# The Denjoy–Wolff set for holomorphic sequences, non-autonomous dynamical
systems and wandering domains

Anna Miriam Benini Partially supported by Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA, INdAM) and by PRIN
2017, Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics.
Dep. of Mathematical, Physical and Computer Sciences, Università di Parma,
Italy. Vasiliki Evdoridou  Supported by the EPSRC grant EP/R010560/1. School
of Mathematics and Statistics, The Open University, Milton Keynes, UK.
Mathematical Sciences Research Institute, Berkeley, CA, USA.  Núria Fagella
Partially supported by the Spanish grant MTM2017-86795-C3-3-P and
PID2020-118281GB-C32, and the Catalan grants 2017SGR1374 and ICREA Academia
2020. Mathematical Sciences Research Institute, Berkeley, CA, USA.  Dep. de
Matemàtiques i Informàtica, Universitat de Barcelona, Catalonia, Spain.
Centre de Recerca Matemàtica, Bellaterra, Catalonia, Spain.  Philip J. Rippon
† School of Mathematics and Statistics, The Open University, Milton Keynes,
UK.  Gwyneth M. Stallard † School of Mathematics and Statistics, The Open
University, Milton Keynes, UK.

###### Abstract

There are many classical results, related to the Denjoy–Wolff Theorem,
concerning the relationship between orbits of interior points and orbits of
boundary points under iterates of holomorphic self-maps of the unit disc. Here
we address such questions in the much more general setting of sequences
$(F_{n})$ of holomorphic maps between simply connected domains. We show that,
while some classical results can be generalised, with an interesting
dependence on the geometry of the domains, a much richer variety of behaviours
is possible. Some of our results are new even in the classical setting.

Our methods apply in particular to non-autonomous dynamical systems, when
$(F_{n})$ are forward compositions of holomorphic maps, and to the study of
wandering domains in holomorphic dynamics.

We construct examples to illustrate the possible behaviours that can occur.
Our proofs use techniques from geometric function theory, measure theory and
ergodic theory, and the examples involve a ‘weak independence’ version of the
second Borel–Cantelli lemma and the concept from ergodic theory of ‘shrinking
targets’.

##  1 Introduction

We consider sequences of holomorphic maps $F_{n}:U\to U_{n},n\in{\mathbb{N}},$
between simply connected domains, and prove several results concerning the
relationship between sequences $(F_{n}(z))$, where $z$ is an interior point of
$U$ and $(F_{n}(\zeta))$, where $\zeta$ lies in the boundary of $U$. Of
course, in general, the maps $F_{n}$ are not defined on the boundary of $U$
but we shall define a ‘radial extension’ of $F_{n}$ which in many cases exists
at almost all boundary points, with respect to harmonic measure – in this
situation we say that $F_{n}$ has a ‘full radial extension’ to $\partial U$.
We discuss radial extensions in detail in Section 2. For simplicity, we use
the notation $F_{n}$ to refer also to the extension of our original map at
these boundary points. We assume throughout the paper that simply connected
domains are proper subsets of the complex plane.

An important case of our general setting is non-autonomous dynamical systems,
where $F_{n}=f_{n}\circ\cdots\circ f_{1}$ is a forward composition of
holomorphic maps $f_{n}:U_{n-1}\to U_{n}$, $n\in{\mathbb{N}}$, where
$U=U_{0}$; some additional results hold in this case that are not true for
general sequences. In the case that $F_{n}$ is a forward composition, our
results concern the relationship between orbits of interior points and orbits
of boundary points. For simplicity, we describe the sequence of images
$(F_{n}(z)$), where $z\in U$ or $z\in\partial U$, as an orbit irrespective of
whether or not $F_{n}$ is defined by forward composition.

There are several natural questions which we address: does the behaviour of
interior orbits determine the behaviour of boundary orbits (and vice-versa)?
If all interior orbits approach the boundary, then do they approach the orbit
of a particular point on the boundary and, if so, how large is the set of
boundary points with similar limiting behaviour? These and other related
questions are investigated in this paper.

A particular type of non-autonomous dynamical system arises in the case that
$U$ is a wandering domain of a holomorphic map $f$ and $F_{n}$ is the $n$-th
iterate $f^{n}$. In fact, obtaining a better understanding of the boundary
dynamics for such domains was the original motivation for this paper. We
refer, for example, to [Ber93, CG93] for background on holomorphic dynamics.
The iteration of $f$ induces a partition of the complex plane into two
completely invariant sets: the Fatou set $F(f)$, on which the sequence
$(f^{n})$ forms a normal family and its complement, the Julia set $J(f)$, a
closed fractal set where chaotic dynamics occur. There is an induced dynamics
on the set of connected components of the Fatou set, called Fatou components,
since they map amongst themselves. A Fatou component $U$ is wandering if
$f^{n}(U)\cap f^{m}(U)=\emptyset$ for all $n\neq m$ and (pre)periodic
otherwise.

Wandering domains are the least understood of all Fatou components, and the
subject of some important and difficult open questions in the field.
Establishing the presence or the absence of wandering domains in a given
system may have consequences in terms of structural stability (or the absence
thereof), by affecting for example the dimension of the Teichmüller space of
the map [FH08]. Understanding the possible dynamics of the map within and on
the boundary of the wandering domains, in the way that we already do for
periodic components, is therefore a natural goal.

A new point of view was taken in [BEF+21], which led to a classification of
all possible interior dynamics inside a simply connected wandering domain from
two different perspectives; firstly in terms of the hyperbolic distances
between orbits and secondly in terms of the behaviour of orbits in relation to
the boundary. We showed that all interior points have the same type of
dynamical behaviour, giving nine different classes of simply connected
wandering domains, all of which are realisable. In this paper, we go one step
further and show how the dynamics of a function on the boundary of a wandering
domain relates to the behaviour of the interior orbits - in particular, in the
case that the interior orbits converge to the boundary. We refer to [Bak02,
BFJK20b, RS11, BRS13, ERS20, MPRW22, MPS20, MBRG13] for other aspects of the
dynamics of wandering domains.

Many authors have studied another special case of our general setting in which
the functions $F_{n}$ are iterates $f^{n}$ of a holomorphic self-map $f$ of
the unit disc ${\mathbb{D}}$. In this setting we have the following classical
result, known as the Denjoy–Wolff Theorem; see, for example, [CG93, p.79].

######  Theorem 1.1 (Denjoy–Wolff Theorem).

Let $f:{\mathbb{D}}\to{\mathbb{D}}$ be holomorphic. Provided $f$ is not a
rotation about a point in ${\mathbb{D}}$, there exists a unique point
$p\in\overline{{\mathbb{D}}}$ (the Denjoy–Wolff point) such that $f^{n}(z)\to
p$ as $n\to\infty$ for all $z\in{\mathbb{D}}$.

In particular, for iterates of self-maps of the unit disc, all interior points
have the same limiting behaviour under iteration whenever there is a
Denjoy–Wolff point. In our general setting, orbits of interior points do not
necessarily converge together in the same way. We do, however, have the
following result which shows that, if the orbit of one interior point
converges towards the boundary, then the orbits of all interior points not
only converge to the boundary but have a common limiting behaviour.

First though we remark that the concept of orbits of points ‘approaching the
boundary’ is delicate to define in our general setting since it depends on the
shape of the sets $U_{n}$, which may become highly distorted. There are
alternative candidates for the definition of convergence to the boundary, but
in [BEF+21] and in this paper we use a definition based on Euclidean distance,
denoted by $\operatorname{dist}(.,.)$; see Section 3 for some further
possibilities.

######  Theorem A.

Let $F_{n}:U\to U_{n},n\in{\mathbb{N}},$ be a sequence of holomorphic maps
between simply connected domains. If there exists $z_{0}\in U$ such that

$\operatorname{dist}(F_{n}(z_{0}),\partial U_{n})\to 0\;\text{ as
}n\to\infty,$

then, for all $z\in U$,

$|F_{n}(z)-F_{n}(z_{0})|\to 0\;\text{ as }n\to\infty.$

Throughout this paper, the reader should bear in mind that, whenever we
consider a sequence $(F_{n})$ of holomorphic maps between simply connected
domains for which interior orbits converge to the boundary, then it follows
from Theorem A that these orbits all have the same limiting behaviour.

We show in Section 3 that Theorem A follows from a hyperbolic metric estimate
used in the proof of one of our classification results in [BEF+21] on the
internal dynamics of wandering domains.

We note that there are sequences of maps for which the orbits of all interior
points converge together without approaching the boundary (see Section 3 for a
classification of such behaviour) and there are also sequences for which the
interior orbits do not converge together; see Example 7.3. In the case where
the orbits of all the interior points converge to the boundary and hence
converge together by Theorem A, it is natural to ask whether there are any
boundary points whose orbits have the same limiting behaviour. This question
motivates the following definition, which is a natural generalisation of the
Denjoy–Wolff point to our setting.

######  Definition 1.2.

Let $F_{n}:U\to U_{n},n\in{\mathbb{N}},$ be a sequence of holomorphic maps
between simply connected domains, for which interior orbits converge to the
boundary. The Denjoy–Wolff set of $(F_{n})$ consists of those points
$\zeta\in\partial U$ such that the orbit $(F_{n}(\zeta))$ is well-defined and,
for all $z\in U$,

$|F_{n}(\zeta)-F_{n}(z)|\to 0\;\text{ as }n\to\infty.$

Note that in our general setting the orbits of boundary points of $U$ need not
lie on the boundaries of the sets $U_{n},n\in{\mathbb{N}}$; that is, for
$\zeta\in\partial U$, they only satisfy $F_{n}(\zeta)\in\overline{U_{n}}$, for
$n\in{\mathbb{N}}$.

Also, note that in the classical case of iterates of holomorphic self-maps of
the unit disc, if the Denjoy–Wolff point $p$ of a function $f$ is in
$\partial{\mathbb{D}}$, then $f$ is defined at $p$ via radial limits and $p$
is a fixed point of $f$ and belongs to the Denjoy–Wolff set defined above.

Our main goal for this paper is to prove results on the size of the
Denjoy–Wolff set for different sequences of holomorphic maps, measuring size
in terms of harmonic measure (discussed further in Section 2).

One motivation for our results is the following remarkable dichotomy, which
originates in the work of Neuwirth [Neu78], Aaronson [Aar78], and Doering and
Mañé [DM91]. This result concerns the special case that $F_{n}$ is the $n$th
iterate of an inner function $f$; that is, $f$ is a holomorphic self-map of
the unit disc ${\mathbb{D}}$ whose radial extension maps
$\partial{\mathbb{D}}$ to $\partial{\mathbb{D}}$, apart from a set of measure
zero. The dichotomy shows that, in this special case, if interior orbits tend
towards the boundary sufficiently quickly, then almost all boundary points are
in the Denjoy–Wolff set, whereas if interior orbits tend towards the boundary
more slowly, then almost no boundary points are in the Denjoy–Wolff set.

The statement below is a modification of the original dichotomy given in
[Aar78] and [DM91], more suited to our present purposes.

######  Theorem 1.3 (ADM dichotomy).

Let $f:{\mathbb{D}}\to{\mathbb{D}}$ be an inner function with a Denjoy–Wolff
point $p\in\overline{\mathbb{D}}$.

  1. (a)

If $\sum_{n\geq 0}(1-|f^{n}(z)|)<\infty$ for some $z\in{\mathbb{D}}$, then
$p\in\partial{\mathbb{D}}$ and $\lim_{n\to\infty}f^{n}(\zeta)=p$ for almost
every $\zeta\in\partial{\mathbb{D}}$.

  2. (b)

If $\sum_{n\geq 0}(1-|f^{n}(z)|)=\infty$ for some $z\in{\mathbb{D}}$, then the
iterates $f^{n}(\zeta),n\in{\mathbb{N}},$ are dense in $\partial{\mathbb{D}}$
for almost every $\zeta\in\partial{\mathbb{D}}$; in particular, the set of
boundary points converging to $p$ has measure 0.

Theorem 1.3, part (a) is in [Aar78, Section 3] and [DM91, Theorem 4.1]. The
proof of part (b) follows from several results in ergodic theory, and we
outline this in Section 8.3, where we also discuss other aspects of this proof
relating to questions that our paper leaves open.

We remark that Theorem 1.3, part (a) was generalised from inner functions to
holomorphic self-maps of $\mathbb{D}$ in [BMS05, Theorem 4.2], which also
includes a slightly weaker version of part (b) for inner functions.

Given these results, it is natural to ask to what extent this ADM dichotomy
holds in our general setting. In Section 4 we show that part (a) does have an
analogue in the general setting. Interestingly, the result that we obtain
depends on the geometry of the boundaries of the domains $(U_{n})$. In the
case that all the domains are the unit disc, then we obtain a direct analogue
of the result for iterates of inner functions.

######  Theorem B.

Let $F_{n},n\in{\mathbb{N}},$ be a sequence of holomorphic self-maps of
$\mathbb{D}$ and suppose that there exists $z_{0}\in\mathbb{D}$ such that

$\sum_{n=0}^{\infty}\left(1-|F_{n}(z_{0})|\right)<\infty.$

Then almost all points in $\partial{\mathbb{D}}$ belong to the Denjoy–Wolff
set of $(F_{n})$.

In our new setting of general simply connected domains $U$ and $U_{n}$ instead
of the unit disc, an analogue of the condition in Theorem B would be that

$\sum_{n=0}^{\infty}\operatorname{dist}(F_{n}(z_{0}),\partial U_{n})<\infty.$
(1.1)

However, the less smooth the boundaries of these domains are, the stronger the
hypothesis we need. Without any smoothness assumptions on the boundaries, we
can prove the following.

######  Theorem C.

Let $F_{n}:U\to U_{n},n\in{\mathbb{N}},$ be a sequence of holomorphic maps
between simply connected domains, each with a full radial extension to
$\partial U$, and suppose that there exists $z_{0}\in U$ such that

$\sum_{n=0}^{\infty}\operatorname{dist}(F_{n}(z_{0}),\partial
U_{n})^{1/2}<\infty.$ (1.2)

Then almost all points in $\partial U$ belong to the Denjoy–Wolff set of
$(F_{n})$.

######  Remarks.

1\. Condition (1.2) in Theorem C is sharp. Indeed, an example in Section 4
shows that the power $1/2$ in (1.2) cannot be increased.
2\. A special case of Theorem C is when $(U_{n})$ is an orbit of simply
connected wandering domains of a transcendental entire or meromorphic function
$f$. Note that in this case $F_{n}=f^{n}$ always has a continuous extension to
the boundary, except perhaps at $\infty$.

Despite this successful generalisation of the first part of the ADM dichotomy
(Theorem 1.3, part (a)), the second part of the dichotomy fails in the general
setting of sequences of functions, and even in the setting of non-autonomous
dynamical systems; indeed, under the same assumption on interior orbits
converging to the boundary, very different types of behaviour can occur on the
boundary itself. We illustrate this in Section 8 with a number of examples.

Having seen that fast convergence of interior points to the boundary implies
that the Denjoy–Wolff set has full measure, in Section 5 we give a result
which shows, perhaps surprisingly, that there are examples for which the
orbits of all interior points converge together towards the boundary and yet
the Denjoy–Wolff set is empty. (This cannot occur in the classical case since
the Denjoy–Wolff point always belongs to the Denjoy–Wolff set.)

######  Theorem D (Empty Denjoy–Wolff set).

Let $(a_{n})$ be an increasing sequence in $[0,1)$ such that $a_{0}=0$ and
$\lim_{n\to\infty}a_{n}=1$. Given any sequence $(\lambda_{n})$ in
$\partial\mathbb{D}$, consider the sequence of Möbius maps $F_{n}$ defined as

$F_{n}(z)=\frac{\lambda_{n}z+a_{n}}{1+\lambda_{n}a_{n}z}.$

Then $F_{n}(z)\to 1$ as $n\to\infty$, for all $z\in{\mathbb{D}}$.

If, in addition, $\sum_{n\geq 0}(1-a_{n})=\infty,$ then there exists
$(\lambda_{n})$ in $\partial{\mathbb{D}}$ such that no point of
$\partial\mathbb{D}$ (even the point 1) has an orbit under $F_{n}$ that
converges to 1; that is, the Denjoy–Wolff set of $(F_{n})$ is empty.

######  Remark.

Note that if $\sum_{n\geq 0}(1-a_{n})<\infty$ in Theorem D, then it follows
from Theorem B that almost all points in $\partial{\mathbb{D}}$ are in the
Denjoy–Wolff set of $(F_{n})$.

Each of the functions $F_{n}$ in Theorem D can in fact be written as a forward
composition of Möbius maps $f_{n}$ with $F_{n}=f_{n}\circ F_{n-1}$. Thus, non-
autonomous systems of holomorphic self-maps of ${\mathbb{D}}$ may have an
empty Denjoy–Wolff set, even when the orbits of interior points converge to
the boundary. It remains an open question as to whether this behaviour can
occur for wandering domains.

So far, we have considered the effect of the behaviour of interior points on
the behaviour of boundary points. In Section 6 we consider the converse
problem. We show that if a sufficiently large set of boundary points have
orbits with a common long-term behaviour then this set must be in the
Denjoy–Wolff set. That is, the orbits of all interior points must have the
same limiting behaviour as the orbits of this set of boundary points. This is
new even for iterates of inner functions.

######  Theorem E.

Let $F_{n}:U\to U_{n},n\in{\mathbb{N}},$ be a sequence of holomorphic maps
between simply connected domains, each with a full radial extension, and
suppose that there exists $\zeta_{0}\in\partial U$ such that
$\operatorname{dist}(F_{n}(\zeta_{0}),\partial U_{n})\to 0$ as $n\to\infty$.
If

$L_{0}:=\\{\zeta\in\partial U:|F_{n}(\zeta)-F_{n}(\zeta_{0})|\to 0\;\;\text{as
}n\to\infty\\}$

has positive harmonic measure with respect to $U$, then $L_{0}$ is in the
Denjoy–Wolff set of $(F_{n})$.

Note that it follows from the conclusion of the theorem that only one such set
of positive measure may exist.

For the later sections of the paper we restrict ourselves to the setting of
non-autonomous dynamical systems. In this setting, each map $F_{n}$ is the
forward composition of holomorphic maps $f_{n}:U_{n-1}\to U_{n}$ between
simply connected domains. Under this assumption, we give a classification
result for internal dynamics in terms of the limiting behaviour of the
hyperbolic distance between orbits of pairs of points, as we did in the
special case of wandering domains [BEF+21] and using the same method of proof.
We give this classification in Section 7, together with an example which shows
that the classification does not hold in general for sequences of holomorphic
maps that are not given by forward compositions. One of the three
possibilities in our classification is that the maps are contracting, defined
as follows. Here we use the notation $\operatorname{dist}_{U}$ to denote the
hyperbolic distance within a simply connected domain $U$.

######  Definition 1.4.

A sequence of maps $F_{n}:U\to U_{n},n\in{\mathbb{N}},$ is contracting if

$\operatorname{dist}_{U_{n}}(F_{n}(z),F_{n}(z^{\prime}))\to 0\;\text{ as
}n\to\infty,\quad\mbox{for all }z,z^{\prime}\in U.$

For contracting forward compositions of holomorphic maps which preserve
boundaries, we prove the strong result that the Denjoy–Wolff set has either
full or zero measure.

