UNBOUNDED FAST ESCAPING WANDERING DOMAINS VASILIKI EVDORIDOU ADI GL UCKSAM AND LETICIA PARDO-SIM ON Abstract. We introduce a new approximation technique into the context of complex

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UNBOUNDED FAST ESCAPING WANDERING DOMAINS

VASILIKI EVDORIDOU, ADI GL ¨

UCKSAM, AND LETICIA PARDO-SIM ´

Abstract. We introduce a new approximation technique into the context of complex

dynamics that allows us to construct examples of transcendental entire functions with

unbounded wandering domains. We provide examples of entire functions with an orbit of

unbounded fast escaping wandering domains, answering a long-standing question of Rip-

pon and Stallard. Moreover, these examples cover all possible types of simply-connected

wandering domains in terms of convergence to the boundary. In relation to a conjecture

of Baker, it was unknown whether functions of order less than one could have unbounded

wandering domains. For any given order greater than 1/2 and smaller than 1, we provide

an entire function of such order with an unbounded wandering domain.

1. Introduction

Let f:C→Cbe a transcendental entire function and for every n≥0, denote by fn

its n-th iterate. The Fatou set of f,F(f), consists of all the points z∈Cfor which the

family {fn}n∈Nis equicontinuous with respect to the spherical metric. Its complement,

J(f).

.=C\F(f), is known as the Julia set. The Fatou set of an entire map is always open

and consists of connected components, called Fatou components. For a Fatou component

Uof fand each n≥0, we denote by Unthe Fatou component containing fn(U), and

call the sequence (Un)n≥0an orbit. A Fatou component Umay be periodic, if Un⊆U

for some n∈N,preperiodic, if Unis periodic for some n∈N,or, otherwise, Uis called a

wandering domain. Equivalently, Uis a wandering domain if Un=Umimplies n=m.

Baker [Bak76] was the ﬁrst to provide an example of a transcendental entire function

with a wandering domain in 1976, while Sullivan [Sul85] showed in 1985 that they do not

occur for rational functions. Since then, many examples of transcendental entire functions

with wandering domains, most of them bounded, have been provided, e.g. [Bak84,Her84,

EL87,Dev90,FH09]. In recent years, with the development of new techniques, much

progress on the understanding of wandering domains has been achieved, e.g. [Bis15,

Laz17,MS20,FJL19,BEF+22,BT21,MRW22].

All limit functions of the family of iterates in a wandering domain are constant [Fat20],

and so they can be classiﬁed into escaping, when the only limit function is inﬁnity, oscillat-

ing, if both inﬁnity and at least one other ﬁnite value are limit functions, and dynamically

bounded, when all limit functions are in C. It remains one of the major open problems in

transcendental dynamics whether dynamically bounded wandering domains exist. In this

paper, we focus on escaping wandering domains.

2020 Mathematics Subject Classiﬁcation. Primary 37F10; secondary 30D05. Key words: entire func-

tions, wandering domains, fast escaping set.

arXiv:2210.13350v2 [math.DS] 2 Feb 2023

2 V. EVDORIDOU, A. GL ¨

UCKSAM, AND L. PARDO-SIM ´

For a transcendental entire function f, points might converge to inﬁnity under iteration

at diﬀerent rates. Of particular importance is the fast escaping set,A(f), of points that

escape to inﬁnity ‘as fast as possible’ under iteration. This set, introduced by Bergweiler

and Hinkkanen [BH99] in connection to permutable functions and absence of wandering

domains, is deﬁned in [RS12] as

A(f).

.={z: there exists `∈Nsuch that |fn+`(z)| ≥ Mn

f(R),for n∈N},

where Mf(r).

.= max|z|=r|f(z)|, for r > 0, is the maximum modulus function, Mn

f(r)

denotes the n-th iterate of the function Mf(r), and R > 0 can be taken to be any value

such that Mf(r)> r, for r≥R. It follows from [RS12, Theorem 1.2] that if Uis a

component of F(f) such that U∩A(f)6=∅, then U⊂A(f), and we say that Uis fast

escaping. We note that all fast escaping components of F(f) must be wandering domains;

see [BH99, Lemma 4].

Multiply connected wandering domains are always bounded and fast escaping, [Bak84,

RS05]. However, only Bergweiler [Ber11] and Sixsmith [Six12] have provided examples

of simply connected, fast escaping wandering domains, which are shown to be bounded

in both cases. This led Rippon and Stallard to ask whether there exist unbounded fast

escaping wandering domains, [RS12, Question 1, p. 802]; see also [BRS16, p. 453] for the

connection of this problem to that of permutable functions having the same Julia set. We

give a positive answer to Rippon and Stallard’s question.

Theorem 1.1. There exists a transcendental entire function with an orbit of unbounded

fast escaping wandering domains.

