilde{z}_{n}$ on the segment
$[0,G_{N}^{n}(\tilde{z}_{N})]$. In particular, for $n>N$,
$\varphi_{n}([0,G_{N}^{n}(\tilde{z}_{N})])$ is a non-trivial loop
$\sigma_{n}\in U_{n}$ satisfying
$\operatorname{ind}(\sigma_{n},p_{n})\to+\infty$, and if $N$ is large enough
then $G_{N}^{n}([0,\tilde{z}_{N}])\approx[0,G_{N}^{n}(\tilde{z}_{N})]$,
meaning that $f^{n}(\gamma)$ and $\sigma_{n}$ are homotopic. This concludes
the proof. ∎

A stronger consequence of essential thinness, namely the existence of
absorbing annuli, follows from Lemma 4.1. As mentioned in the introduction,
the proof is inspired by an argument of Shishikura for entire functions
[Shi21].

######  Theorem 4.2.

Let $f$ be a meromorphic function with an essentially thin wandering domain
$U$. Assume that $U$ is infinitesimally semi-contracting or infinitesimally
eventually isometric. Then, there exist $n_{0}\in\mathbb{N}$ and a sequence
$(A_{n})_{n=n_{0}}^{\infty}$ of topological annuli in $U_{n}$ such that:

  1. (1)

For every compact subset $K\subset U$, there exists $N\in\mathbb{N}$ such that
$f^{n}(K)\subset A_{n}$ for all $n\geq N$;

  2. (2)

$f(A_{n})\subset A_{n+1}$;

  3. (3)

$\operatorname{mod}A_{n}\to+\infty$ as $n\to+\infty$.

###### Proof.

We obtain the relevant annuli by using collars. We will first show that, for
any $z\in U$ and sufficiently large $n$, $f^{n}(z)$ belongs to the collar of
some simple closed geodesic in $U_{n}$, and second that any two non-trivial
loops in $U$ are eventually in the collar of the same closed geodesic in
$U_{n}$.

The first of these two claims follows immediately from Theorem 2.6 and Lemma
4.1, and the fact that Fatou components of meromorphic functions have no
isolated punctures. For the second, we will need “small collars” around our
geodesics. If $\sigma\subset U_{n}$ denotes a simple closed geodesic, let

$A_{\sigma}(\epsilon)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{z\in
A_{\sigma}\colon\operatorname{inj}(z)\leq\epsilon\\},$

where $A_{\sigma}$ denotes the collar around $\sigma$ and $\epsilon<M$ is
fixed, where $M$ is the Margulis constant. Note that $A_{\sigma}$ is a
topological annulus. Indeed, by Theorem 2.6, the injectivity radius on the
boundary of $A_{\sigma}$ is at least $M$. Hence by Theorem 2.4,
$A_{\sigma}(\epsilon)$ is isometric to a corresponding ‘’thin collar” around
the central geodesic of a round annulus, and it is elementary to see that the
latter is itself a round annulus. Now, let $z_{0}\in U$ and let $n_{0}$ be
such that $f^{n}(z_{0})$ belongs to the collar of a simple closed geodesic
$\gamma_{n}$ of $U_{n}$ for $n\geq n_{0}$. We define
$A_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=A_{\gamma_{n}}(\epsilon)$.

Let $K$ be a compact subset of $U$ with $z_{0}\in K$. By the Schwarz–Pick
lemma, $\mathrm{diam}_{U_{n}}(K)$ is bounded from above by
$\mathrm{diam}_{U}(K)$. By Lemma 4.1, for sufficiently large $n$ we have
$f^{n}(z_{0})\in A_{\gamma_{n}}(\epsilon_{n})$, where $\epsilon_{n}\to 0$ as
$n\to+\infty$. It follows that $f^{n}(z_{0})\in A_{n}$ for large $n$, and
$\mathrm{dist}_{U_{n}}(f^{n}(z_{0}),\partial A_{n})\to\infty$. (This again
follows from the collar theorem and the hyperbolic geometry of round annuli.)
Hence $f^{n}(K)\subset A_{n}$ for sufficiently large $n$, which proves 1.