######  Theorem F.

Let $f_{n}:U_{n-1}\to U_{n}$, $n\in{\mathbb{N}}$, be a sequence of holomorphic
maps between simply connected domains, each with a full radial extension, and
let $F_{n}:U_{0}\to U_{n}$ be defined by

$F_{n}=f_{n}\circ\cdots\circ f_{1},\quad\text{for }n\in{\mathbb{N}},$

with the additional assumption that

$f_{n}(\zeta)\in\partial U_{n},\;\text{ for almost all }\zeta\in\partial
U_{n-1}.$ (1.3)

If $(F_{n})$ is contracting and interior orbits converge to the boundary, then
the Denjoy–Wolff set has either full or zero harmonic measure with respect to
$U_{0}$.

######  Remarks.

1\. In Section 2, we show that the composition of functions with full radial
extensions also has a full radial extension; for the moment, we just remark
that this is the case in the special cases where $U_{0}$ is a wandering domain
of an entire function $f$ with $F_{n}=f^{n}$ for $n\in{\mathbb{N}}$ or where
all the functions $f_{n}$, $n\in{\mathbb{N}}$, are inner functions.

2\. Several results in this paper, including Theorem F, can be applied when
the domains $(U_{n})$ are multiply connected and we are concerned with orbits
converging to the ‘outer boundary’; that is, the boundary of the topological
hull of $U_{n}$. This approach is used in [RF21, Theorem 1.4], where a result
analogous to Theorem A is given in the setting of dynamical behaviour near the
outer boundary of multiply connected wandering domains of meromorphic
functions.

In Section 7, we prove a general result, Theorem 7.4, that has Theorem F as a
corollary. Our proof of Theorem 7.4 uses a result of Pommerenke and
generalises one aspect of it. We also give an example to show that Theorem F
does not hold in general if the forward compositions are not contracting.

In Section 8 of the paper we give a number of examples of forward compositions
of self-maps of the unit disc. These illustrate the different types of
behaviour that can occur if interior orbits approach the boundary sufficiently
slowly that the sum in Theorem B is divergent. Our final example, Example 8.3,
is a contracting sequence; for this example, we are able to show that the
Denjoy–Wolff set has measure zero and, in fact, that orbits of almost all
boundary points are dense in $\partial{\mathbb{D}}$. This raises the question
of whether an analogue of part (b) of the ADM dichotomy may exist in the
setting of contracting forward compositions. We discuss this in Section 8.3
and also in the final section of the paper on open questions.

In Section 9, we discuss briefly the extent to which our results hold if we
replace the Euclidean metric as a measure of proximity by the spherical
metric; the latter metric has the advantage of including those points whose
orbits tend to $\infty$ but at the same time we potentially lose control of
some of the more subtle behaviour amongst those orbits that do tend to
$\infty$.

The proofs of Theorems A to F and of our examples use a variety of techniques
from geometric function theory, harmonic analysis and measure theory including
contraction of the hyperbolic metric, Löwner’s lemma, the Milloux–Schmidt
inequality, the Severini–Egorov theorem, and a version of the second
Borel–Cantelli lemma associated with the ergodic theory concept of ‘shrinking
targets’.

The paper is structured as follows. In Section 2 we give several preliminary
results concerning the extension of our maps to certain points on the boundary
and estimates of harmonic measure of parts of the boundary and their images.
In Section 3 we classify the possible behaviours of orbits of interior points
in relation to their distance to the boundary, and prove Theorem A.

In Section 4 we prove our analogue of part (a) of the ADM dichotomy; Theorems
B and C are special cases of this result. The proof of Theorem D, showing that
the Denjoy–Wolff set can be empty, can be found in Section 5, while Section 6
contains the proof of Theorem E.

In Section 7 we consider the case that our holomorphic maps are forward
compositions and give a classification with regards to the hyperbolic distance
between orbits of pairs of points. We prove Theorem F, showing that in a
particular situation when the maps are contracting, the Denjoy–Wolff set has
either full or zero measure.

In Section 8 we give examples related to part (b) of the ADM dichotomy for
self-maps of the unit disc. Whether the interior and boundary dynamics
demonstrated in these examples can be replicated by examples of simply
connected wandering domains is an open question, and will be the subject of
further investigation. In Section 9 we briefly discuss the possible extensions
of our results where the Euclidean metric is replaced by the spherical metric
and finally, in Section 10, we state several additional open questions related
to this work.

##  2 Preliminary results

In this section we record a number of properties of certain boundary
extensions of holomorphic maps between simply connected domains and the
relationship of these boundary extensions to the harmonic measure of sets on
the boundaries of these domains. In doing so, we use a number of classical
results on the boundary behaviour of holomorphic maps, which can mostly be
found in [Pom92].

In this section only, we denote such boundary extensions of a holomorphic map
$f$ by $f^{*}$ for clarity; elsewhere in the paper we drop the * notation for
simplicity.

###  2.1 Boundary extensions

Recall that our general setting is that of sequences of holomorphic maps

$F_{n}:U\longrightarrow U_{n},\quad n\in{\mathbb{N}},$

where $U$ and $U_{n},n\in{\mathbb{N}},$ are simply connected domains, always
assumed to be proper subsets of ${\mathbb{C}}$ and not necessarily distinct,
and that in some cases the functions $F_{n}$ are defined by forward
composition as $F_{n}=f_{n}\circ\cdots\circ f_{1}$, where each
$f_{n}:U_{n-1}\to U_{n}$ is holomorphic and $U_{0}=U$.

In the wandering domains context, where $F_{n}=f^{n}$ with $f$ entire, and
$f:U_{n-1}\to U_{n}$, the maps are naturally defined and continuous on the
whole of $\partial U_{0}$ (except maybe at the point at infinity). In the
inner functions context, however, where $U_{n}={\mathbb{D}}$, for $n\geq 0$,
each $F_{n}$ extends in the sense of radial limits and even nontangential
limits to a full measure subset of $\partial{\mathbb{D}}$ by theorems of Fatou
and Lindelöf; see [CL66, p.19], for example. We now consider whether such
boundary extensions exist in the general setting of this paper.

We shall suppose quite generally that $U$ and $V$ are simply connected proper
subdomains of ${\mathbb{C}}$ and $f:U\to V$ is holomorphic. We wish to define
the concept of a ‘radial extension’ of $f$ to boundary points of $U$, wherever
this extension makes sense.

A simple case is obtained when $U={\mathbb{D}}$, that is, $f:{\mathbb{D}}\to
V$ is a holomorphic map (not necessarily conformal). If $V$ is bounded, then
the radial limit at $\zeta\in\partial{\mathbb{D}}$ exists if and only if the
non-tangential limit at $\zeta$ exists and this occurs if and only the limit
exists along any curve in ${\mathbb{D}}$ ending at $\zeta$, by the theorem of
Lindelöf mentioned above; moreover, all these limits coincide. In the general
case, since the complement of $V$ contains an unbounded continuum, the
function

$g(z)=\sqrt{f(z)-w_{0}}$

where $w_{0}\notin V$ and a suitable branch of the square root is taken, is
holomorphic in ${\mathbb{D}}$ and $g({\mathbb{D}})$ omits a disc, with centre
$w_{1}$ say; thus the function $z\mapsto(g(z)-w_{1})^{-1}$ is holomorphic and
bounded in ${\mathbb{D}}$, so we see that the relationship between radial
limits, nontangential limits and limits along curves remains true for such
$f$. Therefore it makes sense to define the radial extension of such an $f$ as
the map

$f^{*}(\zeta)=\lim_{r\rightarrow 1}f(r\zeta),\;\text{ for
}\zeta\in\operatorname{Rad}(f),$

where $\operatorname{Rad}(f)$ denotes the subset of $\partial{\mathbb{D}}$ at
which $f$ has a finite radial limit. The function $f^{*}$ assigns to each
point of $\partial{\mathbb{D}}$ its radial limit, whenever it exists, and
takes values in the closure of $f({\mathbb{D}})$. Note that even when $f$ has
a nontangential limit at $\zeta\in\partial{\mathbb{D}}$, the function $f$ may
fail to converge to that limit along curves in ${\mathbb{D}}$ ending at
$\zeta$ that are not nontangential.

Moving to the case where $f:U\to V$, with $U$ and $V$ being proper simply
connected domains, we introduce a Riemann map $\varphi:{\mathbb{D}}\to U$. By
the discussion above, $\varphi$ has radial (and equal nontangential) limits
almost everywhere on $\partial{\mathbb{D}}$ and in fact
$\partial{\mathbb{D}}\setminus\operatorname{Rad}(\varphi)$ has capacity zero
by a theorem of Beurling [Pom92, Theorem 9.19].

A point $p\in\partial U$ is called accessible from $U$ if there exists an arc
$\gamma:[0,1)\to U$ such that $\gamma(t)\to p$ as $t\to 1$. We denote the set
of accessible boundary points of $U$ by $\operatorname{AP}(U)$ (there seems to
be no standard notation for this set). Every homotopy class of curves
$\gamma\subset U$ with their endpoints fixed and one endpoint at an accessible
boundary point $p$ is called an access to $p$ from $U$. It follows from
another theorem of Lindelöf [CG93, page 8] that there is a bijection between
the set $\operatorname{Rad}(\varphi)$ and the set of accesses to boundary
points of $U$, induced by $\varphi$; see [BFJK17, Correspondence Theorem], for
example.

In defining the concept of a radial extension of $f:U\to V$, our aim is that
the extension at almost all accessible boundary points of $U$ is consistent
with the limiting behaviour of $f$ along curves to $\partial U$ through all
possible accesses to such points.

######  Definition 2.1 (Radial extensions).

Let $f:U\to V$ be a holomorphic map between simply connected domains, let
$p\in\operatorname{AP}(U)$, and let $\varphi:{\mathbb{D}}\to U$ be a Riemann
map. Then we say that

  * (a)

a curve in $U$ approaches $p$ radially if it is the image of a curve in
${\mathbb{D}}$ that approaches a point of $\partial{\mathbb{D}}$
nontangentially;

  * (b)

$f$ has a radial extension to $p$ with value $f^{*}(p)$ if for every arc
$\gamma(t),t\in[0,1),$ in $U$ such that $\gamma(t)\to p$ radially as $t\to 1$,
the image curve $f(\gamma(t))\to f^{*}(p)$ as $t\to 1$;

  * (c)

$f$ has a full radial extension (to $\partial U$) if the domain of $f^{*}$
includes a subset of $\operatorname{AP}(U)$ of full harmonic measure with
respect to $U$.

We make two remarks about Definition 2.1.

######  Remarks.

1\. We focus on the limiting behaviour of $f$ only on curves in $U$ that
approach a boundary point $p$ radially in order to be consistent with the
definition of radial extension of a holomorphic map $f:{\mathbb{D}}\to V$ in
terms of nontangential limits, given above.

2\. The existence of a full radial extension of $f$ implies that (up to a set
of harmonic measure zero) $f$ maps accesses to a point $p\in\partial U$ to
accesses to $f^{*}(p)\in\partial V$, unless $f^{*}(p)$ belongs to $V$, which
is a priori also possible.

In general, the radial extension of a conformal map $\varphi:{\mathbb{D}}\to
U$ is not an injective map, since boundary points of $U$ can have several
(possibly uncountably many) accesses from $U$. Nevertheless
$(\varphi^{*})^{-1}(p)$ is always a set of measure zero, by the F. and M.
Riesz theorem [Dur70]. Boundary points with a single access from $U$ are
called uniquely accessible and domains for which all their accessible boundary
points are uniquely accessible are said to satisfy the _unique accessibility
property_. In this case the radial extension of $\varphi$ is injective (and
therefore bijective).

There are some situations in which full radial extensions exist in a trivial
way; in particular, this is the case when $U$ and $V$ are Fatou components of
a holomorphic map or when $f$ is an inner function, since $U=V={\mathbb{D}}$
in this case. More generally, we have the following.

######  Proposition 2.2 (Simple cases).

Let $f:U\to V$ be a holomorphic map between simply connected domains. If

  1. (a)

$f$ has a continuous extension to $\partial U$, or

  2. (b)

$U$ has the unique accessibility property,

then $f$ admits a full radial extension $f^{*}$. In case (a), we have
$f^{*}:=f|_{\partial U}$.

It is interesting to ask whether if $f:U\to V$ is a holomorphic map between
simply connected domains and $f$ has a full radial extension to $\partial U$,
then this implies that at almost all $\zeta\in\partial U$, with respect to
harmonic measure, either $f$ has a continuous extension to $\zeta$ or the
point $\zeta$ is uniquely accessible from within $U$? It is straightforward to
construct examples where both types of boundary behaviour occur in sets of
positive harmonic measure.

###### Proof of Proposition 2.2.

If $f$ is continuous in $\overline{U}$, then the conclusion that $f$ has a
full radial extension is trivial.

For case (b) let $\varphi_{U},\varphi_{V}$ be Riemann maps from ${\mathbb{D}}$
to $U,V$ respectively and let $g:{\mathbb{D}}\rightarrow{\mathbb{D}}$ be
defined as $g:=\varphi_{V}^{-1}\circ f\circ\varphi_{U}$.

If $U$ has the unique accessibility property, then $\varphi_{U}^{*}$ is
injective, so its inverse is well defined on
$\varphi^{*}(\partial{\mathbb{D}})$, which has full harmonic measure in $U$
because $\varphi_{U}$ is the Riemann map. Hence we can define

$f^{*}:=(\varphi_{V}\circ g)^{*}\circ(\varphi_{U}^{*})^{-1}$

on the set
$E=\operatorname{AP}(U)\cap\varphi_{U}^{*}(\operatorname{Rad}(\varphi_{V}\circ
g))$, which is of full harmonic measure in $U$ since it is the intersection of
two sets of full harmonic measure. With this definition, if $\gamma(t)$ in $U$
converges radially to $p\in E$, then $\varphi_{U}^{-1}(\gamma(t))$ converges
nontangentially to a point
$\zeta\in\operatorname{Rad}(\varphi_{U})\cap\operatorname{Rad}(\varphi_{V}\circ
g)$, so $f$ has a full radial extension. ∎

We also point out to what extent the radial extension property is preserved
under forward composition.

######  Proposition 2.3.

Let $f_{1}:U_{0}\to U_{1}$ and $f_{2}:U_{1}\to U_{2}$ be holomorphic maps
between simply connected domains, with radial extensions $f_{1}^{*}$ to
$E_{0}\subset\operatorname{AP}(U_{0})$ and $f_{2}^{*}$ to
$E_{1}\subset\operatorname{AP}(U_{1})$, respectively.

Then $f_{2}\circ f_{1}:U_{0}\to U_{2}$ has a radial extension to almost all
points of $E_{0}\setminus(f_{1}^{*})^{-1}(\partial U_{1}\setminus E_{1})$.

###### Proof.

For $\zeta_{0}\in E_{0}\setminus(f_{1}^{*})^{-1}(\partial U_{1}\setminus
E_{1})$, let $\gamma_{0}(t),0\leq t<1,$ be a path in $U_{0}$ that converges
radially to $\zeta_{0}$. Then, by our assumption about $\zeta_{0}$, we have
$\gamma_{1}(t):=f_{1}(\gamma_{0}(t))\to\zeta_{1}:=f_{1}^{*}(\zeta_{0})$ as
$t\to 1$; if $\zeta_{1}\in U_{1}$ we are done, otherwise $\zeta_{1}\in E_{1}$.
Note that $\gamma_{1}(t)$ may or may not converge radially to $\zeta_{1}$.

Now let $\hat{\gamma}_{1}(t),0\leq t<1,$ be a path in $U_{1}$ that does
converge radially to $\zeta_{1}$ and is homotopic in $U_{1}$ to
$\gamma_{1}(t)$. Then
$f_{2}(\hat{\gamma}_{1}(t))\to\hat{\zeta}_{2}:=f_{2}^{*}(\zeta_{1})$ as $t\to
1$. Suppose, if possible, that
$f_{2}(\gamma_{1}(t))=f_{2}(f_{1}(\gamma_{0}(t)))\nrightarrow\hat{\zeta}_{2}$
as $t\to 1$. Since $\gamma_{0}(t)$ converges to $\zeta_{0}$ radially, we know
that apart from a subset of $E_{0}$ of harmonic measure zero
$f_{2}(f_{1}(\gamma_{0}(t)))$ converges to a point, $\zeta_{2}\in\partial
U_{2}$ say, and we can only have $\zeta_{2}\neq\hat{\zeta}_{2}$ for countably
many $\zeta_{0}\in E_{0}$, by Bagemihl’s ambiguous point theorem [Pom92],
applied in $U_{1}$. Therefore, for all $\zeta_{0}\in
E_{0}\cap(f_{1}^{*})^{-1}(E_{1})$ apart from a set of harmonic measure zero,
we have $f_{2}(\gamma_{1}(t))=f_{2}(f_{1}(\gamma_{0}(t)))\to\hat{\zeta}_{2}$,
and so $f_{2}\circ f_{1}$ has a radial extension to $\zeta_{0}$, as required.
∎

###  2.2 Properties of harmonic measure

In order to work with our definition of a radial extension, which involves the
harmonic measure of boundary sets of simply connected domains, we first recall
certain topological properties of boundary sets, and the behaviour of their
images and pre-images under holomorphic mappings. Theorem 2.4, part (a) is
given in [Pom92, Proposition 2.5] and part (b) is due to Cantón, Granados and
Pommerenke [CGP03, Theorem 1].

######  Theorem 2.4 (Borel sets on the boundary).

Let $f:{\mathbb{D}}\to\hat{\mathbb{C}}$ be continuous.

  * (a)

The set $\operatorname{Rad}(f)$ is a Borel set and, for any Borel set
$B\subset\hat{\mathbb{C}}$,

$(f^{*})^{-1}(B)=\\{\zeta\in\operatorname{Rad}(f):f^{*}(\zeta)\in B\\}\;\text{
is a Borel set.}$

  * (b)

Moreover, if $f$ is a conformal map and $A\subset\operatorname{Rad}(f)$ is a
Borel set, then

$f^{*}(A)=\\{f^{*}(\zeta):\zeta\in A\\}\;\text{ is a Borel set}.$

In particular, it follows from Theorem 2.4 that the set of accessible points
of $\partial U$ is a Borel set, since $\operatorname{AP}(U)$ is the image
under a Riemann map of the Borel set
$\operatorname{Rad}(f)\subset\partial{\mathbb{D}}$ (of full measure).

Note that every Borel set is a Souslin set (see [Pom92, page 132]) and hence
universally measurable, and in particular measurable with respect to Lebesgue
measure on $\partial{\mathbb{D}}$ and with respect to harmonic measure, which
we discuss next.

We shall briefly recall the definition of harmonic measure and some properties
of it that we need. Good references for harmonic measure are [Con95, Chapter
2], [GM05], [Pom92] and [BCDM20].

A standard definition of harmonic measure in the unit disc is in terms of the
Poisson integral

$\omega(z,A,{\mathbb{D}})=\frac{1}{2\pi}\int_{A}\frac{1-|z|^{2}}{|\zeta-z|^{2}}\,|d\zeta|,\quad
z\in{\mathbb{D}},$

where $A\subset\partial{\mathbb{D}}$ is measurable. The real function $\omega$
is harmonic in ${\mathbb{D}}$ with $0\leq\omega(z,A,{\mathbb{D}})\leq 1$
there, and has radial limit 1 (and indeed nontangential limit 1) at almost
every point in $A$ and radial limit 0 at almost every point in
$\partial{\mathbb{D}}\setminus A$.