We prove Theorem 1.1 by introducing a new approximation technique into the world of

complex dynamics allowing us to construct functions with unbounded wandering domains.

Our method is based on H¨ormander’s solution to ¯

∂-equations; see H¨ormander’s Theorem

on p.9. In particular, it provides control over the growth of the resulting function while

allowing for ﬂexibility on the choice of domains of approximation, see Section 2for details.

While all orbits within (pre)periodic Fatou components and multiply connected wan-

dering domains behave essentially in the same manner, see [Ber93,BRS13], the internal

dynamics of simply connected wandering domains exhibit a wider range of possibilities,

and have only been completely classiﬁed recently. Namely, in [BEF+22], two classiﬁca-

tions are provided, giving rise to three cases each. The ﬁrst in terms of the long-term

behaviour of orbits with respect to hyperbolic distances, [BEF+22, Theorem A], and the

second with respect to convergence to the boundary, [BEF+22, Theorem C]; for the deﬁni-

tion, see Theorem 2.3. Examples of bounded escaping wandering domains of each possible

type are provided in [BEF+22]; and of bounded oscillating ones in [ERS22]. This prompts

the question of whether there are examples of unbounded wandering domains with all dif-

ferent possible internal dynamics. We provide examples with all the diﬀerent behaviours

in terms of convergence to the boundary, showing that the function from Theorem 1.1

can be chosen to have wandering domains of any of the types arising from [BEF+22,

Theorem C].

Theorem 1.2. For each of the three possible types of simply connected wandering domains

in terms of convergence to the boundary, there exists a transcendental entire function with

an orbit of unbounded fast escaping wandering domains of that type.

UNBOUNDED WANDERING DOMAINS 3

Remark. In Observation 2.4 we explain how we can choose the above wandering domains

to be ‘contracting’ with respect to hyperbolic distances, and discuss the other two types

arising from [BEF+22, Theorem A].

The existence of unbounded wandering domains is related to a conjecture of Baker from

1981, [Bak81]. Part of this conjecture states that a transcendental entire function fof

order less than half cannot have unbounded Fatou components. Recall that the order of

an entire map fis deﬁned as ρ(f).

.= lim supr→∞

log log Mf(r)

log r. Zheng [Zhe00] showed that

such functions have no unbounded periodic or preperiodic Fatou components, and it is still

an open question if such a map can have an unbounded wandering domain. In addition,

the conjecture has been shown to hold in many instances where growth conditions are

imposed on the function, see e.g. [HM09,RS09,RS13,NRS18] and the survey [Hin08].

Absence of unbounded wandering domains has also been shown for further functions

of small order. Recently, Nicks, Rippon and Stallard [NRS18] showed that real entire

functions with only real zeros and of order less than one do not have orbits of unbounded

wandering domains. Note however that there are known examples of functions of order

one with orbits of unbounded wandering domains, as shown by Herman for the map

z7→ z−1 + e−z+ 2πi, [Her84]. This suggests the following question.

Question 1.3 ([HL19, Problem 2.93, p. 61], [NRS18, p. 101]).Let fbe a transcendental

entire function of order less than one. Can fhave (an orbit of) unbounded wandering

domains?

The following theorem answers the question by providing the ﬁrst examples of entire

functions of order less than 1 with unbounded wandering domains.

Theorem 1.4. For every ε∈(0,1/2], there exists an entire function of order 1/2 + ε

with an unbounded fast escaping wandering domain.

Remark. The functions constructed in Theorems 1.1 and 1.2 are of inﬁnite order of growth,

see (2.2) in Corollary 2.2. Therefore, new tools are required to prove Theorem 1.4. We note

that our construction ensures that the resulting function has one unbounded wandering

domain, U0. However, we were not able to show whether any element, Un, in its orbit is

bounded or unbounded; see Corollary 3.2 for details. It remains an open question whether

there exists an entire function of order less than 1 with an orbit of unbounded wandering

domains.

Notation. Given a set U⊂Cand δ > 0, we let

U−δ.

.={z∈U: dist(z, C\U)> δ}and U+δ.

.={z∈C: dist(z, U)< δ},

i.e., the set U−δis the subset of Uwhere all the points are separated from the boundary

of Uby at least δand the set U+δis a ‘fattening’ or ‘padding’ of the set Uby δ.

Given a sequence {an}we will use the notation an&0 to indicate that {an}is monotone

decreasing to 0 and the notation an% ∞ to indicate that {an}is monotone increasing

to ∞. We will denote by Sthe strip of width one centred about the real line, i.e.,

.={z∈C:|Im(z)|<1/2},

4 V. EVDORIDOU, A. GL ¨

UCKSAM, AND L. PARDO-SIM ´

and by dm Lebesgue’s measure on the complex plane. We denote by B(z, r) the disk

centred at zof radius r > 0 and by nB(z, r), n ≥2, the disk centred at zof radius n·r.