To prove 2, we show that any point $z\in A_{\hat{\gamma}_{n}}(\epsilon)$
satisfies $\operatorname{inj}(f(z))\leq\operatorname{inj}(z)$. Indeed, let
$\sigma\subset A_{\hat{\gamma}_{n}}(\epsilon)$ be a non-trivial loop through
$z$. Then, by Lemma 4.13 and the argument principle, $f\circ\sigma$ is also a
non-trivial loop in $U_{n+1}$ through $f(z)$, and by the Schwarz–Pick lemma
implies $\ell_{U_{n+1}}(f\circ\sigma)\leq\ell_{U_{n}}(\sigma)$. It follows
from Definition 2.5 that $\operatorname{inj}(f(z))\leq\operatorname{inj}(z)$,
finishing the proof.

Finally, 3 is a simple consequence of Theorem 2.4 and the fact that the
hyperbolic length of $\gamma_{n}$ tends to $0$. ∎

With this, we can prove Theorems 1.8 and 1.9.

###### Proof of Theorem 1.8.

Most of the work follows immediately from Theorem 4.2; we only have to show
that the domain has eventual connectivity two, whence the rest will follow
from†††Alternatively, one can find the contracting foliation using an approach
similar to the proof of Theorem 1.9, and the eventually isometric foliation by
taking the foliation orthogonal to it. [Fer22, Theorem 1.1]. To this end, let
$N\in\mathbb{N}$ be large enough that $f\colon U_{n}\to U_{n+1}$ is a local
hyperbolic isometry for all $n\geq N$ (which must eventually happen since $U$
is infinitesimally eventually isometric). Denote by $\gamma_{n}\subset U_{n}$,
$n\geq N$, the hyperbolic geodesics inside the absorbing annuli
$A_{n}=A_{\gamma_{n}}(\epsilon)$, $\epsilon\in(0,M)$ fixed. Assume that $U$
has connectivity greater than two, so that there exists another simple closed
geodesic $\sigma\neq\gamma_{N}$. By Theorem 4.21, we have that
$f^{n}(\sigma)\subset A_{\gamma_{n}}$ for large enough $n$. Since $f^{n}\colon
U_{N}\to U_{N+n}$ is a local hyperbolic isometry, $f^{n}(\sigma)$ is a closed
geodesic, which is a contradiction since the only closed geodesic inside
$A_{\gamma_{n}}(\epsilon)$ is $\gamma_{n}$. ∎

Finally, we prove Theorem 1.9.

###### Proof of Theorem 1.9.

Most of the claims have already been proved, but it remains to construct the
contracting foliation. We take uniformising maps
$\varphi_{n}\colon\mathbb{D}\to U_{n}$ normalised so that
$\varphi_{n}(0)=f^{n}(z_{0})$, and lifts $g_{n}\colon\mathbb{D}\to\mathbb{D}$
given by Lemma 2.3. We normalise these lifts so that $g_{n}^{\prime}(0)>0$.
Let $\Gamma_{n}\subset\mathcal{A}(\mathbb{D})$, $n\geq 0$, denote the group of
deck transformations of $\varphi_{n}$. We claim that the sequence to show that
$\Gamma_{n}$ converges in the geometric topology to a continuous one-parameter
group $\Gamma$.

For $n,\ell\geq 0$, consider the functions
$H_{n}^{n+\ell}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n+\ell}\circ\dots\circ
g_{n}$. By the normalisation of the maps $\varphi_{n}$, each $H_{n}^{n+\ell}$
fixes $0$ with non-negative real derivative. Moreover, since $U$ is
infinitesimally semi-contracting, the derivative
$(H_{n}^{n+\ell})^{\prime}(0)$ converges to $1$ as $n\to\infty$, uniformly in
$\ell$. By normality of the set of holomorphic self-maps of the disc and the
Schwarz lemma, it follows that $H_{n}^{n+\ell}$ converges to the identity,
uniformly in $\ell$ and locally uniformly on $\mathbb{D}$, as $n\to\infty$.