It follows that for any such set $A$ we have
$\omega(0,A,{\mathbb{D}})=\lambda(A)$, where $\lambda$ is normalised Lebesgue
measure on $\partial{\mathbb{D}}$, and more generally
$\omega(z,A,{\mathbb{D}})=\lambda(M(A))$, where $M$ is a Möbius map of
${\mathbb{D}}$ onto ${\mathbb{D}}$ taking $z$ to 0; see [Pom92, page 85].

We now define harmonic measure in a general simply connected domain $U$ in
terms of the pullback under a Riemann map of the normalized Lebesgue measure
on the unit circle, following the approach in [BCDM20, Chapter 7].

######  Definition 2.5 (Harmonic measure in a simply connected domain).

Let $U\neq{\mathbb{C}}$ be a simply connected domain, $z\in U$, and let
$\varphi:{\mathbb{D}}\rightarrow U$ be a conformal map such that
$\varphi(0)=z$. The _harmonic measure_ of a Borel set $A\subset\partial U$ is
defined as

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

(Note that $(\varphi^{*})^{-1}(A)$ is a Borel set in $\partial{\mathbb{D}}$ by
Theorem 2.4, part (a).)

Since $\lambda$ is invariant under rotation, this definition is independent of
the choice of $\varphi$ provided it satisfies $\varphi(0)=z$.

The harmonic measure $\omega(z,A,U)$ can also be interpreted as the
probability of hitting the boundary at a point in $A$ while following a
Brownian path in $U$ starting at the point $z$; see [MP10, Section 3.4] and
[GM05, Chapter III], for example.

By the theorem of Beurling mentioned earlier, the set $\operatorname{AP}(U)$
has full harmonic measure with respect to $U$. Also, for any Borel set
$A\subset\partial U$, the function $\omega(z,A,U)$ is harmonic in $U$ and
takes values in the interval $[0,1]$. Moreover, for almost all points
$\zeta\in\operatorname{AP}(U)$ there is a path $\gamma(t)\subset U,0\leq t<1$,
the image of a radius in ${\mathbb{D}}$, such that $\gamma(t)\to\zeta$ and
$\omega(\gamma(t),A,U)\to 1_{A}(\zeta)$ as $t\to 1-$. In this sense, the
function $\omega(z,A,U)$ is the solution of the Dirichlet problem in $U$ with
boundary values $\omega(\zeta,A,{\mathbb{D}})=1_{A}(\zeta)$.

###  2.3 Radial extensions expand harmonic measure

In this section we show that the radial extension of a holomorphic map as
defined in Section 2.1 expands the harmonic measure of sets in the boundary.
Our result depends on the following version of Löwner’s lemma, which can be
found in [Pom92, Proposition 4.15] and [BCDM20, Theorem 7.1.8 and Proposition
7.1.4 part (4)].

######  Theorem 2.6 (Löwner’s lemma).

Let $f$ be a holomorphic self-map of ${\mathbb{D}}$, $f^{*}$ be its radial
extension, and let $S\subset\partial{\mathbb{D}}$ be a Borel set. Then

$\omega(z,(f^{*})^{-1}(S),{\mathbb{D}})\leq\omega(f(z),S,{\mathbb{D}}),\;\text{
for }z\in{\mathbb{D}}.$ (2.1)

######  Remark.

If $f$ is a Möbius map of ${\mathbb{D}}$ onto ${\mathbb{D}}$, or even an inner
function, then equality holds in (2.1) throughout ${\mathbb{D}}$; see [DM91,
Corollary 1.5(b)].

In our proofs we use the following extension of Theorem 2.6 to simply
connected domains.

######  Theorem 2.7 (Expansion on the boundary of simply connected domains).

Let $f:U\to V$ be a holomorphic map between simply connected domains with full
radial extension $f^{*}$ and let $S\subset\partial V$ be a Borel set. Then
$(f^{*})^{-1}(S)$ is a Borel set and, for $z\in U$,

$\omega(z,(f^{*})^{-1}(S),U)\leq\omega(f(z),S,V).$ (2.2)

In particular, if $(f^{*})^{-1}(S)$ has positive (respectively, full) harmonic
measure with respect to $U$, then $S$ has positive (respectively, full)
harmonic measure with respect to $V$.

###### Proof.

First, let $\varphi_{U}$ and $\varphi_{V}$ be Riemann maps from ${\mathbb{D}}$
to $U$ and $V$, respectively. Then $(f^{*})^{-1}(S)$ is a Borel set because it
is the image under $\varphi_{U}^{*}$ of the preimage
under$(f\circ\varphi_{U})^{*}$ of $S$, apart from a possible set of harmonic
measure 0. This follows from our definition of radial extensions and by
applying both parts of Theorem 2.4.

Now define

$g(w)=\varphi_{V}^{-1}\circ f\circ\varphi_{U}(w),\;\text{ for
}w\in{\mathbb{D}},$

so $g$ is a self-map of ${\mathbb{D}}$. Then $(\varphi_{V}^{*})^{-1}(S)$ is a
Borel set, by Theorem 2.4 (a). Also, by our definition of radial extension,
and also the bijection between accesses and non-tangential limits of Riemann
maps, mentioned earlier, it is straightforward to check that apart from a set
of harmonic measure 0 on $\partial{\mathbb{D}}$, we have

$((f\circ\varphi_{U})^{*})^{-1}(S)\subset(g^{*})^{-1}((\varphi_{V}^{*})^{-1}(S)).$

Therefore, by Theorem 2.6, we know that, for $w\in{\mathbb{D}}$,

$\omega(w,((f\circ\varphi_{U})^{*})^{-1}(S),{\mathbb{D}})\leq\omega(w,(g^{*})^{-1}((\varphi_{V}^{*})^{-1}(S)),{\mathbb{D}})\leq\omega(g(w),(\varphi_{V}^{*})^{-1}(S),{\mathbb{D}}).$
(2.3)

Finally, we take $z\in U$ and assume that the Riemann maps have been chosen so
that

$\varphi_{U}(0)=z\quad\text{and}\quad\varphi_{V}(0)=f(z).$

Then $g(0)=0$ and we have, by the definition of harmonic measure,

$\omega(0,((f\circ\varphi_{U})^{*})^{-1}(S),{\mathbb{D}})=\omega(z,(f^{*})^{-1}(S),U)\;\;\text{and}\;\;\omega(0,(\varphi_{V}^{*})^{-1}(S),{\mathbb{D}})=\omega(f(z),S,V),$

so the inequality (2.2) follows by applying (2.3) with $w=0$. ∎

Combining Theorem 2.7 with Proposition 2.3, we obtain the following useful
result about composing full radial extensions.

######  Corollary 2.8.

Let $f_{1}:U_{0}\to U_{1}$ and $f_{2}:U_{1}\to U_{2}$ be holomorphic maps
between simply connected domains, with full radial extensions $f_{1}^{*}$ and
$f_{2}^{*}$, respectively.

Then $f_{2}\circ f_{1}:U_{0}\to U_{2}$ has a full radial extension given by
$f_{2}^{*}\circ f_{1}^{*}$.

###### Proof.

By hypothesis the sets $E_{0}$ and $E_{1}$ specified in Proposition 2.3 have
full harmonic measure with respect to $U_{0}$ and $U_{1}$, respectively, and
we can deduce that $(f_{1}^{*})^{-1}(\partial U_{1}\setminus E_{1})$ must have
zero harmonic measure with respect to $U_{0}$ by Theorem 2.7. It follows that
$E_{0}\setminus(f_{1}^{*})^{-1}(\partial U_{1}\setminus E_{1})$ has full
harmonic measure with respect to $U_{0}$, as required. ∎

##  3 Boundary convergence and proof of Theorem A

Before proving Theorem A, we give a classification of sequences $F_{n}:U\to
U_{n},n\in{\mathbb{N}},$ of holomorphic maps between simply connected domains
in terms of whether orbits $F_{n}(z)$ of interior points $z\in U$ converge
towards the boundary or not. We gave such a classification in the special case
that $U$ is a wandering domain of a holomorphic map $f$ in [BEF+21, Theorem
C]. The same proof holds in our general setting and the techniques used also
lead to a proof of Theorem A.

######  Theorem 3.1 (Boundary convergence classification).

Let $F_{n}:U\to U_{n}$ be a sequence of holomorphic maps between simply
connected domains. Then exactly one of the following holds:

  * (a)

$\liminf_{n\to\infty}\operatorname{dist}(F_{n}(z),\partial U_{n})>0$ for all
$z\in U$, that is, all orbits stay away from the boundary;

  * (b)

there exists a subsequence $n_{k}\to\infty$ for which
$\lim_{k\to\infty}\operatorname{dist}(F_{n_{k}}(z),\partial U_{n_{k}})=0$ for
all $z\in U$, while for a different subsequence $m_{k}\to\infty$ we have that

$\liminf_{k\to\infty}\operatorname{dist}(F_{m_{k}}(z),\partial
U_{m_{k}})>0,\quad\text{for }z\in U;$

  * (c)

$\lim_{n\to\infty}\operatorname{dist}(F_{n}(z),\partial U_{n})=0$ for all
$z\in U$, that is, all orbits converge to the boundary.

Our proof of this result in [BEF+21] in the special case that $U$ is a
wandering domain of a holomorphic function $f$ with $F_{n}=f^{n}$ depends on
the following lemma [BEF+21, Lemma 4.1], relating hyperbolic density
$\rho_{U}$ to hyperbolic distance in a simply connected domain $U$. Hence the
original proof of Theorem 3.1 for simply connected wandering domains also
works in our general setting.

######  Lemma 3.2 (Estimate of hyperbolic quantities).

Let $U\subset\mathbb{C}$ be a simply connected domain. Then, for all
$z,z^{\prime}\in U$,

$\exp(-2\operatorname{dist}_{U}(z,z^{\prime}))\leq\frac{\rho_{U}(z^{\prime})}{\rho_{U}(z)}\leq\exp(2\operatorname{dist}_{U}(z,z^{\prime})).$

Theorem A strengthens part (c) of the boundary convergence classification by
showing that all orbits converge to the boundary with the same limiting
behaviour, and its proof is also based on Lemma 3.2. It is useful here to
state a more precise result than Theorem A.

######  Theorem 3.3.

Let $F_{n}:U\to U_{n},n\in{\mathbb{N}},$ be a sequence of holomorphic maps
between simply connected domains and let $z,z_{0}\in U$ with
$d=\operatorname{dist}_{U}(z,z_{0})$. Then

$|F_{n}(z)-F_{n}(z_{0})|\leq 2de^{2d}\operatorname{dist}(F_{n}(z_{0}),\partial
U_{n}),\;\text{ for }n\in{\mathbb{N}}.$ (3.1)

Therefore, if we also have

$\operatorname{dist}(F_{n}(z_{0}),\partial U_{n})\to 0\;\text{ as
}n\to\infty,$

then, for all $z\in U$,

$|F_{n}(z)-F_{n}(z_{0})|\to 0\;\text{ as }n\to\infty.$

###### Proof.

Let $z,z_{0}$ and $d$ be as in the statement and let

$\delta_{n}:=\operatorname{dist}(F_{n}(z_{0}),\partial U_{n}),\;\text{ for
}n\in{\mathbb{N}}.$

By the standard estimates of the hyperbolic density on simply connected
domains (see [CG93, page 13] or [BM07, Theorem 8.6]),

$\rho_{U_{n}}(F_{n}(z_{0}))\geq\frac{1}{2\operatorname{dist}(F^{n}(z_{0}),\partial
U_{n})}=\frac{1}{2\delta_{n}},\;\text{ for }n\in{\mathbb{N}}.$

Hence, by Lemma 3.2, for any hyperbolic disc $\Delta_{n}$ of radius $d$
centred at $F_{n}(z_{0})$ we have

$\rho_{U_{n}}(w)\geq\frac{e^{-2d}}{2\delta_{n}},\quad\text{for
}w\in\Delta_{n},n\in{\mathbb{N}}.$ (3.2)

By the contracting property of the hyperbolic metric, we have
$F_{n}(z)\in\overline{\Delta_{n}}$ for $n\in{\mathbb{N}}$.

Now let $\gamma\subset\Delta_{n}$ be a hyperbolic geodesic joining $F_{n}(z)$
to $F_{n}(z_{0})$; then the Euclidean distance apart of these two points is
bounded from above by the Euclidean length
${\ell_{\operatorname{Eucl}}}(\gamma)$. Since $\gamma$ is a hyperbolic
geodesic, we deduce from (3.2) that

$d\geq\ell_{U_{n}}(\gamma)=\int_{\gamma}\rho_{U_{n}}(z)\,|dz|\geq\frac{e^{-2d}}{2\delta_{n}}{\ell_{\operatorname{Eucl}}}(\gamma),$

so

$|F_{n}(z)-F_{n}(z_{0})|\leq{\ell_{\operatorname{Eucl}}}(\gamma)\leq
2\delta_{n}de^{2d},\quad\text{for }n\in{\mathbb{N}},$

which gives (3.1) and completes the proof. ∎

######  Remark.

As noted in the introduction, the matter of convergence to the boundary is
somewhat delicate in that it is closely related to the shape of the domains
$U_{n}$, and there may be situations where it is more appropriate to use an
alternative definition. For example, if the domains shrink with Euclidean
diameters tending to 0, then the Denjoy–Wolff set is automatically the whole
of $\partial U$. This geometric problem was discussed briefly in [BEF+21],
where the following observation was made.

Recall that the Euclidean distance of a point $z$ from the boundary of a
hyperbolic domain $U$ is closely related to the hyperbolic density at the
point in the domain. Indeed, when $U_{n}$ are simply connected domains, the
standard estimates for hyperbolic density mentioned above imply that

$\operatorname{dist}(F_{n}(z),\partial U_{n})\to
0\iff\rho_{U_{n}}(F_{n}(z))\to\infty.$

The proof of the boundary convergence classification and of Theorem 3.3 use
the hyperbolic densities $\rho_{U_{n}}(F_{n}(z))$ and, as in [BEF+21, Section
4], we point out that these results remain true if we replace
$\rho_{U_{n}}(F_{n}(z))$ by $a_{n}\rho_{U_{n}}(F_{n}(z))$, where the positive
sequence $(a_{n})$ is chosen in some way depending on the geometry of the
domains $U_{n}$, and the Euclidean distances within $U_{n}$ are also scaled
appropriately. For example, if the domains $U_{n}$ are shrinking, then it may
make sense to say that $F_{n}(z)$ converges to the boundary if
$a_{n}\rho_{U_{n}}(F_{n}(z))\to\infty$ as $n\to\infty$ where

$a_{n}=\sup_{D}\\{\operatorname{diam}D:D\;\text{is a disc contained
in}\;U_{n}\\}.$

##  4 Fast convergence to the boundary

In this section we prove a general result, Theorem 4.1, of which Theorem B and
Theorem C are special cases. This result shows that if orbits of points
converge sufficiently quickly to the boundary, then the Denjoy–Wolff set has
full measure, thus generalising the ADM dichotomy Theorem 1.3, part (a).

######  Theorem 4.1.

Let $F_{n}:U\to U_{n},n\in{\mathbb{N}},$ be a sequence of holomorphic maps
between simply connected domains, each with a full radial extension, and
suppose that the domains $U_{n},n\in{\mathbb{N}},$ all satisfy an
$\alpha$_-harmonic measure condition_ for some $\frac{1}{2}\leq\alpha\leq 1$
and some $C(r),r>0,$ independent of $n$.

Suppose that there exists $z_{0}\in U$ such that

$\sum_{n=0}^{\infty}\operatorname{dist}(F_{n}(z_{0}),\partial
U_{n})^{\alpha}<\infty.$ (4.1)

Then, for almost every $\zeta\in\partial U$,

$|F_{n}(\zeta)-F_{n}(z_{0})|\to 0\;\text{ as $n\to\infty$},$ (4.2)

and hence, in view of Theorem A, the Denjoy–Wolff set has full measure in
$\partial U$.

######  Remarks.

1\. The estimate (3.1) in the statement of Theorem 3.3 shows that condition
(4.1) is independent of the choice of $z_{0}\in U$.

2\. If we replaced (4.1) by the hypothesis that
$\operatorname{dist}(F_{n}(z_{0}),\partial U_{n})\to 0$ as $n\to\infty$, then
we would obtain the same result for any subsequence $(F_{n_{k}})$ for which
the corresponding series in (4.1) is convergent.

We give a formal definition of the $\alpha$-harmonic measure condition at the
beginning of Subsection 4.1. This is a geometric condition on the boundaries
of the domains which determines the rate of convergence required in (4.1).

We shall see that a domain with $C^{2}$ boundary, such as the unit disc,
satisfies the $\alpha$-harmonic measure condition with $\alpha=1$. On the
other hand, a general simply connected domain $U$, under no assumptions on the
regularity of its boundary, satisfies an $\alpha$-harmonic measure condition
with $\alpha=1/2$. Hence

  * (a)

Theorem 1.3, part (a) is a special case of Theorem 4.1, with
$U=U_{n}=\mathbb{D}$ and $F_{n}=f^{n}$, where $f$ is an inner function, for
$n\in\mathbb{N}$;

  * (b)

Theorem B is a special case of Theorem 4.1, with $U_{n}=\mathbb{D}$ and
$F_{n}:{\mathbb{D}}\rightarrow{\mathbb{D}}$ holomorpic self maps of the disc,
for $n\in\mathbb{N}$;

  * (c)

Theorem C follows from Theorem 4.1 if we make no assumptions on the geometric
nature of the domains and take $\alpha=\frac{1}{2}$ in (4.1).

In Section 4.3 we give an example which shows that, in one sense, Theorem 4.1
is best possible. For this example, where the domains are all cardioids, the
conclusion holds if we take $\alpha=1/2$ in (4.1), but does not hold for any
larger value of $\alpha$.

###  4.1 Domains with the $\alpha$-harmonic measure condition

We begin with a formal definition and discussion of our geometric condition.

######  Definition 4.2.

Let $\frac{1}{2}\leq\alpha\leq 1$. A simply connected domain $U$ satisfies an
$\alpha$_-harmonic measure condition_ if there exists a function $C(r)>0$,
$r>0$, such that, for every $\zeta\in\partial U$ and $r>0$, we have that

$\omega(z,\partial V\cap U,V)\leq C(r)|z-\zeta|^{\alpha},\;\text{ for }z\in
U\cap D(\zeta,r),$ (4.3)

where $V$ is the component of $U\cap D(\zeta,r)$ that contains $z$.

######  Remark.

Observe that the left-hand side of (4.3) can be interpreted as the probability
of a Brownian path in $U$ starting at the point $z$ first exiting $V$ through
$\partial V\cap U$, which tends to be larger or smaller according to the
relative size of $\partial V\cap U$ in $\partial V$. As shown by our Example
4.5, the largest case for $\omega(z,\partial V\cap U,V)$ occurs when the
boundary has an external cusp at $\zeta$.

The following geometric criteria for satisfying the $\alpha$-harmonic measure
condition are well known and we give only a brief indication of where proofs
can be found.