Given a set T⊂Cand a constant a∈C, we denote by T+athe translation of Tby a,

i.e., T+a.

.={z+a:z∈T}. The Euclidean distance is denoted by dist, the hyperbolic

distance in a simply connected domain Uis denoted by distUand the symbol 4denotes

the end of the proof of a claim within the proof of a theorem. By numerical constant we

mean a constant number that does not depend on any parameter.

Acknowledgements. We thank Dave Sixsmith for helpful comments and the referee for

many comments and suggestions which greatly improved the presentation of the paper.

This material is partially based upon work supported by the National Science Foundation

under Grant No. DMS-1928930 while the authors participated in programs hosted by the

Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2022

semester.

2. Orbits of unbounded wandering domains

Our main goal in this section is to prove the following ﬂexible construction theorem,

from which Theorems 1.1 and 1.2 will follow readily. Roughly speaking, it tells us that

given a collection of holomorphic maps, hk, deﬁned on the horizontal strip S, there exists

an entire function fthat approximates, for each k≥1, a translation of hkfrom S−εk+iτk

to S+iτk+1 for sequences εk&0 and τk% ∞, up to a prescribed error, δk. Moreover, f

will map a strip of points in between each copy of Sclose to the point iτ0; see Figure 1.

In particular, we will show in Corollary 2.2 that fhas an orbit, {Un}, of unbounded

wandering domains, asymptotically equal to horizontal strips and separated by preimages

of a basin of attraction containing iτ0, that are fast escaping whenever the translation

parameters, τk, diverge to inﬁnity suﬃciently fast.

Theorem 2.1. Given

(i) A sequence {εk} ⊂ 0,1

4with εk&0;

(ii) A sequence τk% ∞ with τ0≥3

2,τk+1 > τk+1+εk+εk+1 and such that

δk.

.=3

εk·τ2

< εk+1 →0; (2.1)

(iii) The map h0:S→Cdeﬁned as h0(z).

.=i(τ0−τ1)and a sequence {hk}k≥1of

holomorphic maps, hk:S→S, such that

|hk(z)|≤|z|nk+bk,

where {bk},{nk} ⊂ Nare non-decreasing sequences;

There exists a monotone increasing sequence {ak}, where akdepends only on {hj}k+1

j=0 ,

{εj}k+1

j=0 and {τj}k+1

j=0 , and an entire function f:C→Cwith the following properties.

(1) For every k≥0, for every z∈S−εk+iτk,

|f(z)−(hk(z−iτk) + iτk+1)|< δk.

UNBOUNDED WANDERING DOMAINS 5

(2) For every k≥0and z∈Wk.

.={w:τk+ 1/2 + εk<Im(w)< τk+1 −1/2−εk+1},

|f(z)−iτ0|< δk.

(3) If Im(z)≤τk+1 −3

2, then

|f(z)| ≤ 4 exp akexp π

εk|Re(z)|.

iτ2S−ε2+iτ2

iτ1S−ε1+iτ1

S−ε0+iτ0

h0(z−iτ0) + iτ1

h1(z−iτ1) + iτ2

iτ0

Figure 1. Schematic of the domains and functions in the statement of

Theorem 2.1 and Corollary 2.2. The entire function fwill map the domains

S−ε0+iτ0and Wn(in light gray) to a circle around iτ0, which belongs to an

attracting basin, A. The wandering domains {Un}n≥1lie in the complement

of these domains, and for each n,Uncontains S−εn+iτn, in dark gray. In

white, between S−εn+iτnand Wn, the strips where there is no control over

fother than an upper bound.

The following corollary shows that the function fconstructed in Theorem 2.1 has an

orbit of unbounded wandering domains, which converge to horizontal strips. Moreover,

the translations, τk, can be chosen to be large enough, in a precise way, so that the

wandering domains are fast escaping.

Corollary 2.2. Let hk:S→Sbe holomorphic maps satisfying that for every k,

hk(S−εk)⊂S−2εk+1 and let fbe the entire function resulting from Theorem 2.1. Then,

fhas an orbit of unbounded escaping wandering domains {Uk}k≥1such that for every

k≥1,

(1) S−εk+iτk⊆Uk⊆S+εk+iτk.

(2) If, in addition, τk→ ∞ fast enough, namely, if for all k≥1

τk+1 >4 exp ak−1exp π

εk−1τk−3

2+3

2,(2.2)

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UNBOUNDEDFASTESCAPINGWANDERINGDOMAINSVASILIKIEVDORIDOU,ADIGLUCKSAM,ANDLETICIAPARDO-SIMONAbstract.Weintroduceanewapproximationtechniqueintothecontextofcomplexdynamicsthatallowsustoconstructexamplesoftranscendentalentirefunctionswithunboundedwanderingdomains.Weprovideexamplesofentirefunctionswithano...

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