Now suppose that $(\gamma_{n_{k}})_{k=0}^{\infty}$ is a sequence of deck
transformations $\gamma_{n_{k}}\in\Gamma_{n_{k}}$ that converges to some
Möbius self-map $\gamma$ of the disc, as $k\to\infty$. Recall that this is
equivalent to saying that $\gamma_{n_{k}}(0)\to\gamma(0)$ and
$\gamma_{n_{k}}^{\prime}(0)\to\gamma^{\prime}(0)$. We claim that the sequence
can be extended to a sequence $(\gamma_{n})_{n=n_{0}}^{\infty}$ with
$\gamma_{n}\in\Gamma_{n}$, and $\gamma_{n}\to\gamma$. Indeed, let $n_{k}\leq
n<n_{k+1}$. By the lift relation from Lemma 2.3ii, it follows that there is
$\gamma_{n}\in\Gamma_{n}$ with
$H_{n_{k}}^{n}\circ\gamma_{n_{k}}=\gamma_{n}\circ H_{n_{k}}^{n}$. Since
$H_{n_{k}}^{n}$ converges to the identity as $k\to\infty$, and since
$\gamma_{n_{k}}(0)$ is contained in a compact subset of the disc, we conclude
that $\gamma_{n}(0)\to\gamma(0)$ and
$\gamma_{n}^{\prime}(0)\to\gamma^{\prime}(0)$, so $\gamma_{n}\to\gamma$ as
claimed.

By Lemma 2.8, $(\Gamma_{n})_{n\in\mathbb{N}}$ has a convergent subsequence
$(\Gamma_{n_{k}})_{k=0}^{\infty}$, converging geometrically to a group
$\Gamma\in\mathcal{S}(\mathbb{D})$. By the above, and the definition of
geometric convergence in Definition 2.7, it follows that this limit is unique,
so $\Gamma_{n}\to\Gamma$ as $n\to\infty$. The limit $\Gamma$ is a continuous
one-parameter group is an immediate consequence of [MT98, Theorem 7.1] and the
fact that $U$ is essentially thin. Indeed, the latter implies that there is a
sequence $(\gamma_{n})_{n\in\mathbb{N}}$ that converges to the identity
(recall the proof of Proposition 1.4).

To define the singular foliation, note first that the limit group $\Gamma$ has
no fixed points, since (by the same argument as in the proof of Proposition
1.4) any limit function of a sequence of elements in $\Gamma_{n}$ is either
the identity or fixed-point free. In other words, $\Gamma$ is either a
hyperbolic group or a parabolic group (see [Aba23, Definition 5.6.4]). Either
way, the action of $\Gamma$ on $\mathbb{D}$ induces a foliation
$\mathcal{F}^{\prime}$ of $\mathbb{D}$ (which contains exactly one geodesic if
$\Gamma$ is hyperbolic, and only horocycles if $\Gamma$ is parabolic). Let
$G\colon\mathbb{D}\to\mathbb{D}$ be the limit of $H_{0}^{n}$ as $n\to\infty$,
which exists and is non-constant by the considerations above. We claim that
the pullback (singular) foliation
$\mathcal{F}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=G^{*}\mathcal{F}^{\prime}$
is the desired contracting foliation.

To verify this, take two points $z$ and $w$ on the same leaf of $\mathcal{F}$
so that, as $n\to+\infty$, their orbits $G_{n}(z)$ and $G_{n}(w)$ converge to
$G(z)$ and $G(w)$, which are points on the same leaf of
$\mathcal{F}^{\prime}$. Now, for $n\in\mathbb{N}$, let $\tilde{w}_{n}$ denote
the point on the $\Gamma_{n}$-orbit of $G_{n}(z)$ that is closest to
$G_{n}(w)$; it is clear that
$d_{U_{n}}\left(f^{n}\circ\varphi_{0}(z),f^{n}\circ\varphi_{0}(w)\right)=d_{U_{n}}\left(\varphi_{n}(G_{n}(z)),\varphi_{n}(G_{n}(w))\right)=d_{\mathbb{D}}(G_{n}(w),\tilde{w}_{n})$.
Since $\Gamma_{n}$ converges to $\Gamma$ geometrically and the action of
$\mathcal{A}(\mathbb{D})$ on $\mathbb{D}$ is smooth, it follows from
Definition 2.7ii that $\tilde{w}_{n}\to w$, which completes the proof. ∎

##  5\. Proof of Theorem 1.5

We begin by considering case 1.

######  Proposition 5.1.

Any infinitesimally contracting wandering domain $U$ of a meromorphic function
is globally contracting.

###### Proof.

This proof is mostly implicit in the proof of [Fer24, Theorem 1.1(a)]; we
outline it here for completeness.