######  Lemma 4.3.

Assuming the notation of Definition 4.2:

  1. (a)

for any simply connected domain $U$ and $\zeta\in\partial U$, we have

$\omega(z,\partial V\cap U,V)\leq C(r)|z-\zeta|^{\frac{1}{2}},\;\text{ for
}z\in V,$

so we can take $\alpha=\tfrac{1}{2}$;

  2. (b)

if every $\zeta\in\partial U$ is the vertex of a sector of angle $\beta$,
$0<\beta\leq\pi$, and side-length $s>0$, contained in ${\mathbb{C}}\setminus
U$, then

$\omega(z,\partial V\cap U,V)\leq
C(r)|z-\zeta|^{\frac{\pi}{2\pi-\beta}},\;\text{ for }z\in V,$

where $C(r)$ also depends on $\beta$ and $s$, so we can take
$\alpha=\pi/(2\pi-\beta)$;

  3. (c)

if every $\zeta\in\partial U$ lies on the boundary of some disc of radius
$s>0$ contained in ${\mathbb{C}}\setminus U$, then

$\omega(z,\partial V\cap U,V)\leq C(r)|z-\zeta|,\;\text{ for }z\in V,$

where $C(r)$ also depends on $s$, so we can take $\alpha=1$.

Property (a) can be proved by applying the Milloux–Schmidt inequality [Hay89,
page 289] to the subharmonic function

$u(z):=\begin{cases}\omega(z,\partial V\cap U,V),\quad z\in V,\\\ 0,\quad z\in
D(\zeta,r)\setminus V;\end{cases}$

we state a version of the Milloux–Schmidt inequality here, since we will use
it again in Section 6.

######  Lemma 4.4 (Milloux–Schmidt inequality).

Suppose that $u$ is subharmonic and continuous in $\\{z:|z|\leq r\\}$ and that

$\min_{|z|=\rho}u(z)=0,\;\;\text{for }0<\rho<r.$ (4.4)

Then

$u(z)\leq\frac{4}{\pi}\max_{|z|=r}u(z)\tan^{-1}(|z|/r)^{1/2},\;\;\text{for
}0<|z|<r.$

Lemma 4.4 shows that in Lemma 4.3 part (a) we can take $C(r)=4/(\pi r^{1/2})$.

Parts (b) and (c) of Lemma 4.3 can be proved by applying standard techniques
such as conformal mappings and the maximum principle or Ahlfors’ distortion
theorem. In fact, using Ahlfors’ distortion theorem the values of $\alpha$
stated in (b) and (c) can be obtained with slightly smaller exterior domains
than sectors and discs, respectively, expressed in terms of a so-called Dini
condition; more details can be found in [Pom92, Section 3.4] and [Ahl10,
Chapter 3-3, p. 41].

######  Remark.

Using results from [BT21], it is possible to construct individual wandering
domains which satisfy (sharply) any of the geometries described in Lemma 4.3.

###  4.2 Proof of Theorem 4.1

We now prove our main result. The first part of the argument has its origins
in the proof of [RS11, Theorem 1.1] but the very general context here, with
maps not being surjective, introduces significant new challenges; see also
[OS16, Lemma 4.1] for a related generalisation of [RS11, Theorem 1.1].

###### Proof of Theorem 4.1.

We claim that we need only prove the result in the case that $U={\mathbb{D}}$
and $z_{0}=0$. Indeed, if we know that the result holds for a sequence of the
form $\hat{F}_{n}=F_{n}\circ\varphi_{U}:{\mathbb{D}}\to U_{n}$, where
$\varphi_{U}$ is a Riemann map from ${\mathbb{D}}$ onto $U$ with
$\varphi_{U}(0)=z_{0}$, then the result holds for $F_{n}:U\to U_{n}$ since
almost every $\xi\in\partial{\mathbb{D}}$ has the properties that the radial
limit $\zeta:=\varphi_{U}(\xi)$ exists and the full radial extension
$F_{n}(\zeta)$ exists (by hypothesis and the invariance of harmonic measure
under a conformal map), so

$F_{n}(\zeta)-F_{n}(z_{0})=\hat{F}_{n}(\xi)-\hat{F}_{n}(0)\to 0\;\text{ as
}n\to\infty,$

for almost every $\xi\in\partial{\mathbb{D}}$ and hence for almost every
$\zeta\in\partial U$, with respect to harmonic measure.

So, from now on we assume that $U={\mathbb{D}}$ and $z_{0}=0$. We have to show
that for almost all $\zeta\in\partial{\mathbb{D}}$ we have
$F_{n}(\zeta)-F_{n}(0)\to 0$ as $n\to\infty$. For each $n\in{\mathbb{N}}$, let
$\zeta_{n}$ be a point in $\partial U_{n}$ that satisfies

$|F_{n}(0)-\zeta_{n}|=\operatorname{dist}(F_{n}(0),\partial U_{n}).$

By hypothesis, $F_{n}(0)-\zeta_{n}\to 0$ as $n\to\infty$.

For every $r>0$, we define the sequence of open sets $S_{n}(r)=U_{n}\cap
D(\zeta_{n},r)$, for $n\in{\mathbb{N}}$. We then define the set $E(r)$ of
points $\zeta\in\partial{\mathbb{D}}$ such that $F_{n}(\zeta)$ eventually lies
in $\overline{S_{n}(r)}$, that is,

$E(r)=\bigcup_{N\geq 0}\bigcap_{n\geq
N}\left(F_{n}^{-1}\left(\overline{S_{n}(r)}\right)\cap\partial{\mathbb{D}}\right).$

Notice that by the definition of $S_{n}(r)$ we have

$|F_{n}(\zeta)-\zeta_{n}|\rightarrow 0\text{ as $n\rightarrow\infty$,\quad for
}\zeta\in\bigcap_{r>0}E(r),$ (4.5)

and, since the sets $E(r)$ decrease as $r$ decreases, for any sequence of
radii $r_{k}\to 0$ as $k\to 0$ we have $\bigcap_{r>0}E(r)=\bigcap_{k\geq
1}E(r_{k})$. To prove our result it is sufficient to show that the latter set
has full harmonic measure in $\partial{\mathbb{D}}$.

First note that
$\overline{U_{n}}\setminus\overline{S_{n}(r)}\subset\overline{U_{n}\setminus
S_{n}(r)}$, so

$\partial{\mathbb{D}}\setminus E(r)\subset\bigcap_{N\geq 0}\bigcup_{n\geq
N}(F_{n}^{-1}(\overline{U_{n}\setminus S_{n}(r)})\cap\partial{\mathbb{D}}).$
(4.6)

We shall use (4.6) to show that

$\omega(0,\partial{\mathbb{D}}\setminus E(r),{\mathbb{D}})=0,\quad\text{for
}r>0.$ (4.7)

To prove this, we take $N(r)\in{\mathbb{N}}$ so large that $F_{n}(0)\in
S_{n}(r)$ for all $n\geq N(r)$, and then for all $n\geq N(r)$ we define

  * •

$V_{n}(r)$ to be the component of $S_{n}(r)$ that contains $F_{n}(0)$,

  * •

$W_{n}(r)$ to be the component of $F_{n}^{-1}(V_{n}(r))\cap{\mathbb{D}}$ that
contains $0$, and

  * •

$\Gamma_{n}(r):=\partial V_{n}(r)\cap U_{n}$.

${\mathbb{D}}$$U_{n}$$0$$W_{n}(r)$$F_{n}(0)$$r$$U_{n}\setminus
V_{n}(r)$$V_{n}(r)$$\Gamma_{n}(r)$$\zeta_{n}$$F_{n}$ Figure 1:  Illustration
of the proof of Theorem 4.1. The set $V_{n}(r)$ and the connected component
$W_{n}(r)$ of its preimage which contains $0$ are shaded (seagreen online).
The arcs $\Gamma_{n}(r)$ and some of their preimages are drawn thicker (pink
online). Notice that there may also be preimages of $\Gamma_{n}(r)$ in
$\partial{\mathbb{D}}$, since iterates of points in $\partial{\mathbb{D}}$ may
fall in the interior of $U_{n}$. Points in ${\mathbb{D}}\setminus W_{n}(r)$,
which are shown in white, may be mapped under $F_{n}$ to either
$U_{n}\setminus V_{n}(r)$ or $V_{n}(r)$, since the preimage of $V_{n}(r)$
under $F_{n}$ may have more than one connected component. Finally, recall that
the points $(\zeta_{n})_{n\in{\mathbb{N}}}$ do not form an orbit.

Note that $V_{n}(r)$ and $W_{n}(r)$ are both simply connected domains, by the
maximum principle, and $\Gamma_{n}(r)$ consists of arcs of $\partial
D(\zeta_{n},r)$. Moreover, $W_{n}(r)$ is a Jordan domain since the preimage of
$\Gamma_{n}(r)$ in ${\mathbb{D}}$ consists of at most countably many curves
that accumulate at no point of ${\mathbb{D}}$, and the ends of each such
preimage curve consist of single points of $\partial{\mathbb{D}}$.

The latter property follows from a theorem of G. MacLane [Mac63, Theorem 1],
applied here to the function $z\mapsto F_{n}(z)-\zeta_{n}$, which has finite
radial limits almost everywhere on $\partial{\mathbb{D}}$. MacLane’s theorem
states that the level curves of any function holomorphic in ${\mathbb{D}}$
accumulate at singleton points of $\partial{\mathbb{D}}$ (rather than at arcs
of $\partial{\mathbb{D}}$) if and only if the function has asymptotic values
at all points of a dense subset of $\partial{\mathbb{D}}$, which is certainly
the case for the function $z\mapsto F_{n}(z)-\zeta_{n}$.

In particular, $W_{n}(r)$ has the unique accessibility property and so the
restricted map $F_{n}:W_{n}(r)\to V_{n}(r)$ has full radial extension; see
Proposition 2.2.

We now apply Theorem 2.7 to the map $F_{n}:W_{n}(r)\to V_{n}(r)$ to deduce
that

$\omega(0,F_{n}^{-1}(\Gamma_{n}(r))\cap\partial
W_{n}(r),W_{n}(r))\leq\omega(F_{n}(0),\Gamma_{n}(r),V_{n}(r)).$ (4.8)

Next we deduce from the fact that
$\overline{V_{n}(r)}\subset\overline{S_{n}(r)}$ that, for all
$z\in{\mathbb{D}}$,

$\omega(z,F_{n}^{-1}(\overline{U_{n}\setminus
S_{n}(r)})\cap\partial{\mathbb{D}},{\mathbb{D}})\leq\omega(z,F_{n}^{-1}(\overline{U_{n}\setminus
V_{n}(r)})\cap\partial{\mathbb{D}},{\mathbb{D}}).$ (4.9)

Using the interpretation of harmonic measure as the exit distribution of
Brownian motion, it is clear that, for all $z\in W_{n}(r)$, we have

$\omega(z,F_{n}^{-1}(\overline{U_{n}\setminus
V_{n}(r)})\cap\partial{\mathbb{D}},{\mathbb{D}})\leq\omega(z,F_{n}^{-1}(\Gamma_{n}(r))\cap\partial
W_{n}(r),W_{n}(r)),$ (4.10)

since any Brownian path in ${\mathbb{D}}$ originating at 0 can only exit
${\mathbb{D}}$ at a point of $F_{n}^{-1}(\overline{U_{n}\setminus
V_{n}(r)})\cap\partial{\mathbb{D}}$ by first exiting $W_{n}(r)$ at a point of
$F_{n}^{-1}(\Gamma_{n}(r))\cap\partial W_{n}(r)$.

Alternatively, we can apply the maximum principle in the Jordan domain
$W_{n}(r)$ to justify (4.10), by arguing that this inequality holds everywhere
on $\partial W_{n}(r)$, apart from at most countably many points. Indeed,
apart from the endpoints of the preimage curves of $\Gamma_{n}(r)$, these two
bounded positive harmonic functions in $W_{n}(r)$ have the same radial
boundary extensions everywhere on $\partial W_{n}(r)\cap\partial{\mathbb{D}}$
and the positive harmonic function on the right takes the value 1 everywhere
on $F_{n}^{-1}(\Gamma_{n}(r))\cap\partial W_{n}(r)=\partial
W_{n}(r)\cap{\mathbb{D}}$.

Using the inequalities (4.6), (4.9), (4.10) and (4.8), in this order, we
deduce that

$\displaystyle\omega(0,\partial{\mathbb{D}}\setminus E(r),{\mathbb{D}})$
$\displaystyle\leq$ $\displaystyle\inf_{N\geq N(r)}\sum_{n\geq
N}\omega(0,F_{n}^{-1}\left(\overline{U_{n}\setminus
S_{n}(r)}\right)\cap\partial{\mathbb{D}},{\mathbb{D}})$ (4.11)
$\displaystyle\leq$ $\displaystyle\inf_{N\geq N(r)}\sum_{n\geq
N}\omega(0,F_{n}^{-1}\left(\overline{U_{n}\setminus
V_{n}(r)}\right)\cap\partial{\mathbb{D}},{\mathbb{D}})$ $\displaystyle\leq$
$\displaystyle\inf_{N\geq N(r)}\sum_{n\geq
N}\omega(0,F_{n}^{-1}(\Gamma_{n}(r))\cap\partial W_{n}(r),W_{n}(r))$
$\displaystyle\leq$ $\displaystyle\inf_{N\geq N(r)}\sum_{n\geq
N}\omega(F_{n}(0),\Gamma_{n}(r),V_{n}(r)).$

Since the domains $U_{n}$ are all assumed to satisfy the $\alpha$-harmonic
measure condition, we deduce from (4.3) that

$\omega(F_{n}(0),\Gamma_{n}(r),V_{n}(r))\leq
C(r)|F_{n}(0)-\zeta_{n}|^{\alpha},$

for some positive function $C(r),r>0$. Hence, by (4.1), the right-hand side of
(4.11) is equal to 0 for every $r>0$ and (4.7) follows.

Since a countable union of sets of zero measure has zero measure, we deduce
from (4.7) that for any sequence of radii $r_{k}\to 0$ as $k\to\infty$

$\omega(0,\bigcup_{k\geq 1}(\partial{\mathbb{D}}\setminus
E(r_{k})),{\mathbb{D}})=0.$

Therefore, by (4.5), we have $|F_{n}(\zeta)-\zeta_{n}|\rightarrow 0$ as
$n\to\infty$ for $\zeta$ in a subset of $\partial{\mathbb{D}}$ of full
harmonic measure, namely, $\bigcap_{k\geq 1}E(r_{k})$.

Since $|\zeta_{n}-F_{n}(0)|\rightarrow 0$ by assumption, this implies that

$|F_{n}(\zeta)-F_{n}(0)|\leq|F_{n}(\zeta)-\zeta_{n}|+|\zeta_{n}-F_{n}(0)|\rightarrow
0\;\text{ as }n\to\infty,$

for $\zeta$ in a set of full harmonic measure, as claimed. ∎

######  Remark.

It can be seen from the proof of Theorem 4.1 that if the domains $U_{n}$
satisfy an $\alpha$-harmonic measure condition, $1/2\leq\alpha\leq 1$, but the
orbit $(F_{n}(0))$ actually approaches parts of the boundaries of the $U_{n}$
that satisfy a $\beta$-harmonic measure condition, where $\beta>\alpha$, then
the convergence condition (4.1) can be weakened to replace the exponent
$\alpha$ by $\beta$.

###  4.3 Example of self-maps of a cardioid domain

We give an example which has the property that in Theorem 4.1 the exponent in
condition (4.1) cannot be taken to be greater than $\frac{1}{2}$. We consider
holomorphic self-maps of the domain

$U=\varphi({\mathbb{D}}),\quad\text{where }\varphi(z)=(z-1)^{2},$

which is the conformal image of ${\mathbb{D}}$ under $\varphi$, meeting the
real axis in the interval $\varphi((-1,1))=(0,4)$, with $\partial U$ a
cardioid having its inward pointing cusp at $\varphi(1)=0$.

By part (a) of Lemma 4.3 the domain $U$ satisfies an $\alpha$-harmonic measure
condition for $\alpha=\frac{1}{2}$. On the other hand, $U$ does not satisfy
such a condition for any $\alpha>\frac{1}{2}$. Indeed, we can check that the
harmonic measure $\omega(z,\partial V\cap U,V)$, where $V=U\cap D(0,r),r>0$,
satisfies

$\omega(x,\partial V\cap U,V)\sim C(r)x^{1/2}\;\ \text{ as }x\to 0+,$

for some constant $C(r)>0$, by using $\varphi^{-1}$ to map this harmonic
measure to a boundary neighbourhood of $1$ in ${\mathbb{D}}$ within which the
corresponding harmonic measure behaves near $1$ like a multiple of the
distance to $\partial{\mathbb{D}}$.

######  Example 4.5.

There exists a sequence of holomorphic maps $F_{n}:U\to U$ such that

  * (a)

$F_{n}(1)\to 0\in\partial U$ as $n\to\infty$ (note that $1=\varphi(0)\in U$);

  * (b)

$\sum\operatorname{dist}(F_{n}(1),\partial
U)^{\alpha}\begin{cases}<\infty,\quad\text{for }\alpha>1/2,\\\
=\infty,\quad\text{for }\alpha=1/2;\end{cases}$

  * (c)

$\\{\zeta\in\partial U:F_{n}(\zeta)\to 0\;\text{as}\;n\to\infty\\}$ has
measure 0; that is, the Denjoy–Wolff set has measure zero.

###### Proof.

Let $B_{n}$ be the sequence of Blaschke products defined as
$B_{n}(z)=M_{n}(z^{2^{n}})$, with $M_{n}(z)=\frac{z+1-1/{n}}{1+(1-1/n)z}$. We
then have that

$B_{n}(0)=1-\frac{1}{n}\rightarrow 1\;\text{ as }n\rightarrow\infty,$ (4.12)

and

$B_{n}(\zeta)\nrightarrow 1\;\;\text{as }n\to\infty,\quad\text{for almost
every }\zeta\in\partial\mathbb{D},$ (4.13)

where (4.13) follows by applying Example 8.3 with
$a_{n}=1-1/n,n\in{\mathbb{N}}$, according to which the orbit of almost every
point $\zeta\in\partial\mathbb{D}$ is dense in $\partial{\mathbb{D}}$.

Now define the self-maps $F_{n}:U\rightarrow U$ as

$F_{n}=\varphi\circ B_{n}\circ\varphi^{-1}.$

Then, by (4.12) and the fact that $\varphi(0)=1$, we have

$F_{n}(1)=\varphi(B_{n}(0))=\varphi(1-1/n)=1/n^{2}\to 0\;\text{ as
}n\to\infty,$ (4.14)

which gives property (a) and also property (b), since
$\operatorname{dist}(F_{n}(1),\partial U)=1/n^{2}$ for $n\in{\mathbb{N}}$.
Finally, property (c) holds since $F_{n}(\zeta)\nrightarrow 0$ as $n\to\infty$
for almost all $\zeta\in\partial U$, by (4.13). ∎

Note that we can write the maps $F_{n}:U\to U$ as a forward composition
$F_{n}=f_{n}\circ\cdots\circ f_{1}$, where $f_{n}=\varphi\circ
b_{n}\circ\varphi^{-1}$ and $b_{n}$ are the maps given in Example 8.3 such
that $B_{n}=b_{n}\circ\cdots\circ b_{1}$.

##  5 Empty Denjoy–Wolff sets – the proof of Theorem D

In this section we prove Theorem D. In the proof we need the following basic
estimate for the mapping properties of Möbius transformations, which is also
used in Section 8.