We start by applying Lemma 2.3, obtaining functions
$g_{n}\colon\mathbb{D}\to\mathbb{D}$ that lift $f|_{U_{n}}$. By Proposition
1.1 and (3), the hyperbolic derivative of
$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\dots\circ
g_{0}$ tends to $0$ for all $z\in\mathbb{D}$. It follows that $G_{n}\to 0$
locally uniformly on $\mathbb{D}$, and the conclusion follows since
$\mathrm{dist}_{U_{n}}(f^{n}(z),f^{n}(w))\leq\mathrm{dist}_{\mathbb{D}}(G_{n}(\tilde{z}),G_{n}(\tilde{w}))$,
where $\tilde{z}$ and $\tilde{w}$ are lifts of $z$ and $w$ under
$\varphi_{0}$. (See also [BEF+22, Theorem 2.1] and [Fer24, Lemma 3.3].) ∎

By [Fer22, Theorem 1.1], an infinitesimally contracting or semi-contracting
wandering domain cannot have a finite eventual connectivity greater than $1$.
Hence, in view of Proposition 5.1 and Theorems 1.8 and 1.9, to prove Theorem
1.5, it only remains to consider the case where $U$ is essentially thick and
either infinitesimally semi-contracting or infinitesimally eventually
isometric.

First, suppose that $U$ is infinitesimally eventually isometric and pick
$z_{0}\in U$. Then, if $\varphi_{n}\colon\mathbb{D}\to U_{n}$ are uniformising
maps with $\varphi_{n}(0)=f^{n}(z_{0})$, we know that there exists $c>0$ such
that the hyperbolic ball $D_{\mathbb{D}}(0,c)$ is mapped isometrically onto
$D_{U_{n}}(f^{n}(z_{0}),c)$ by $\varphi_{n}$ for all $n\in\mathbb{N}$.
Applying Lemma 2.3 gives us lifts $g_{n}\colon\mathbb{D}\to\mathbb{D}$ of
$f|_{U_{n}}$ fixing the origin, and since $U$ is infinitesimally eventually
isometric we see that any limit function of
$G_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{n}\circ\cdots\circ
g_{1}$ is non-constant. In particular, because $\varphi_{n}$ is isometric on
the disc $D_{\mathbb{D}}(0,c)$, we see that $f|_{D_{U}}(z_{0},c)$ satisfies
the hypotheses of [Fer24, Theorem 1.1]. This proves that $U$ is locally
eventually isometric.

Moreover, the connectivity of $U_{n}$ is non-decreasing as $n\to\infty$. If
$U_{n}$ has finite connectivity greater than one, and $f\colon U_{n}\to
U_{n+1}$ is not a conformal isomorphism, then the connectivity of $U_{n+1}$ is
strictly less than that of $U_{n}$. It follows that either $U$ is globally
eventually isometric, and has an eventual connectivity, or $U$ has eventual
connectivity $\infty$.

If $U$ is essentially thick and infinitesimally semi-contracting, then it
follows similarly from [Fer24, Theorem 1.1] that it is globally semi-
contracting. As already mentioned, $U$ cannot have a finite eventual
connectivity greater than $1$. This completes the proof of Theorem 1.5.

##  6\. Examples

Let us first consider the case of entire functions. Examples of simply
connected globally contracting, globally semi-contracting and globally
eventually isometric wandering domains are discussed in [BEF+22]. An entire
function having a wandering domain of eventual connectivity $\infty$ was first
constructed by Baker [Bak63]. A wandering domain with eventual connectivity
$2$ was constructed by Kisaka and Shishikura [KS08]. This covers all cases of
Theorem 1.5 that can be realised by entire functions, and in particular gives
examples of cases 3 and 6 of Theorem 1.5.

Globally eventually isometric wandering domains of any eventual connectivity
$\geq 2$ are constructed in [BKL90]; their existence also follows from Theorem
2.1 or [MPRW23, Theorem 1.3].

Examples of essentially thick globally contracting or semi-contracting
wandering domains with eventual connectivity $\infty$ are given in [RS08,
Example 1] and [Fer24, Theorem 1.2], respectively.

Examples of wandering domains without an eventual connectivity are constructed
in [Fer22, Theorem 1.3] and [HWZ22, Theorem 4]. By suitably modifying the
construction of [Fer22, Theorem 1.3], one can easily ensure that this domain
is either infinitesimally contracting or infinitesimally semi-contracting;
notice that, in the latter case, the domain must be essentially thick by
Theorem 1.53. Different modifications can also make the domain infinitesimally
contracting and essentially thin.

It remains to prove Theorem 1.6. To do so, we construct a suitable model map
to which we can apply Theorem 2.1.

######  Proposition 6.1.

There exist sequences $(b_{n})_{n=0}^{\infty}$ of degree $2$ Blaschke products
$\mathbb{D}\to\mathbb{D}$ and $(U_{n})_{n=0}^{\infty}$ of subdomains of
$\mathbb{D}$ with the following properties:

  1. (1)

$b_{n}\colon U_{n}\to U_{n+1}$ is a degree two covering map with $b_{n}(0)=0$
and $0\in U_{n}$ for all $n$;

  2. (2)

there is an increasing sequence $(r_{n})_{n=0}^{\infty}$ with $r_{n}\nearrow
1$ such that $b_{n}|_{D(0,r_{n})}$ is injective, for all $n\geq 0$.

###### Proof.

For $0<a<1$, define

$b_{a}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=z\cdot\frac{z+a_{n}}{1+a_{n}z}.$

The maps $b_{n}$ in the proposition will be of the form $b_{n}=b_{a_{n}}$,
where $a_{n}\nearrow 1$ is a suitable sequence of positive numbers.

The functions $b_{a}\colon\mathbb{D}\to\mathbb{D}$ have degree two, fix the
origin, and have a single critical point at

$c_{a}=\frac{-1+\sqrt{1-a^{2}}}{a}.$

The critical point approaches $-1$ as $a\to 1$, as does its value
$v_{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=b_{a}(c_{a})$.
Clearly $b_{a}$ converges to the identity as $a\to 1$; in particular, for any
$r<1$, $b_{a}$ is injective on $D(0,r)$ for $a$ sufficiently close to $1$. (In
fact, it is easy to see that $b_{a}$ is injective on $D(0,\lvert
c_{a}\rvert)$.)

We now construct the numbers $a_{n}$ inductively. As part of the construction,
we also define the increasing sequence $r_{n}$, as well as compact subsets
$A_{k}^{n}\subset\mathbb{D}$, for $0\leq k\leq n+1$, for $n\geq 0$, each of
which is a finite union of topological discs. These are chosen to have the
following properties for all $n\geq 0$.

  1. (i)

$b_{n}$ is injective on $D(0,r_{n})$.

  2. (ii)

$0\notin A_{k}^{n}$ for all $k\leq n+1$ and
$c_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=c_{a_{n}}\in
A_{k}^{n}$ for all $k\leq n$.

  3. (iii)

$b_{k}^{-1}(A_{k+1}^{n})=A_{k}^{n}$ for $0\leq k\leq n$.

  4. (iv)

If $n\geq 1$, every connected component of $A_{k}^{n-1}$ is also a connected
component of $A_{k}^{n}$ and $(A_{k}^{n+1}\setminus A_{k}^{n})\cap
D(0,r_{n})=\emptyset$ for $0\leq k\leq n$.

Suppose that $m\geq 0$ is such, for all $n<m$, the numbers $a_{n}$ and sets
$A_{k}^{n}$ have been defined with the above properties. (This inductive
hypothesis is empty for $m=0$.) If $m=0$, set
$r_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=1/2$; otherwise,
choose $r_{m}\in(r_{m-1},1)$ such that $A_{m}^{m-1}\subset D(0,r_{m})$. Then
choose $a_{m}$ close enough to $1$ to ensure that $c_{m}<-r_{m}$. With this
choice, $b_{m}$ is injective on $D(0,r_{m})$, so i holds for $n=m$.

Next, choose $\epsilon>0$ so small that $D(c_{m},\epsilon)\cap
b_{m}(D(0,r_{m}))=\emptyset$, and set

$A_{m+1}^{m}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\overline{D(c_{m},\epsilon)}\cup
b_{m}(A_{m}^{m-1}).$ (4)

We define $A_{k}^{m}$, for $k=0,\dots,m$ recursively according to iii.