######  Lemma 5.1 (Preimages under Möbius maps).

Consider a Möbius map of the form

$M_{a}(z)=\frac{z+a}{1+az},\ 0<a<1,\ a=1-\varepsilon,$

and let $T\subset\partial{\mathbb{D}}$ be an arc $(e^{i\theta},e^{i\varphi})$,
with $0<\theta<\varphi<2\pi$. Then there exists $c(\theta,\varphi)>0$ such
that

$|M_{a}^{-1}(T)|\sim c(\theta,\varphi)\varepsilon\ \ \text{ as
$\varepsilon\rightarrow 0$}.$ (5.1)

###### Proof.

The proof is a direct computation. For $z,w\in\partial{\mathbb{D}}$ we have
that

$M_{a}^{-1}(z)-M_{a}^{-1}(w)=\frac{z-a}{1-az}-\frac{w-a}{1-aw}=\frac{(1-a^{2})(z-w)}{(1-az)(1-aw)},$

so

$\displaystyle|M_{a}^{-1}(e^{i\varphi})-M_{a}^{-1}(e^{i\theta})|$
$\displaystyle=\frac{(1-a^{2})|e^{i\varphi}-e^{i\theta}|}{|(1-ae^{i\varphi})(1-ae^{i\theta})|}$
$\displaystyle=\frac{\varepsilon(2-\varepsilon)|e^{i\varphi}-e^{i\theta}|}{(|1-e^{i\varphi}|+O(\varepsilon))(|1-e^{i\theta}|+O(\varepsilon))}$
$\displaystyle=\left(\frac{2|e^{i\varphi}-e^{i\theta}|}{|(1-e^{i\varphi})(1-e^{i\theta})|}\right)\varepsilon+O(\varepsilon^{2})\;\text{
as }\varepsilon\rightarrow 0.$

Since $|M_{a}^{-1}(T)|\sim|M_{a}^{-1}(e^{i\varphi})-M_{a}^{-1}(e^{i\theta})|$
as $\varepsilon\rightarrow 0$, the claim follows. ∎

###### Proof of Theorem D.

Recall that

$F_{n}(z)=\frac{\lambda_{n}z+a_{n}}{1+\lambda_{n}a_{n}z},\;\text{ for
}n\in{\mathbb{N}},$

where $(a_{n})$ is an increasing sequence in $[0,1)$, with $a_{0}=0$ and
$a_{n}\to 1$ as $n\to\infty$, and $(\lambda_{n})$ is a sequence in
$\partial{\mathbb{D}}$ to be chosen. It is clear that $F_{n}(0)=a_{n}$ for
$n\in{\mathbb{N}}$, whatever choice is made of $\lambda_{n}$, so

$F_{n}(z)\to 1\;\text{ as }n\to\infty,\text{ for all }z\in{\mathbb{D}},$

by Theorem A.

We now put $\varepsilon_{n}=1-a_{n}$ for $n\in{\mathbb{N}}$, and assume that

$\sum_{n=1}^{\infty}(1-a_{n})=\sum_{n=1}^{\infty}\varepsilon_{n}=\infty.$
(5.2)

Then we write

$F_{n}(z)=B_{n}(\lambda_{n}z),\quad\text{where}\quad
B_{n}(z)=\frac{z+a_{n}}{1+a_{n}z},\;\text{ for }n\in{\mathbb{N}}.$

Let $S=(e^{-i\theta},e^{i\theta})$ be any arc, where $0<\theta<\pi/4$. We show
how to choose $(\lambda_{n})$ so that no $\zeta\in\partial{\mathbb{D}}$ has an
orbit that eventually lies in $S$. We have that $B_{n}^{-1}(S^{c})$ is an arc
in $\partial{\mathbb{D}}$ with centre $-1$ and it follows by Lemma 5.1 that

$|B_{n}^{-1}(S^{c})|\sim c(S)\varepsilon_{n}\;\text{ as }n\to\infty.$ (5.3)

Therefore, for any choice of $(\lambda_{n})$, the preimage $F_{n}^{-1}(S^{c})$
is an arc with centre $-\lambda_{n}^{-1}$ and with $|F_{n}^{-1}(S^{c})|\sim
c(S)\varepsilon_{n}$ as $n\to\infty$. We introduce intervals
$I_{n}=[\theta_{n},\theta_{n+1}],n=0,1,\dots$, such that $\theta_{0}=0$ and

$\theta_{n+1}-\theta_{n}=|B_{n}^{-1}(S^{c})|,\quad\text{so}\quad\bigcup_{n=0}^{\infty}I_{n}=[0,\infty),$

by (5.2) and (5.3).

Now, we choose $\lambda_{n}\in\partial\mathbb{D}$ such that

$-\lambda_{n}^{-1}=e^{i(\theta_{n}+\theta_{n+1})/2}.$

With this choice we know that if $\theta_{n}\leq\theta\leq\theta_{n+1}$, then

$e^{i\theta}\in F_{n}^{-1}(S^{c}),\quad\text{so}\quad\lambda_{n}e^{i\theta}\in
B_{n}^{-1}(S^{c})$

and hence $B_{n}(\lambda_{n}e^{i\theta})\in S^{c}$. But any
$\zeta\in\partial\mathbb{D}$ (including $\zeta=1$) is of the form
$\zeta=e^{i\varphi_{n}}$, where $\varphi_{n}\in I_{n}$ for infinitely many
$n$, so

$F_{n}(\zeta)=B_{n}(\lambda_{n}e^{i\varphi_{n}})\in S^{c},\quad\text{for such
}n.$

Hence with this choice of $\lambda_{n}$ we have $F_{n}(\zeta)\nrightarrow 1$
as $n\to\infty$, as required. ∎

######  Remark.

A small modification to the example in Theorem D gives a sequence of degree
$p$ Blaschke products, where $p\geq 2$, of the form

$F_{n}(z)=\frac{\lambda_{n}z^{p}+a_{n}}{1+\lambda_{n}a_{n}z^{p}},\quad
n\in{\mathbb{N}},$

such that $F_{n}(z)\to 1$ as $n\to\infty$ for all $z\in{\mathbb{D}}$, but the
Denjoy–Wolff set of $(F_{n})$ is empty. However, note that, as opposed to the
Möbius example of Theorem D, the sequence $(F_{n})$ cannot be written as a
forward composition sequence $F_{n}=f_{n}\circ\cdots\circ f_{1}$ where the
$f_{n}$ are self-maps of the unit disc.

##  6 Proof of Theorem E

In this section we prove Theorem E. The first part of the proof involves
similar techniques to that of Theorem 4.1, so we omit some of the details
here.

First, we can assume that $\zeta_{0}\in\partial U_{0}$ is accessible since
$L_{0}$ has positive harmonic measure. We claim we can assume that
$U_{0}={\mathbb{D}}$. To see this, let
$L^{\prime}_{0}=\varphi_{0}^{-1}(L_{0})$, where $\varphi_{0}:{\mathbb{D}}\to
U_{0}$ is a Riemann map. Then $L^{\prime}_{0}\subset\partial{\mathbb{D}}$ has
positive measure and the sequence
$\hat{F}_{n}=F_{n}\circ\varphi_{0}:{\mathbb{D}}\to U_{n}$, $n\in{\mathbb{N}}$,
satisfies $\hat{F}_{n}(\xi)-F_{n}(\zeta_{0})\to 0$ as $n\to\infty$ for all
$\xi\in L^{\prime}_{0}$. Thus if the result holds for
$\hat{F}_{n}:{\mathbb{D}}\to U_{n}$, that is,
$\hat{F}_{n}(z)-F_{n}(\zeta_{0})\to 0$ as $n\to\infty$ for all
$z\in{\mathbb{D}}$ or equivalently $F_{n}(\varphi_{0}(z))-F_{n}(\zeta_{0})\to
0$ as $n\to\infty$ for all $z\in{\mathbb{D}}$, then
$F_{n}(z)-F_{n}(\zeta_{0})\to 0$ as $n\to\infty$ for all $z\in U_{0}$, as
required.

So, from now on we assume that $U={\mathbb{D}}$ and
$L_{0}\subset\partial{\mathbb{D}}$ has positive measure. Let
$\zeta_{n}:=F_{n}(\zeta_{0})$ for $n\in{\mathbb{N}}$, so by hypothesis
$\operatorname{dist}(\zeta_{n},\partial U_{n})\to 0$ as $n\to\infty$
(possibly, $\operatorname{dist}(\zeta_{n},\partial U_{n})=0$ for all
$n\in{\mathbb{N}}$ if $\zeta_{n}\in\partial U_{n}$ for all
$n\in{\mathbb{N}}$). Then take any $z_{0}\in{\mathbb{D}}$ and put
$z_{n}=F_{n}(z_{0})\in U_{n}$, for $n\in{\mathbb{N}}$. To obtain our result it
is sufficient, by Theorem A, to prove that

$|z_{n}-\zeta_{n}|\to 0\;\text{ as }n\to\infty.$ (6.1)

By the Severini–Egorov theorem (see [Roy88, Chapter 3] or [Tao11, Theorem
1.3.26 and Exercise 1.4.31], for example), there is a compact set
$E_{0}\subset L_{0}\subset\partial{\mathbb{D}}$ such that

$\omega(z_{0},E_{0},{\mathbb{D}})\geq\tfrac{1}{2}\omega(z_{0},L_{0},{\mathbb{D}})\;\text{
and }\;|F_{n}(\zeta)-\zeta_{n}|\to 0\text{ as }n\to\infty,\;\text{ uniformly
for }\zeta\in E_{0}.$ (6.2)

Hence there exists a positive sequence $r_{n}\to 0$ as $n\to\infty$ such that

$F_{n}(E_{0})\subset D(\zeta_{n},r_{n})\quad\text{and}\quad
D(\zeta_{n},r_{n})\cap\partial U_{n}\neq\emptyset,\;\text{ for
}n\in{\mathbb{N}}.$ (6.3)

We prove (6.1) by showing that, for some positive absolute constant $C>1$ we
have

$z_{n}\in D(\zeta_{n},Cr_{n}),\;\text{ for }n\in{\mathbb{N}}.$ (6.4)

It is sufficient to consider only those values of $n$ for which $z_{n}\notin
D(\zeta_{n},r_{n})$.

For such $n$, we let $V_{n}$ denote the component of
$U_{n}\setminus\overline{D(\zeta_{n},r_{n})}$ that contains $z_{n}$, $W_{n}$
denote the component of $F_{n}^{-1}(V_{n})\subset{\mathbb{D}}$ that contains
$z_{0}$, and put $\Gamma_{n}=\partial V_{n}\cap U_{n}$. Note that $V_{n}$ and
$W_{n}$ are both simply connected domains, by the maximum principle. Moreover,
as in the proof of Theorem 4.1, $W_{n}$ is a Jordan domain since the part of
the preimage of $\Gamma_{n}$ that lies in ${\mathbb{D}}$ consists of at most
countably many curves that accumulate at no point of ${\mathbb{D}}$, and the
ends of each such preimage curve consist of single points of
$\partial{\mathbb{D}}$.

We can now apply Theorem 2.7 to $F_{n}$ in $W_{n}$ to deduce that

$\omega(z_{0},F_{n}^{-1}(\Gamma_{n})\cap\partial
W_{n},W_{n})\leq\omega(z_{n},\Gamma_{n},V_{n}).$ (6.5)

Next observe that $E_{0}$ lies in $\partial{\mathbb{D}}\setminus\partial
W_{n}$, by the definitions of $L_{0}$, $V_{n}$ and $W_{n}$. Therefore, since
any Brownian path in $W_{n}$ originating at $z\in W_{n}$ can only exit
${\mathbb{D}}$ at a point of $E_{0}$ by first exiting $W_{n}$ at a point of
$\partial W_{n}\cap{\mathbb{D}}$, we have

$\omega(z,E_{0},{\mathbb{D}})\leq\omega(z,F_{n}^{-1}(\Gamma_{n})\cap\partial
W_{n},W_{n}),\;\text{ for }z\in W_{n},n\in{\mathbb{N}}.$ (6.6)

Alternatively, we can prove (6.6) by using the maximum principle, because the
harmonic measure on the left has radial boundary extension 0 almost everywhere
on $\partial{\mathbb{D}}\cap\partial W_{n}$ (since
$E_{0}\subset\partial{\mathbb{D}}\setminus\partial W_{n}$) and the harmonic
measure on the right takes the value 1 almost everywhere on
${\mathbb{D}}\cap\partial W_{n}$.

Now we estimate the right-hand side of (6.5) from above. To do this we choose
an unbounded closed connected set $K_{n}$ which lies in ${\mathbb{C}}\setminus
U_{n}$ and contains a point in $D(\zeta_{n},r_{n})\cap\partial U_{n}$. This
set exists since $U_{n}$ is a simply connected proper subset of
${\mathbb{C}}$. Then let $\Omega_{n}$ denote the component of the complement
of $K_{n}\cup\overline{D(\zeta_{n},r_{n})}$ that contains $z_{n}$, which is a
simply connected domain such that $V_{n}\subset\Omega_{n}$ and
$\Gamma_{n}\subset\partial\Omega_{n}$. Then, by the maximum principle again,
this time applied first in $V_{n}$ and second in $\Omega_{n}$, we have

$\omega(z,\Gamma_{n},V_{n})\leq\omega(z,\Gamma_{n},\Omega_{n})\leq\omega(z,\partial\Omega_{n}\cap
U_{n},\Omega_{n}),\;\text{ for }z\in V_{n},n\in{\mathbb{N}}.$ (6.7)

In particular, (6.7) holds for $z=z_{n}$.

Now the right-hand function in (6.7) can be estimated from above by first
mapping $\Omega_{n}$ to a bounded domain using
$w=\psi_{n}(z):=1/(z-\zeta_{n})$ and then applying the Milloux-Schmidt
inequality to the harmonic function

$u(w)=\omega(\psi_{n}^{-1}(w),\partial\Omega_{n}\cap U_{n},\Omega_{n}),\quad
w\in\psi_{n}(\Omega_{n}),n\in{\mathbb{N}}.$

Note that the domain $\psi_{n}(\Omega_{n})$ lies in $D(0,R_{n})$, where
$R_{n}:=r_{n}^{-1}$, and $u$ vanishes on the part of the boundary of
$\psi_{n}(\Omega_{n})$ that lies in $D(0,R_{n})$. After extending $u$ to be 0
throughout $D(0,R_{n})\setminus\psi_{n}(\Omega_{n})$ and hence subharmonic in
$D(0,R_{n})$, we deduce from the Milloux–Schmidt inequality, Lemma 4.4, that
for $n\in{\mathbb{N}},$ we have

$u(w)\leq\frac{4}{\pi}\tan^{-1}\left(\frac{|w|}{R_{n}}\right)^{\frac{1}{2}}\leq\frac{4}{\pi}\left(\frac{|w|}{R_{n}}\right)^{\frac{1}{2}},\;\text{
for }w\in\psi_{n}(\Omega_{n}),$

and in particular, taking $w=\psi_{n}(z_{n})$, that

$\omega(z_{n},\partial\Omega_{n}\cap
U_{n},\Omega_{n})=u(\psi_{n}(z_{n}))\leq\frac{4}{\pi}\left(\frac{1}{R_{n}|z_{n}-\zeta_{n}|}\right)^{\frac{1}{2}}=\frac{4}{\pi}\left(\frac{r_{n}}{|z_{n}-\zeta_{n}|}\right)^{\frac{1}{2}}.$

Combining this estimate with (6.2), (6.5), (6.6) and (6.7), we deduce that,
for $n\in{\mathbb{N}}$,

$\omega(z_{0},L_{0},{\mathbb{D}})\leq
2\omega(z_{0},E_{0},{\mathbb{D}})\leq\frac{8}{\pi}\left(\frac{r_{n}}{|z_{n}-\zeta_{n}|}\right)^{\frac{1}{2}},\quad\text{so}\quad|z_{n}-\zeta_{n}|\leq\frac{64r_{n}}{(\pi\omega(z_{0},L_{0},{\mathbb{D}}))^{2}}.$

Since $r_{n}\to 0$ as $n\to\infty$ and $\omega(z_{0},L_{0},{\mathbb{D}})>0$ by
hypothesis, this proves (6.4), as required.

##  7 Classifying forward compositions and proof of Theorem F

In this section we focus on forward compositions of holomorphic maps between
simply connected domains. We first classify such sequences in terms of
hyperbolic distances, extending the classification of wandering domains given
in [BEF+21]. We then prove a general result concerning contracting sequences
which implies Theorem F. At the end of the section we give an example which
shows that the conclusion of Theorem F does not hold if the hypothesis of
contracting is omitted.

###  7.1 Hyperbolic classification

In [BEF+21, Theorem A], we classified simply connected wandering domains into
three distinct types: contracting, semi-contracting or eventually isometric,
according to whether hyperbolic distances between orbits of points, for almost
all points, tend to 0, decrease but do not tend to 0, or are eventually
constant, respectively. In Section 1 of this paper we extended the definition
of ‘contracting’ to the more general context of sequences of holomorphic maps
$F_{n}:U\to U_{n}$ between simply connected domains; see Definition 1.4.

In this section, we extend the concepts of ‘semi-contracting’ and ‘eventually
isometric’ from wandering domains to sequences of holomorphic maps that can be
expressed as forward compositions of holomorphic maps between simply connected
domains; that is, sequences of the form $F_{n}=f_{n}\circ\cdots\circ f_{1}$,
where $f_{n}:U_{n-1}\to U_{n}$ are holomorphic maps.

We also give a criterion to discriminate between the three types of forward
compositions based on the concept of _hyperbolic distortion_ [BM07, Sect.
5,11].

######  Definition 7.1 (Hyperbolic distortion).

If $f:U\rightarrow V$ is a holomorphic map between two hyperbolic domains $U$
and $V$, then the _hyperbolic distortion_ of $f$ at $z$ is

$\|Df(z)\|_{U}^{V}:=\lim_{z^{\prime}\rightarrow
z}\frac{\operatorname{dist}_{V}(f(z^{\prime}),f(z))}{\operatorname{dist}_{U}(z^{\prime},z)}=\frac{\rho_{V}(f(z))|f^{\prime}(z)|}{\rho_{U}(z)},$

where $\rho_{U}(z)$ denotes the hyperbolic density at $z\in U$.

Note that in the following result the holomorphic maps need not be
surjections.

######  Theorem 7.2 (Hyperbolic classification theorem).

Let $U_{n},n\geq 0$ be a sequence of simply connected domains (not necessarily
distinct or disjoint) and $f_{n}:U_{n-1}\to U_{n},n\in{\mathbb{N}}$ be a
sequence of holomorphic maps. Define the forward compositions
$F_{n}=f_{n}\circ\cdots\circ f_{1},n\in{\mathbb{N}},$ and the set

$E=\\{(z,z^{\prime})\in U_{0}\times U_{0}:F_{n}(z)=F_{n}(z^{\prime})\text{\
for some $n\in{\mathbb{N}}$}\\}.$

For $n\in{\mathbb{N}}$, let $\lambda_{n}(z)$ denote the hyperbolic distortion
$\|Df_{n}(F_{n-1}(z))\|_{U_{n-1}}^{U_{n}}$.

Then exactly one of the following holds:

  * (1)

$\operatorname{dist}_{U_{n}}(F_{n}(z),F_{n}(z^{\prime}))\underset{n\to\infty}{\longrightarrow}c(z,z^{\prime})=0$
for all $z,z^{\prime}\in U_{0}$, that is, the sequence $(F_{n})$ is
contracting on $U_{0}$; this case occurs if and only if
$\sum_{n=1}^{\infty}(1-\lambda_{n}(z))=\infty$.

  * (2)

$\operatorname{dist}_{U_{n}}(F_{n}(z),F_{n}(z^{\prime}))\underset{n\to\infty}{\longrightarrow}c(z,z^{\prime})>0$
and $\operatorname{dist}_{U_{n}}(F_{n}(z),F_{n}(z^{\prime}))\neq
c(z,z^{\prime})$ for all $(z,z^{\prime})\in(U_{0}\times U_{0})\setminus E$,
$n\in{\mathbb{N}}$, in which case we say that the sequence $(F_{n})$ is semi-
contracting on $U_{0}$.

  * (3)

There exists $N>0$ such that for all $n\geq N$,
$\operatorname{dist}_{U_{n}}(F_{n}(z),F_{n}(z^{\prime}))=c(z,z^{\prime})>0$
for all $(z,z^{\prime})\in(U\times U)\setminus E$, in which case we say that
the sequence $(F_{n})$ is eventually isometric on $U_{0}$; this case occurs if
and only if $\lambda_{n}(z)=1$, for $n$ sufficiently large.

By the Schwarz-Pick Lemma, the eventually isometric case occurs if and only if
$f_{n}:U_{n-1}\to U_{n}$ is univalent for all sufficiently large $n$.

The proof of Theorem 7.2 is essentially identical to that of the corresponding
result for simply connected wandering domains, [BEF+21, Theorem A], depending
as it does on using Riemann maps to reduce the situation to classifying non-
autonomous systems of forward compositions of self-maps of the unit disc that
fix the point 0, as discussed in [BEF+21, Section 2].

The hyperbolic classification theorem does not hold in general for sequences
of holomorphic maps between simply connected domains that are not defined by
forward composition, as the following example shows.

######  Example 7.3.

Consider the sequence $(F_{n})$ of degree 2 Blaschke products defined as
follows. For $z\in{\mathbb{D}}$ and $n\in{\mathbb{N}}$,

$F_{n}(z):=z\varphi_{2/n}(\varphi_{1/2}(z)),\quad\text{where}\quad\varphi_{a}(z):=\frac{z-a}{1-\overline{a}z},\;\;a\in{\mathbb{D}}.$

Then, $\operatorname{dist}_{{\mathbb{D}}}(F_{n}(1/2),F_{n}(0))\to 0\text{ as
}n\to\infty,$ but

$\operatorname{dist}_{{\mathbb{D}}}(F_{n}(z),F_{n}(z^{\prime}))\nrightarrow
0\;\text{ as }n\to\infty,\;\text{ for all
}z,z^{\prime}\in{\mathbb{D}},\;(z,z^{\prime})\neq(0,1/2).$ (7.1)

###### Proof.

For $n\in{\mathbb{N}}$, $F_{n}(0)=0,F_{n}(1/2)=-1/n$, so
$\operatorname{dist}_{{\mathbb{D}}}(F_{n}(1/2),F_{n}(0))\to 0\text{ as
}n\to\infty,$ and

$F_{n}(z)=z\frac{\varphi_{1/2}(z)-2/n}{1-(2/n)\varphi_{1/2}(z)}\to
z\varphi_{1/2}(z)\;\text{ as }n\to\infty,\;\;\text{for all }z\in{\mathbb{D}},$

from which the property (7.1) readily follows. ∎

The form of the functions in Example 7.3 was suggested by work of P. Mercer
([Mer97]).

###  7.2 Proof of Theorem F

Theorem F is a special case of part (b) of the following result, obtained by
including the assumption that the interior orbits converge to the boundary.

######  Theorem 7.4.

Let $F_{n}:U_{0}\to U_{n}$, $n\in{\mathbb{N}}$, be a sequence of holomorphic
maps between simply connected domains, each having a full radial extension,
with the additional hypothesis that

$F_{n}(\zeta)\in\partial U_{n},\;\text{ for almost all }\zeta\in\partial
U_{0}.$ (7.2)

Also, suppose that $(F_{n})$ is contracting in $U_{0}$.

  * (a)

If there are measurable subsets $L\subset\partial U_{0}$ and
$L_{n}\subset\partial U_{n}$, $n\in{\mathbb{N}}$, such that
$L=F_{n}^{-1}(L_{n})$, for $n\in{\mathbb{N}}$, up to a set of harmonic measure
0, then $L$ has either full or zero harmonic measure with respect to $U_{0}$.

  * (b)

If in addition $F_{n}=f_{n}\circ\cdots\circ f_{1},$ where $f_{n}:U_{n-1}\to
U_{n}$, $n\in{\mathbb{N}}$, are holomorphic maps between simply connected
domains, each with a full radial extension, then for any $\zeta_{0}\in\partial
U_{0}$ the set

$\\{\zeta\in\partial U_{0}:|F_{n}(\zeta)-F_{n}(\zeta_{0})|\rightarrow 0\\}$
(7.3)

either has full or zero harmonic measure with respect to $U_{0}$.

For the proof of Theorem 7.4 we need the following ‘strong mixing’ result due
to Pommerenke [Pom81, Theorem 1].

######  Lemma 7.5 (Pommerenke).

Let $(G_{n})_{n\geq 1}$ be a sequence of inner functions such that

$G_{n}(0)=0\quad\text{and}\quad G_{n}(z)\to 0\;\text{ as }n\to\infty,\;\text{
for all }z\in{\mathbb{D}},$

and suppose there are measurable subsets $E$ and $E_{n}$, $n\in{\mathbb{N}}$,
of $\partial{\mathbb{D}}$ such that $E=G_{n}^{-1}(E_{n})$, for
$n\in{\mathbb{N}}$, up to a set of measure 0. Then, for all arcs
$A\subset\partial{\mathbb{D}}$, we have

$|A\cap E|=\frac{1}{2\pi}|A|.|E|.$ (7.4)

It follows that the set $E$ has either full or zero Lebesgue measure.

######  Remark.

The statement of Lemma 7.5 is different from that of [Pom81, Theorem 1] in
several respects. First, for simplicity, we have normalised the inner
functions to fix 0. Second, the proof of the identity (7.4) is obtained by
following the proof of [Pom81, Theorem 1] while exchanging the roles of the
sets $E_{n}$ and $E$ in that proof. Third, the final statement of Lemma 7.5
follows by applying (7.4) to a nested sequence of arcs converging to a point
of density of $E$, when such a point exists.

###### Proof of Theorem 7.4.

Let $z_{0}$ be an arbitrary point in $U_{0}$ and let $h_{n}$ be a univalent
map from $U_{n}$ onto $U_{0}$ with $h_{n}(F_{n}(z_{0}))=z_{0}$. Then the map
$h_{n}\circ F_{n}$ is a self-map of $U_{0}$ fixing $z_{0}$ (see Figure 2). Now
let $\varphi:\mathbb{D}\to U_{0}$ be a Riemann map such that
$\varphi(0)=z_{0}$. Accessible boundary points of $U_{0}$ correspond to points
of $\partial\mathbb{D}$ where $\varphi$ has radial limits (see Section 2.1).

${\mathbb{D}}$$E$$E_{n}$$G_{n}$$L_{n}$$L$$F_{n}$$h_{n}$$h_{n}\circ
F_{n}$$U_{0}$$U_{n}$$h_{n}(L_{n})$$\varphi$$\ldots$$0$$z_{0}$$F_{n}(z_{0})$
Figure 2:  Illustration of the proof of Theorem 7.4. The sets $E_{i}$ and
$L_{i}$ are represented as arcs for simplicity but are in fact just
measurable.

For $n\geq 1$ we define the map $G_{n}:{\mathbb{D}}\rightarrow{\mathbb{D}}$ as

$G_{n}=\varphi^{-1}\circ h_{n}\circ F_{n}\circ\varphi,$

and note that $G_{n}(0)=0$ for $n\in{\mathbb{N}}$. Since the sequence
$(F_{n})$ is contracting we have that
$\operatorname{dist}_{U_{n}}(F_{n}(z),F_{n}(z_{0}))\rightarrow 0$ as
$n\to\infty$ for all $z\in U_{0}$. Also, since $\varphi^{-1}\circ
h_{n}:U_{n}\rightarrow{\mathbb{D}}$ is a conformal map, we have, for every
$w\in{\mathbb{D}}$,

$\displaystyle\operatorname{dist}_{\mathbb{D}}(G_{n}(w),G_{n}(0))$
$\displaystyle=\operatorname{dist}_{\mathbb{D}}(\varphi^{-1}\circ h_{n}\circ
F_{n}\circ\varphi(w),\varphi^{-1}\circ h_{n}\circ F_{n}(z_{0}))$
$\displaystyle=\operatorname{dist}_{U_{n}}(F_{n}(\varphi(w)),F_{n}(z_{0}))\rightarrow
0\;\text{ as }n\to\infty,$

which means that $(G_{n})$ is also contracting.

Next define the sets $E_{n}\subset\partial{\mathbb{D}}$ as

$E=\varphi^{-1}(L)\quad\text{and}\quad
E_{n}=\varphi^{-1}(h_{n}(L_{n})),\;\text{ for }n\in{\mathbb{N}}.$ (7.5)

By (7.2) and (7.5) we have, for $n\in{\mathbb{N}}$, up to a set of measure
zero (by an argument similar to the one used in the proof of Proposition 2.3),
that

$\displaystyle G_{n}^{-1}(E_{n})$ $\displaystyle=\varphi^{-1}\circ
F_{n}^{-1}\circ h_{n}^{-1}\circ\varphi(E_{n})$
$\displaystyle=\varphi^{-1}\circ F_{n}^{-1}\circ h_{n}^{-1}\circ h_{n}(L_{n})$
$\displaystyle=\varphi^{-1}(L)=E.$

Suppose now that $L$ has positive harmonic measure in $U_{0}$, so $E$ has
positive Lebesgue measure in $\partial{\mathbb{D}}$. Since $F_{n}$ maps
$\partial U_{0}$ to $\partial U_{n}$, $G_{n}$ is an inner function for every
$n$. We can now apply Lemma 7.5 to the sequence $(G_{n})$ and deduce that
$|E|=2\pi$, and so $L\subset\partial U_{0}$ has full harmonic measure.

For part (b), let $L$ be the set in (7.3), and assume that for some
$\zeta_{0}\in\partial U_{0}$ the set $L$ has positive harmonic measure in
$U_{0}$. Then define the set $L_{n}\subset\partial U_{n}$, for
$n\in{\mathbb{N}}$, as

$L_{n}:=\\{\zeta\in\partial U_{n}:|f_{n+m}\circ\cdots\circ
f_{n+1}(\zeta)-F_{m+n}(\zeta_{0})|\to 0\;\text{ as }m\to\infty\\}.$ (7.6)

By definition,

$L=\\{\zeta\in\partial U_{0}:|F_{n}(\zeta)-F_{n}(\zeta_{0})|\rightarrow
0\\}=F_{n}^{-1}(L_{n}),\;\text{ for }n\in{\mathbb{N}},$ (7.7)

up to a set of harmonic measure 0, so we can apply part (a) of the theorem to
deduce that $L$ has full harmonic measure in $U_{0}$. ∎

###  7.3 An example related to Theorem F

The following example shows that Theorem F (and Theorem 7.4) fail without the
hypothesis which requires the sequence of maps to be contracting. The example
consists of a sequence of Möbius self-maps $M_{n}$ of ${\mathbb{D}}$, and
hence it can also be viewed as a forward composition
$M_{n}=m_{n}\circ\cdots\circ m_{1}$ of Möbius self-maps by defining
$m_{n}=M_{n}\circ M^{-1}_{n-1}$.

######  Example 7.6.

There exists a sequence $(M_{n})$ of Möbius self-maps of ${\mathbb{D}}$ such
that $M_{n}(1)=1$ for all $n$, and

  1. (a)

$M_{n}(z)\to 1$ as $n\to\infty$, for all $z\in{\mathbb{D}}$;

  2. (b)

$M_{n}(\zeta)\to 1$ as $n\to\infty$, for all $\zeta=e^{i\theta}$,
$|\theta|<\pi/2$; but

  3. (c)

$M_{n}(\zeta)\not\to 1$ as $n\to\infty$, for all $\zeta=e^{i\theta}$,
$\pi/2\leq|\theta|\leq\pi$.

###### Proof.

We define the maps first on the right half-plane. For points
$\zeta\in\partial\mathbb{H}$ and $a>1$ we shall consider affine self-maps of
$\mathbb{H}$ of the form

$\widehat{M}(z;\zeta,a)=a(z-\zeta)+\zeta.$

Clearly $\widehat{M}(\infty)=\infty$, $\widehat{M}(\zeta)=\zeta$.

For $j=1,2,\ldots$ we divide the segment $[-i,i]$ in the imaginary axis into
$j$ segments $I_{j,k}$, $k=0,\ldots,j-1$, each of length $2/j$, that is

$I_{j,k}=\left[-1+\frac{2k}{j},-1+\frac{2(k+1)}{j}\right]i,\quad
k=0,\ldots,j-1,$

with midpoint $\zeta_{j,k}=\left(-1+\frac{2k+1}{j}\right)i.$ Hence, for all
$k=0,\ldots,j-1$,

$\left|I_{j,k}\right|=\frac{2}{j}\text{ \ \ and \ \
}\bigcup_{k=0}^{j-1}I_{j,k}=[-i,i].$

Now, for all $k=0,\ldots,j-1$, we define the maps

$\widehat{M}_{j,k}(z)=\widehat{M}(z;\zeta_{j,k},j)=j(z-\zeta_{j,k})+\zeta_{j,k},\quad
z\in\overline{\mathbb{H}}.$

Then, on the one hand, for $\zeta\in I_{j,k}$ we have

$|\widehat{M}_{j,k}(\zeta)|\leq j|\zeta-\zeta_{j,k}|+1\leq j(1/j)+1=2,$ (7.8)

and on the other hand, for all $|z|>1$ and since $|\zeta_{j,k}|<1$,

$|\widehat{M}_{j,k}(z)|\geq j|z-\zeta_{j,k}|-1\geq j(|z|-1)-1,\quad 0\leq
k\leq j-1,j=1,2\ldots.$ (7.9)

Now we arrange the maps $\widehat{M}_{j,k}$ into a single sequence
$\widehat{M}_{n}=\widehat{M}_{j,k}$, where $n=\frac{1}{2}j(j+1)+k$, $0\leq
k\leq j-1$, $j=1,2\ldots$, and observe that for all
$z\in\overline{{\mathbb{H}}}$ with $|z|>1$ we have that
$\widehat{M}_{n}(z)\to\infty$ as $n\to\infty$, by (7.9). On the other hand,
every $z\in[-i,i]$ belongs to an infinite number of segments $I_{j,k}$, so
there exists an infinite number of distinct values of $n$ for which
$|\widehat{M}_{n}(z)|<2$ by (7.8)and therefore
$\widehat{M}_{n}(z)\not\to\infty$ as $n\to\infty$.

Finally, we define the corresponding maps on the closed unit disc by
$M_{n}(z)=\beta^{-1}\circ\widehat{M}_{n}\circ\beta(z)$, where
$\beta:{\mathbb{D}}\to\mathbb{H}$ is the Möbius map such that
$\beta(1)=\infty$, $\beta(-1)=0$ and $\beta(\pm i)=\pm i$. By the discussion
above, properties (b) and (c) are immediate. Property (a) follows immediately
from Theorem A, for example. ∎

##  8 Examples related to the ADM dichotomy

In this section we give three examples related to Theorem 1.3, part (b).
Recall that this states that for an inner function
$f:{\mathbb{D}}\to{\mathbb{D}}$ with a Denjoy–Wolff point and such that
$\sum_{n\in{\mathbb{N}}}(1-|f^{n}(z)|)=\infty$, for some $z\in{\mathbb{D}}$,
almost all boundary orbits of $f$ are dense in $\partial{\mathbb{D}}$. Here we
show that this result fails to hold in general if iterates of inner functions
are replaced by forward compositions of inner functions.

In Section 8.1, we give two examples which show this failure: the first
consists of forward compositions of Möbius self-maps of ${\mathbb{D}}$ and is
therefore an isometric sequence in the sense of Theorem 7.2, and the second
consists of forward compositions of Blaschke products of degree 2, which turns
out to be a semi-contracting sequence. Then, in Section 8.2 we give an example
of a forward composition of distinct Blaschke products of degree 2 which is
contracting and for which the conclusion of Theorem 1.3, part (b) does hold –
the techniques used in the proof of this example have some independent
interest.

These examples raise the question of whether part (b) of the ADM dichotomy
holds for contracting forward compositions of inner functions (or indeed for
more general contracting sequences), which we discuss at the end of this
section.

###  8.1 Forward compositions for which Theorem 1.3, part (b) fails

Our first example is a sequence of Möbius self-maps of ${\mathbb{D}}$.

######  Example 8.1.

Let $(a_{n})$ be an increasing sequence on $[0,1)$ such that $a_{0}=0$ and
$\lim_{n\to\infty}a_{n}=1$, let $m_{n}$, $n\in{\mathbb{N}}$, be the Möbius map
such that $m_{n}(\pm 1)=\pm 1$ and $m_{n}(a_{n-1})=a_{n}$, and define
$M_{n}=m_{n}\circ\cdots\circ m_{1}$, that is,

$M_{n}(z)=\frac{z+a_{n}}{1+a_{n}z},\quad\text{for }n\geq 1.$

Then, every point $\zeta\in\partial\mathbb{D}\setminus\\{-1\\}$ satisfies that
$\lim_{n\to\infty}M_{n}(z)=1$, regardless of whether $\sum_{n\geq 0}(1-a_{n})$
converges or not.

###### Proof.

We consider the cross-ratio

$\frac{(z_{1}-z_{3})(z_{4}-z_{2})}{(z_{3}-z_{2})(z_{1}-z_{4})},$

which is invariant under Möbius transformations. Take
$z_{1}=1,z_{2}=-1,z_{3}=0,z_{4}=\zeta\in\partial\mathbb{D}\setminus\\{\pm
1\\}.$ Then the invariance under $M_{n}$ gives

$0\neq\frac{(1-0)(\zeta+1)}{(0+1)(1-\zeta)}=\frac{(1-a_{n})(M_{n}(\zeta)+1)}{(a_{n}+1)(1-M_{n}(\zeta))}.$

Since $a_{n}\to 1$ we have $\frac{1-a_{n}}{a_{n}+1}\to 0$ as $n\to\infty$, so

$\frac{1-M_{n}(\zeta)}{M_{n}(\zeta)+1}\to 0\quad\text{as}\;n\to\infty,$

as required. ∎

As announced, a similar effect to that in Example 8.1 can be achieved with a
forward composition of Blaschke products $(b_{n})$ of degree two. In this
example, $B_{n}=b_{n}\circ\cdots\circ b_{1}\to 1$ as $n\to\infty$ in
${\mathbb{D}}$ and the summability condition of Theorem 1.3, part (b) fails,
and yet $B_{n}\to 1$ as $n\to\infty$ on a whole arc of $\partial{\mathbb{D}}$
with centre 1.