Let us consider the connected components of
$A_{m}^{m}=g_{m}^{-1}(A_{m+1}^{m})$.One such component is given by
$b_{m}^{-1}(D(c_{m},\epsilon))$, which is disjoint from $D(0,r_{m})$ by our
choice of $\epsilon$. Every other component of $A_{m}^{m}$ is a connected
component of $b_{m}^{-1}(b_{m}(A))$, where $A$ is a connected component of
$A_{m}^{m-1}\subset D(0,r_{m})$. Since $b_{m}$ is injective on $D(0,r_{m})$,
there are two such components: $A$ itself, and a second component
$A^{\prime}\subset\mathbb{D}\setminus D(0,r_{m})$. This establishes iv for
$n=k=m$. For $k<m$ iv follows inductively from iii and the Schwarz lemma.

Furthermore, $0\notin A_{m+1}^{m}$ by (4) and the inductive hypothesis for
$n=m-1$, so also $0\notin A_{k}^{m}$ for $k<m+1$ by iii (recall that all the
functions $b_{k}$ fix $0$). Moreover, $c_{m}\in A_{m}^{m}$, and by the
inductive hypothesis and iv we also have $c_{k}\in A_{k}^{m}$ for $k<m$. This
completes the recursive construction.

For $n\geq 0$, we define

$U_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{D}\setminus\bigcup_{k=n}^{\infty}A_{n}^{k}.$

Observe that, by iv, $U_{n}$ is a subdomain of $\mathbb{D}$, and furthermore
$0\in U_{n}$, $c_{n}\notin U_{n}$, and $b_{n}^{-1}(U_{n+1})=U_{n}$. It follows
that $b_{n}\colon U_{n}\to U_{n+1}$ is indeed a degree two covering map. The
proof of the proposition is complete. ∎

Now fix the sequences $(b_{n})_{n=0}^{\infty}$ and $(U_{n})_{n=0}^{\infty}$,
and define

$T_{n}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=z+4n,\quad
K_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\overline{T_{n}(U_{n})}\quad\text{and}\quad
f_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=T_{n+1}\circ
b_{n}\circ(T_{n})^{-1}.$

By Theorem 2.1, there exists now a meromorphic function
$g\colon\mathbb{C}\to\widehat{\mathbb{C}}$, compact sets $\widetilde{K}_{n}$,
and homeomorphisms $\theta_{n}\colon K_{n}\to\widetilde{K}_{n}$ such that
$\mathrm{int}(\widetilde{K}_{n})$ is a Fatou component of $g$ for each $n\geq
0$, and $g\colon\widetilde{K}_{n}\to\widetilde{K}_{n+1}$ is conjugate to
$f_{n}\colon K_{n}\to K_{n+1}$ by the conformal maps $\theta_{n}$. It follows
that $g\colon\widetilde{K}_{n}\to\widetilde{K}_{n+1}$ is an unbranched
covering of degree two for all $n\geq 0$.

Finally, let $r\leq r_{0}$ be sufficiently small to ensure that the hyperbolic
ball in $U_{n}$ of radius $r$ around $0$ is simply connected. Then, since
$g\colon\mathrm{int}(\widetilde{K}_{n})\to\mathrm{int}(\widetilde{K}_{n+1})$
is a degree two covering map, we know from the Schwarz–Pick lemma that the
wandering domains
$V_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{int}(\widetilde{K}_{n})$
are an orbit of infinitesimally isometric wandering domains. Furthermore, the
functions $b_{n}$ are injective in the disc $D(0,r_{n})\supset D(0,r)$ by
Proposition 6.1, and therefore $f_{n}\colon K_{n}\to K_{n+1}$ is injective on
the hyperbolic disc $D_{\mathrm{int}(K_{n})}(4n,r)$ (notice that, since
$K_{n}\subset D(4n,1)$, the hyperbolic disc
$D_{\mathrm{int}(K_{n})}(4n,r_{n})\supset D_{\mathrm{int}(K_{n})}(4n,r)$ is
contained in the Euclidean disc $D(4n,r_{n})$ where $b_{n}$ is injective). We
conclude that $g$ is injective in the discs $D_{V_{n}}(\theta_{n}(4n),r)$,
whence $d_{V_{n}}(g^{n}(z),g^{n}(w))=d_{V_{0}}(z,w)$ for all $n\in\mathbb{N}$
for any pair $z,w\in D_{V_{0}}(\theta_{0}(0),r)$ since $g$ is a local
hyperbolic isometry. Theorem 1.6 now follows from [Fer24, Theorem 1.1].

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