######  Example 8.2.

There exists a sequence of degree two Blaschke products $(b_{n})$ of the form

$b_{n}(z)=\left(\frac{z+\mu_{n}}{1+\mu_{n}z}\right)^{2},$ (8.1)

where $(\mu_{n})$ is a decreasing sequence such that $\mu_{n}\to 1/3$ as
$n\to\infty$, such that

  1. (a)

$B_{n}:=b_{n}\circ\cdots\circ b_{1}$ satisfies
$\sum_{n=1}^{\infty}(1-B_{n}(0))=\infty,$

  2. (b)

$\lim_{n\to\infty}B_{n}(z)=1$, for
$z\in{\mathbb{D}}\cup\\{\zeta\in\partial{\mathbb{D}}:|\zeta-1|<\rho\\},$ for
some $\rho>0.$

###### Proof.

First note that the map $z\mapsto\left(\frac{z+\mu}{1+\mu z}\right)^{2}$ has
an attracting fixed point at 1 for $\mu\in(1/3,1)$ and a parabolic fixed point
at 1 for $\mu=1/3$. It is convenient to construct the sequence $(\mu_{n})$ by
making a change of variables to the right half-plane
$\mathbb{H}=\\{z:\operatorname{Re}z>0\\}$ via the map
$\alpha(z)=\frac{1+z}{1-z}$.

Now $z\mapsto\frac{z+\mu_{n}}{1+\mu_{n}z}$ is conjugate to
$z\mapsto\lambda_{n}z$ under $\alpha$, where
$\lambda_{n}=\frac{1+\mu_{n}}{1-\mu_{n}}$ and we require $(\lambda_{n})$ to be
a decreasing sequence tending to 2 as $n\to\infty$. Also $z\mapsto z^{2}$ is
conjugate to $z\mapsto\frac{1}{2}(z+1/z)$ under $\alpha$. We deduce that
$b_{n}$ in ${\mathbb{D}}$ is conjugate to the map
$z\mapsto\frac{1}{2}(\lambda_{n}z+\frac{1}{\lambda_{n}z})$ in ${\mathbb{H}}$.
Abusing notation, we shall denote this map also by $b_{n}$, and also write
$B_{n}=b_{n}\circ\cdots\circ b_{1}$ in ${\mathbb{H}}$. The sum in statement
(a) now becomes

$\sum_{n=1}^{\infty}\left|1-\alpha^{-1}(B_{n}(1))\right|=\sum_{n=1}^{\infty}\frac{2}{B_{n}(1)+1},$

which diverges if and only if $\sum_{n=1}^{\infty}1/B_{n}(1)$ diverges.

We choose $(\lambda_{n})$ in such a way that $B_{n}(1)=n+1$, for
$n\in{\mathbb{N}}$, so that this divergence is assured. To achieve this we
need

$B_{n}(1)=b_{n}(n)=\frac{1}{2}\left(\lambda_{n}n+\frac{1}{\lambda_{n}n}\right)=n+1,$
(8.2)

so we define

$\lambda_{n}:=\frac{n+1+\sqrt{n^{2}+2n}}{n}=1+\frac{1}{n}+\left(1+\frac{2}{n}\right)^{1/2}=2\left(1+\frac{1}{n}\right)+O(1/n^{2})\;\text{
as }n\to\infty.$ (8.3)

We have $B_{n}(z)\to\infty$ for all $z$ in $\mathbb{H}$, by Theorem A, since
this property holds for $z=1$. So to prove (b) we need to show that, for all
sufficiently large $|y|$:

$B_{n}(iy)\to\infty\;\text{ as }n\to\infty.$ (8.4)

Now

$b_{n}(iy)=\frac{1}{2}\left(\lambda_{n}iy+\frac{1}{\lambda_{n}iy}\right)=\frac{1}{2}\left(\lambda_{n}iy-\frac{i}{\lambda_{n}y}\right)=:i\beta_{n}(y),$
(8.5)

say. By symmetry we need only to consider $y>0$. It is easy to check that the
maps $y\mapsto\beta_{n}(y)$ have a unique positive fixed point
$y^{*}_{n}=\frac{1}{2}\sqrt{n}+o(1)$ and, if $y>y_{n}^{*}$ then
$\beta_{n}(y)>y$. Hence we choose to check that if $y_{n}>n^{\frac{3}{4}}$,
for some $n\geq N$, say, then $y_{n+1}=\beta_{n}(y_{n})>(n+1)^{\frac{3}{4}}$
also.

We have

$\displaystyle y_{n+1}$
$\displaystyle=\beta_{n}(y_{n})=\frac{1}{2}\left(\lambda_{n}n^{3/4}-\frac{1}{\lambda_{n}n^{3/4}}\right)$
$\displaystyle=\frac{1}{2}\left((2(1+1/n)+O(1/n^{2}))n^{3/4}-\frac{1}{(2(1+1/n)+O(1/n^{2}))n^{3/4}}\right)$
$\displaystyle=\frac{n+1}{n^{1/4}}+O(1/n^{5/4})-\frac{n^{1/4}}{4(n+1+O(1/n))}\;\text{
as }n\to\infty.$

We claim that for $n$ large enough, $y_{n+1}>(n+1)^{3/4}.$ This holds since

$\frac{n+1}{n^{1/4}}-(n+1)^{3/4}>\frac{n^{1/4}}{4(n+1+O(1/n))}+O(1/n^{5/4}),$

that is,

$\left(\frac{n+1}{n}\right)^{1/4}-1>\frac{n^{1/4}}{4(n+1)^{3/4}(n+1+O(1/n))}+O\left(1/n^{2}\right),$

which follows from

$\frac{1}{4n}>\frac{n^{1/4}}{4(n+1)^{7/4}+O(1/n^{1/4})}+O\left(\frac{1}{n^{2}}\right),$

true for $n$ large enough. Hence there exists $N\in\mathbb{N}$ such that
$y_{N}>N^{3/4}$ implies that $y_{n}>n^{3/4}$, for $n\geq N$. We deduce from
this and from (8.5) that, for all $y>N^{3/4}$,

$b_{n}\circ\cdots\circ b_{N}(iy)\to\infty\;\text{ as }n\to\infty$

and so (8.4) holds if we relabel the $b_{n}$ starting with $b_{N}$. ∎

######  Remark.

In Example 8.2 it is natural to ask how large is the subset of
$\partial{\mathbb{D}}$ of points such that $B_{n}\to 1$ as $n\to\infty$. It is
plausible that this set is dense in $\partial{\mathbb{D}}$. However, the
sequence of Blaschke products in Example 8.2 is semi-contracting, as we now
show, so we cannot use Theorem 7.4 to deduce that
$\lim_{n\to\infty}B_{n}(\zeta)=1$ for almost all
$\zeta\in\partial{\mathbb{D}}$. Indeed, if we use Definition 7.1 and (8.3) to
compute

$\displaystyle\|\operatorname{D}b_{n}(n)\|_{\mathbb{H}}^{\mathbb{H}}$
$\displaystyle=\frac{\rho_{{\mathbb{H}}}(n+1)}{\rho_{{\mathbb{H}}}(n)}\,b^{\prime}_{n}(n)=\frac{1/(n+1)}{1/n}\,\frac{1}{2}\left(\lambda_{n}-\frac{1}{\lambda_{n}n^{2}}\right)$
$\displaystyle=\frac{n}{n+1}\left(1+1/n+O(1/n^{2})\right)$
$\displaystyle=1+O(1/n^{2})\;\text{ as }n\to\infty,$

we see that

$\sum_{n=1}^{\infty}\left(1-\|\operatorname{D}b_{n}(a_{n})\|_{\mathbb{H}}^{\mathbb{H}}\right)<\infty,$

which by Theorem 7.2 implies that the sequence $(B_{n})$ is semi-contracting.

###  8.2 Forward compositions for which Theorem 1.3, part (b) holds

In contrast to the two previous examples, where we saw that the second part of
the ADM dichotomy does not hold in general, we now show that this second part
does hold for certain types of forward compositions that form contracting
sequences. Note that the following example is the basis of Example 4.5.

######  Example 8.3.

Let $(a_{n})$ be any increasing sequence on $[0,1)$ such that $a_{0}=0$ and
$\lim_{n\to\infty}a_{n}=1$, and for $n\geq 0$ let
$M_{n}(z)=\frac{z+a_{n}}{1+a_{n}z}.$ For $n\in{\mathbb{N}}$ we define the
Blaschke products

$b_{n}(z):=M_{n}\left((M_{n-1}^{-1}(z))^{2}\right),$

and

$B_{n}(z):=b_{n}\circ\cdots\circ
b_{1}(z)=M_{n}(z^{2^{n}})=\frac{z^{2^{n}}+a_{n}}{1+a_{n}z^{2^{n}}}.$ (8.6)

Then, $B_{n}(z)\to 1$ for all $z\in{\mathbb{D}}$. If, in addition,

$\sum_{n\geq 0}(1-B_{n}(0))=\sum_{n\geq 0}(1-a_{n})=\infty,$ (8.7)

then the orbit of almost every point of $\partial{\mathbb{D}}$ under $(B_{n})$
is dense in $\partial{\mathbb{D}}$.

We need some preliminaries before we can prove Example 8.3. We first state a
‘weak independence’ version of the second Borel–Cantelli lemma which (in the
setting of forward compositions, rather than iterates) acts as a substitute
for techniques from the proof of the Poincaré recurrence theorem, used by
Doering and Mañé to prove Theorem 1.3, part (b).

######  Lemma 8.4 (Borel–Cantelli).

If $E_{n}$ are measurable subsets of $[0,1]$ such that $\sum|E_{n}|=\infty$
and

$|E_{m}\cap E_{n}|\leq C|E_{m}|.|E_{n}|,\;\text{ for }n>m\geq L,$

where $C\geq 1$ and $L\in\mathbb{N}$, then

$|\limsup_{n\to\infty}E_{n}|=|\\{x\in[0,1]:x\in E_{n}\text{ \rm{infinitely
often}}\\}|=\left|\bigcap_{N\geq 0}\bigcup_{n\geq N}E_{n}\right|>0.$

Lemma 8.4 is due to Ciesielski and Taylor [CT62] and independently to Lamperti
[Lam63], whose proof is admirably short. Another version is due to Petrov
[Pet02], which has the same hypotheses but the stronger conclusion that
$|\limsup_{n\to\infty}E_{n}|\geq 1/C$. Closely related results of this type
are due Kochen and Stone [KS64], and Yan [Yan06].

We use Lemma 8.4 to prove a ‘shrinking target’ result. This description of a
result that concerns the size of the set of points where a dynamical system
visits infinitely often a sequence of balls whose size shrinks to 0 with $n$
seems to originate with Hill and Velani [HV95] in the context of metric
Diophantine approximation.

######  Lemma 8.5 (Shrinking target).

Let $g(z)=z^{2}$ and let $(\varepsilon_{n})$ be any positive decreasing
sequence in $(0,1]$ such that $\sum_{n=1}^{\infty}\varepsilon_{n}=\infty$.
Further assume that the arcs
$I_{n}\subset\\{e^{i\theta}:|\theta+\pi|\leq\tfrac{1}{2}\varepsilon_{n}\\}$
satisfy $|I_{n}|\geq c\varepsilon_{n}$ for $n\in{\mathbb{N}}$, where $c>0$ is
independent of $n$. Then

$|\\{\zeta\in\partial{\mathbb{D}}:g^{n}(\zeta)\in I_{n}\text{ \rm{infinitely
often}}\\}|>0.$ (8.8)

Lemma 8.5 can in fact be deduced from a more general and more precise result
due to Philipp [Phi67, Theorem 2A], which is based on another version of the
second Borel–Cantelli lemma [Phi67, Theorem 3]. Since the details of the proof
in [Phi67] are involved and we do not need the extra precision here, we
include a proof of Lemma 8.5 based solely on Lemma 8.4.

###### Proof of Lemma 8.5.

First, $g^{-n}(\\{e^{i\theta}:|\theta+\pi|\leq\tfrac{1}{2}\varepsilon_{n}\\})$
consists of $2^{n}$ arcs, each of length $\varepsilon_{n}/2^{n}$ and centred
at the $2^{n}$-th roots of $-1$. Let $A_{n}$ denote the union of these $2^{n}$
arcs; see Figure 3.

$-1$$-i$$i$$A_{1}$$A_{0}$$A_{2}$$A_{3}$ Figure 3:  The sets $A_{n}$.

Then

$|A_{n}|=2^{n}\times\frac{\varepsilon_{n}}{2^{n}}=\varepsilon_{n},\quad n\geq
1.$ (8.9)

We will now find an upper bound for the length $|A_{m}\cap A_{n}|$ whenever
$n>m\geq 1$. We consider two cases. First, take $n=m+p$, where
$p\in{\mathbb{N}}$ is so small that $\frac{1}{2^{p+1}}\geq\varepsilon_{m}$.
The centre points of the arcs of $A_{m}$ are the $2^{m}$-th roots of $-1$,
which are of the form $\exp(2\pi i\theta_{k,m}),$ where
$\theta_{k,m}=k/2^{m+1}$ and $k\in\\{1,\ldots,2^{m+1}-1\\}$ is odd. Therefore
the centre points of the arcs of $A_{m+p}$ that are closest to the arc of
$A_{m}$ with centre $\exp(2\pi i\theta_{k,m})$, where
$k\in\\{1,\ldots,2^{m+1}-1\\}$ is odd, are

$\exp(2\pi i((2^{p}k\pm 1)/2^{m+p+1})=\exp(2\pi i(\theta_{k,m}\pm
1/2^{m+p+1})).$

Since $\frac{1}{2^{p+1}}\geq\varepsilon_{m}$ and the sequence
$(\varepsilon_{n})$ is decreasing, we deduce that

$\frac{1}{2}.\frac{\varepsilon_{m}}{2^{m}}+\frac{1}{2}.\frac{\varepsilon_{m+p}}{2^{m+p}}<\frac{\varepsilon_{m}}{2^{m}}\leq\frac{1}{2^{m+p+1}},$

so the arcs of $A_{m+p}$ closest to $\exp(2\pi i\theta_{k,m})$ do not meet the
arc of $A_{m}$ centred at that point. Hence, for this range of values of $n$
we have $A_{m}\cap A_{n}=\emptyset$.

Now we consider the case where $n=m+p,p\in{\mathbb{N}}$ and
$\frac{1}{2^{p+1}}<\varepsilon_{m}$. Recall that each of the $2^{m}$ arcs of
$A_{m}$ has length $\varepsilon_{m}/2^{m}$ and the angle between the centres
of the arcs $A_{m+p}$ is $2\pi/2^{m+p}$. Hence the number of arcs of $A_{m+p}$
that can meet each arc of $A_{m}$ is at most

$\frac{\varepsilon_{m}/2^{m}}{2\pi/2^{m+p}}+1=\frac{2^{p}\varepsilon_{m}}{2\pi}+1,$

we deduce in this case, and hence in all cases, that

$\displaystyle|A_{m}\cap A_{n}|$ $\displaystyle=|A_{m}\cap A_{m+p}|$
$\displaystyle\leq
2^{m}\left(\frac{2^{p}\varepsilon_{m}}{2\pi}+1\right)\frac{\varepsilon_{m+p}}{2^{m+p}}$
$\displaystyle=\varepsilon_{m}\varepsilon_{m+p}\left(\frac{1}{2\pi}+\frac{1}{2^{p}\varepsilon_{m}}\right)$
$\displaystyle\leq 3|A_{m}|.|A_{n}|.$ (8.10)

We now map the sets $A_{m}$ by $z\mapsto\arg z/(2\pi)$ and apply Lemma 8.4 to
the resulting subsets, $E_{m}$ say, of $[0,1]$. Since

$|E_{m}\cap E_{n}|\leq 6\pi|E_{m}|.|E_{n}|,\quad\text{for }n>m\geq 1,$

we deduce by (8.9) and Lemma 8.4 that

$|\limsup_{n\to\infty}E_{n}|>0,$

and hence (8.8) holds in the case when
$I_{n}=\\{e^{i\theta}:|\theta+\pi|\leq\tfrac{1}{2}\varepsilon_{n}\\}$, for
$n\in{\mathbb{N}}$.

In the general case that
$I_{n}\subset\\{e^{i\theta}:|\theta+\pi|\leq\tfrac{1}{2}\varepsilon_{n}\\}$,
for $n\in{\mathbb{N}}$, let $A^{\prime}_{n}=g^{-n}(I_{n})$. Then
$A^{\prime}_{n}$ consists of $2^{n}$ arcs, each one contained in a different
arc of $A_{n}$. Note that the arcs in $A^{\prime}_{n}$ have length at least
$c|A_{n}|=c\,\varepsilon_{n}/2^{n}$, so $|A^{\prime}_{n}|\geq
c\varepsilon_{n}$. Then, by (8.2), we have for all $n\in{\mathbb{N}}$

$\displaystyle|A^{\prime}_{m}\cap A^{\prime}_{n}|$
$\displaystyle\leq|A_{m}\cap A_{n}|$ $\displaystyle\leq 3|A_{m}|.|A_{n}|$
$\displaystyle\leq(3/c^{2})|A^{\prime}_{m}|.|A^{\prime}_{n}|.$

Therefore, we can once again apply Lemma 8.4 to deduce that (8.8) holds in
general. ∎

To complete the proof of Example 8.3 we need the following property of the
sequence $(B_{n})$ defined in (8.6).

######  Lemma 8.6.

The sequence of maps $B_{n}:{\mathbb{D}}\to{\mathbb{D}},n\geq 1,$ is
contracting.

###### Proof.

We evaluate the hyperbolic distortion along the orbit of 0 under $B_{n}$. By
(8.6), for each $n\in{\mathbb{N}}$:

$\displaystyle\|\operatorname{D}B_{n}(0)\|_{{\mathbb{D}}}^{{\mathbb{D}}}$
$\displaystyle=$ $\displaystyle\lim_{z\to
0}\frac{\operatorname{dist}_{{\mathbb{D}}}(B_{n}(z),B_{n}(0))}{\operatorname{dist}_{{\mathbb{D}}}(z,0)}$
$\displaystyle=$ $\displaystyle\lim_{z\to
0}\frac{\operatorname{dist}_{{\mathbb{D}}}(z^{2^{n}},0)}{\operatorname{dist}_{{\mathbb{D}}}(z,0)}$
$\displaystyle=$ $\displaystyle\lim_{z\to
0}\log\frac{1+|z|^{2^{n}}}{1-|z|^{2^{n}}}\Big{/}\log\frac{1+|z|}{1-|z|}=0.$

It follows that

$\sum_{n=1}^{\infty}(1-\|\operatorname{D}B_{n}(0)\|_{\mathbb{D}}^{\mathbb{D}})=\infty,$

and so $(B_{n})$ is indeed contracting, by Theorem 7.2. ∎

###### Proof of Example 8.3.

Since $B_{n}(0)=a_{n}\to 1$ as $n\to\infty$, it is clear that $B_{n}(x)\to 1$
for all $x\in(0,1)$, so this convergence holds throughout ${\mathbb{D}}$ by
Theorem A.

Next we assume that (8.7) holds. To complete the proof it is sufficient to
show that if $T$ is any non-trivial closed arc of
$\partial{\mathbb{D}}\setminus\\{1\\}$, then almost every point of
$\partial{\mathbb{D}}$ has an orbit under $(B_{n})$ that visits $T$ infinitely
often. To do this we choose an arc $S\subset\partial{\mathbb{D}}$ with centre
1 so that $T\subset S^{c}$. It follows from Lemma 5.1 that there are constants
$c_{S}>c_{T}>0$ such that, for all $n\geq 1$ large enough, the preimage
$M_{n}^{-1}(S^{c})$ contains an arc of $\partial{\mathbb{D}}$ centred at $-1$
of length $c_{S}(1-a_{n}):=\varepsilon_{n},$ say, and the preimage
$M_{n}^{-1}(T)$ contains an arc
$I_{n}\subset\\{e^{i\theta}:|\pi+\theta|\leq\tfrac{1}{2}\varepsilon_{n}\\}$ of
length at least $c_{T}(1-a_{n})=(c_{T}/c_{S})\varepsilon_{n}$. By hypothesis,
$\sum_{n=1}^{\infty}\varepsilon_{n}=\infty$, and also $(\varepsilon_{n})$ is
decreasing since $(a_{n})$ is increasing.

Now, since $M_{n}(I_{n})\supset T$,

$\displaystyle\limsup_{n\to\infty}B_{n}^{-1}(T)$
$\displaystyle=\\{\zeta\in\partial\mathbb{D}:B_{n}(\zeta)\in T\text{
infinitely often}\\}$
$\displaystyle\supset\\{\zeta\in\partial\mathbb{D}:g^{n}(\zeta)\in I_{n}\text{
infinitely often}\\},$

so, by Lemma 8.5,

$|\limsup_{n\to\infty}B_{n}^{-1}(T)|>0.$ (8.11)

We now use Lemma 8.6 and Theorem 7.4, part (a) to deduce from (8.11) that
$\limsup_{n\to\infty}B_{n}^{-1}(T)$ actually has full measure with respect to
$\partial\mathbb{D}$, which implies that $B_{n}(\zeta)\in T$ infinitely often
for almost every $\zeta\in\partial\mathbb{D}$, as required.

To do this we define, for $n\geq 0$,

$L_{n}:=\limsup_{m\to\infty}(b_{n+m}\circ\cdots\circ b_{n+1})^{-1}(T),$ (8.12)

that is, the set of $\zeta\in\partial{\mathbb{D}}$ whose orbit under the
sequence of maps $b_{n+m}$, $m\geq 1$, visits $T$ infinitely often; in
particular,

$L_{0}:=\limsup_{m\to\infty}B_{n}^{-1}(T).$

By construction we have $L_{0}=B_{n}^{-1}(L_{n})$, for $n\geq 0$, and so the
set $\limsup_{n\to\infty}B_{n}^{-1}(T)$ has full measure, by (8.11), Lemma 8.6
and Theorem 7.4, part (a), as required. ∎

######  Remark.

Example 8.3 can be generalised considerably; for example, in Lemma 8.5 we can
replace the map $g(z)=z^{2}$ by $g(z)=z^{p}$ for any integer $p\geq 2$ by
applying [Phi67, Theorem 2A] instead of the direct proof of the lemma given
here. And it seems plausible that in Example 8.3 we may be able to replace the
function $g(z)=z^{2}$ used in the definition of $b_{n}$ by any contracting
Blaschke product, or even inner function, that fixes 0, by using a recent
result [RR20, Theorem 5.11] which gives a shrinking target result for inner
functions that are ergodic on $\partial{\mathbb{D}}$, and hence contracting by
[DM91, Theorem 3.1].

###  8.3 Part (b) of the ADM dichotomy

Earlier in this section we showed that part (b) of the ADM dichotomy does not
hold in general in the setting of forward compositions, at least in the
isometric and semi-contracting cases. This leaves open the possibility that it
holds in the contracting case.

To provide some evidence that this may be the case (apart from our failure to
produce a contracting counter-example!), we outline briefly how to prove
Theorem 1.3, part (b) by using results of Aaronson, Doering and Mañé, and
Neuwirth, most of which appear in the text [Aar97]; see also [BFJK20a, Section
2] for a convenient summary of the concepts from ergodic theory mentioned here
and the relationships amongst them.

All the steps in the proof concern the behaviour of the inner function $f$ on
the boundary of ${\mathbb{D}}$. (Recall that an inner function is defined at
almost all boundary points.)

  * •

the condition that $\sum_{n\geq 0}(1-|f^{n}(z)|)=\infty$ for some
$z\in{\mathbb{D}}$ implies that $f$ is conservative and hence ergodic on
$\partial{\mathbb{D}}$, by [Aar97, Propositions 6.1.7 and 6.1.8];

  * •

since $f$ is an inner function it is non-singular with respect to the Lebesgue
measure on $\partial{\mathbb{D}}$, by [Aar97, Proposition 6.1.1];

  * •

hence $f$ is non-singular, conservative and ergodic on $\partial{\mathbb{D}}$,
which implies, by [Aar97, Proposition 1.2.2], that for any set $E$ of positive
measure in $\partial{\mathbb{D}}$, the iterates $f^{n}(\zeta)$ visit $E$
infinitely often for almost every $\zeta\in\partial{\mathbb{D}}$, a strong
recurrence property which implies the density of orbits property stated in
Theorem 1.3, part (b).

The reader may wish to compare these arguments to those in the proof of
[BFJK20a, Theorem C] which uses a slightly different path to arrive at the
same conclusion.

The relevance of the proof outlined above to our question about contracting
sequences is as follows. We see from the first step in the proof that, for an
inner function $f$, the divergence condition given in Theorem 1.3, part (b)
implies the ergodic property, and we know from [DM91, Theorem 3.1] that the
iterates of an inner function which is ergodic on $\partial{\mathbb{D}}$ form
a contracting sequence in the disc.

So, it is reasonable to ask whether some version of part (b) of the ADM
dichotomy holds for any contracting composition (or indeed sequence) of
holomorphic functions between simply connected domains. However, concepts like
invariance, ergodicity, recurrence and density of orbits only make sense when
the domains $U_{n}$ are all identical so that boundary points can be
identified. In particular, these concepts do not make sense for wandering
domains.

We can, however, ask whether a weaker version of part (b) of the dichotomy
holds, with a different type of proof. More precisely, suppose that
$F_{n}:U\to U_{n}$ is a contracting forward composition (or perhaps sequence)
of holomorphic functions between simply connected domains for which interior
orbits converge to the boundary sufficiently slowly (in some precise sense
that takes the geometry of the domains into account). Then it is plausible
that the Denjoy-Wolff set may have measure zero.

To prove such a result, possible tools include versions of the second
Borel–Cantelli lemma (such as the one used to prove Example 8.3) and Löwner’s
lemma (the case of equality, mentioned in the remark following Lemma 2.6),
together with Pommerenke’s strong boundary mixing result [Pom81, Theorem 1] in
the case of forward compositions. Such an approach would complement our proof
of the generalisation of part (a) of the ADM dichotomy, where we used the
Löwner’s lemma inequality, together with an argument similar to that used to
prove the first Borel–Cantelli lemma. The Borel–Cantelli lemmas have a
dichotomy similar to Theorem 1.3, in which convergent and divergent series
lead to sets of zero measure or positive measure, respectively, and in that
sense they are a natural tool to attack this problem.

##  9 Versions of results with the spherical metric

It is natural to ask to what extent our results hold when the Euclidean metric
for measuring distances to the boundary of a domain is replaced by the
spherical metric. Using the spherical metric has the advantage that sequences
tending to $\infty$ are included but the disadvantage that points may be close
together in the spherical metric but far apart in the Euclidean metric. As an
illustration, consider the sequence

$F_{n}(z)=M_{n}(z)+n,\quad z\in{\mathbb{D}},\;n=1,2,\ldots,$

where $(M_{n})$ is the sequence of Möbius self-maps of ${\mathbb{D}}$ in
Example 7.6. In this case, the spherical distances between all pairs of points
of $\overline{F_{n}({\mathbb{D}})}$ tends to 0 as $n\to\infty$, but the subtle
behaviour of $F_{n}$ on $\partial{\mathbb{D}}$ is only seen when using the
Euclidean metric.

Also, our hypotheses on the functions $F_{n}:U\to U_{n}$ are so general that
Theorem A does not hold when expressed in terms of the spherical metric. For
example, sequences of the form

$F_{n}(z)=C_{n}B(z)+z,\quad z\in{\mathbb{D}},\;n=1,2,\ldots,$ (9.1)

where $B$ is a Blaschke product, $C_{n}\to\infty$ as $n\to\infty$,
$U={\mathbb{D}}$ and $U_{n}=\\{z:|z|<|C_{n}|+1\\}$, and ones of the form

$F_{n}(z)=(2z)^{p_{n}},\quad z\in{\mathbb{D}},\;n=1,2,\ldots,$ (9.2)

where $p_{n}\in{\mathbb{N}}$ and $p_{n}\to\infty$ as $n\to\infty$,
$U={\mathbb{D}}$ and $U_{n}=\\{z:|z|<2^{p_{n}}\\}$, show that we can have
$F_{n}(z)\to\infty$ as $n\to\infty$ for some $z\in{\mathbb{D}}$ but not all.
However, it follows easily from the Euclidean version of Theorem A that with
the extra hypothesis of normality we have the following spherical version;
here we denote spherical distance by $\chi$.

######  Theorem 9.1.

Let $F_{n}:U\to U_{n}$ be a sequence of holomorphic maps between simply
connected domains and suppose that $(F_{n})$ is normal in $U$. If there exists
$z_{0}\in U$ such that

$\chi(F_{n}(z_{0}),\partial U_{n})\to 0\;\text{ as }n\to\infty,$

then, for all $z\in U$,

$\chi(F_{n}(z),F_{n}(z_{0}))\to 0\;\text{ as }n\to\infty.$

The role of the normality hypothesis in Theorem 9.1 is to ensure that if
$\chi(F_{n_{k}}(z_{0}),\infty)\to 0$ as $k\to\infty$ for any subsequence
$(F_{n_{k}})$, then $\chi(F_{n_{k}}(z),\infty)\to 0$ as $k\to\infty$ for all
$z\in U$.

Next we state a spherical version of Theorem C in which the assumption of
normality is not required.

######  Theorem 9.2.

Let $F_{n}:U\to U_{n}$ be a sequence of holomorphic maps between simply
connected domains, each with a full radial extension to $\partial U$, and
suppose that there exists $z_{0}\in U$ such that

$\sum_{n=0}^{\infty}\operatorname{dist_{\chi}}(F_{n}(z_{0}),\partial
U_{n})^{1/2}<\infty.$ (9.3)

Then for almost all points $\zeta\in\partial U$ we have

$\chi(F_{n}(\zeta),F_{n}(z_{0}))\to 0\;\;\text{as}\;n\to\infty.$

######  Remarks.

  * 1.

Examples of the form (9.1), with $C_{n}=n^{3}$ for instance, show that in the
spherical setting the hypothesis (9.3) is not independent of $z_{0}\in U$, in
contrast to the independence of condition (1.2) in Theorem C; see the first
remark after Theorem 4.1.

  * 2.

A special case of Theorem 9.2 can be found in [RS11, remark after Theorem
1.1], where it is observed that if $U$ is a Baker domain of an entire function
$f$ such that for some $z_{0}\in U$ and all sufficiently large $n$ we have

$|f^{n+1}(z_{0})|\geq k|f^{n}(z_{0})|,\;\text{where }k>1,$ (9.4)

then, for almost all $\zeta\in\partial U$, we have

$f^{n}(\zeta)\to\infty\;\text{ as }n\to\infty.$

Indeed, since for a point $z\in{\mathbb{C}}$ we have that
$\operatorname{dist}_{\chi}(z,\infty)\sim 1/|z|$, the assumption (9.4) implies
that the series in (9.3) is convergent by comparing it to a geometric series.
Similarly, by observing for example that the $p$-series
$\sum_{n=0}^{\infty}\frac{1}{n^{p}}<\infty$ for $p>1$, we obtain that the
claim of Theorem 9.2 holds whenever for some $z_{0}\in U$ and all sufficiently
large $n$ we have $|f^{n}(z_{0})|\geq n^{2p}$ for $p>1$, or analogously,

$|f^{n+1}(z_{0})|\geq\left(1+\frac{1}{n}\right)^{2p}|f^{n}(z_{0})|,\;\text{where
}p>1.$ (9.5)

There is also a spherical version of Theorem E in which we again need an extra
hypothesis, one that implies normality.

######  Theorem 9.3.

Let $F_{n}:U\to U_{n}$ be a sequence of holomorphic maps between simply
connected domains, each with a full radial extension, and suppose that there
exists $R>0$ such that

$(\hat{\mathbb{C}}\setminus U_{n})\cap\\{z:|z|=R\\}\neq\emptyset,\;\text{ for
}n\in{\mathbb{N}}.$ (9.6)

If there exists $\zeta_{0}\in\partial U$ such that
$\chi(F_{n}(\zeta_{0}),\partial U_{n})\to 0$ as $n\to\infty$ and

$L_{0}:=\\{\zeta\in\partial U:\chi(F_{n}(\zeta),F_{n}(\zeta_{0}))\to
0\;\;\text{as }n\to\infty\\}$

has positive harmonic measure with respect to $U$, then, for all $z\in U$, we
have

$\chi(F_{n}(z),F_{n}(\zeta_{0}))\to 0\;\;\text{as }n\to\infty.$

The extra hypothesis (9.6) ensures that each point of $\partial U_{n}$,
$n\in{\mathbb{N}}$, lies in a continuum that is exterior to $U_{n}$ and
contains a point of $\\{z:|z|\leq R\\}$. This continuum is required because
replacing the Euclidean metric by the spherical metric means that the
spherical discs needed in the proof sometimes contain $\infty$, and in this
circumstance the condition (9.6) enables us to replace the continuum to
$\infty$ by a continuum to $\\{z:|z|=R\\}$ in the final stage of the proof
where we apply the Milloux–Schmidt inequality; we omit the details.

Note that the condition (9.6) implies that $(F_{n})$ forms a normal family by
using Carathéodory’s form of Montel’s theorem; see [Car54, p. 202].

Finally, Theorem F holds with no modifications when the Euclidean metric is
replaced by the spherical metric, since the deduction of part (b) of Theorem
7.4 from part (a) is essentially unchanged in the spherical setting.

######  Remark.

As this paper was being completed we learnt of very interesting recent work by
Martí-Pete, Rempe and Waterman in [MPRW22]. Of relevance to our current paper,
[MPRW22, Theorem 1.10] concerns the dynamical behaviour of boundary points of
a wandering domain $U$ of an entire function. They prove that the set of
points $z\in\partial U$ such that
$\limsup_{n\to\infty}\chi(f^{n}(z),f^{n}(z_{0}))>0$, where $z_{0}\in U$, which
they call maverick points, forms a set of harmonic measure zero. In the
setting of wandering domains and the spherical metric, their result is
stronger than the results given in this section, though the results in this
section hold in a more general setting. Our results in earlier sections
capture the differing behaviours of orbits of boundary points tending to
infinity.

##  10 Open questions

In this final section we discuss several interesting questions, which arise in
connection with our new results. The first relates to a possible
generalisation of the ADM dichotomy, part (b), discussed in Section 8.3.
Recall that the Denjoy–Wolff set is only defined if interior orbits converge
to the boundary, and that it follows from Theorem A that, in this case, all
interior orbits have the same limiting behaviour. For simplicity, the question
is stated for inner functions; in a more general version, the geometry of the
domain’s boundary would play a role.

######  Question 10.1.

(a) Must the Denjoy–Wolff set have zero measure for any contracting forward
composition of inner functions $F_{n}=f_{n}\circ\cdots\circ f_{1}$,
$n\in{\mathbb{N}}$, such that interior orbits converge to the boundary
sufficiently slowly that $\sum_{n\in{\mathbb{N}}}(1-|F_{n}(0)|)$ is divergent?

(b) More generally, we can ask this question for contracting sequences
$(F_{n})$ of inner functions such that interior orbits converge to the
boundary sufficiently slowly that this series is divergent.

Including the contracting hypothesis in Question 10.1 is in some sense natural
since for iteration of a single inner function the divergence condition
implies the contracting property; see the discussion in Section 8.3.

If the answer to Question 10.1, part (b) does turn out to be ‘yes’, this would
imply that a version of Theorem F holds for sequences of inner functions, not
just forward compositions of them, since if the sum is convergent we can apply
Theorem B, which gives full measure for the Denjoy–Wolff set.

Next we have a couple of questions relating specifically to our examples. In
connection with Theorem D, concerning a sequence of Möbius self-maps of
${\mathbb{D}}$ with empty Denjoy–Wolff set, the following question arises.

######  Question 10.2.

Does there exist an example of a forward composition of non-Möbius self-maps
of ${\mathbb{D}}$ for which interior orbits converge to the boundary and the
Denjoy–Wolff set is empty? Similarly, does there exist an orbit of wandering
domains, univalent or not, for which the interior orbits converge to the
boundary and the Denjoy-Wolff set is empty?

Another of our examples suggests a further question.

######  Question 10.3.

In Example 8.2, in which an entire boundary neighbourhood of 1 converges to 1,
can we deduce that the set of boundary points which converge to 1 is dense in
$\partial{\mathbb{D}}$ or even has full measure there?

Finally, we learnt the following related and interesting question from Marco
Abate: it is known that there is no straightforward generalisation of the
Denjoy–Wolff theorem from ${\mathbb{D}}$ to a general simply connected domain
$U$; see [Mil06, Problem 5-a] for an example of a comb domain with non-locally
connected boundary within which orbits of a univalent self-map accumulate at a
continuum in the boundary.

######  Question 10.4.

Let $f$ be a holomorphic self-map of a simply connected domain $U$. What non-
trivial conditions on $f$ and $U$ are sufficient to ensure that a Denjoy–Wolff
point exists in $\overline{U}$ for $f$?

First note that our Theorem A implies that if $f$ is a holomorphic self-map of
a simply connected domain $U$ and one orbit converges to a point $p\in\partial
U$, then all others must do the same and hence $p$ is the unique Denjoy–Wolff
point.

A trivial sufficient condition for Question 10.4 is that $U$ is a Jordan
domain, and one might conjecture that if $f$ has a full radial extension to
$\partial U$, possibly mapping $\partial U$ to itself almost everywhere, then
this property would also be sufficient. However, it is straightforward to
modify the example in [Mil06, Problem 5-a], replacing the teeth of the comb by
thin triangles, to ensure that the univalent self-map has a full radial
extension. Moreover, the univalent self-map of such a domain $U$ can be
replaced by a self-map of the domain that is conjugate via a Riemann map to a
degree 2 hyperbolic Blaschke product with an attracting fixed point on the
boundary, and within $U$ orbits under this self-map also accumulate at a
continuum.

######  Remark.

There are versions of the Denjoy–Wolff theorem for self-maps of hyperbolic
Riemann surfaces in which the boundary limit set is in general a single point
or a continuum; see [Hei41] and [Hei88]. See also [CS21] for results
concerning the stability of the Denjoy–Wolff theorem under small perturbations
of a holomorphic self-map of ${\mathbb{D}}$ and [AC21] for generalisations of
such results to self-maps of hyperbolic Riemann surfaces.

## References

  * [Aar78] Jon Aaronson.  Ergodic theory for inner functions of the upper half plane.  Ann. Inst. H. Poincaré Sect. B (N.S.), 14(3):233–253, 1978\.
  * [Aar97] Jon Aaronson.  An introduction to infinite ergodic theory, volume 50 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 1997.
  * [AC21] Marco Abate and Argyrios Christodoulos.  Random iteration on hyperbolic Riemann surfaces.  arXiv:2109.10029, 2021.