The escaping set of transcendental selfmaps of the punctured plane
Abstract.
We study the different rates of escape of points under iteration by holomorphic selfmaps of for which both and are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either 0, or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points each of which has as its boundary.
1. Introduction
Complex dynamics concerns the iteration of a holomorphic function on a Riemann surface . Given a point , we consider the sequence given by its iterates and study the possible behaviours as tends to infinity. We partition into the Fatou set, or set of stable points,
and the Julia set , consisting of chaotic points. If is holomorphic and consists of essential singularities, then there are three interesting cases:

and is a rational map;

and is a transcendental entire function;

and both and are essential singularities.
We study this third class of maps, which we call transcendental selfmaps of . Note that if has three or more omitted points, then Montel’s theorem tells us that is constant. In particular, has no omitted values. A basic reference on iteration theory in one complex variable is [milnor06]. See [bergweiler93] for a survey on transcendental entire and meromorphic functions.
The iteration of transcendental (entire) functions dates back to the times of Fatou [fatou26]. However, we have to wait until 1953 to find a paper about the iteration of holomorphic selfmaps of [radstrom53]. Bhattacharyya in his PhD thesis [bhattacharyya] under the supervision of Prof. Noel Baker showed that such maps are all of the form
(1.1) 
where and are nonconstant entire functions. Since then many other people have studied them; for example, see [baker87], [kotus87], [makienko87], [keen86], [liping91], [mukhamedshin91], [hinkkanen94], [bergweiler95] and [bakerdominguez98]. Such maps arise in a natural way when you complexify circle maps. For instance, the socalled Arnol’d standard family of perturbations of rigid rotations
has as its complexification [fagella99].
In this paper we study the escaping set of transcendental selfmaps of . For an entire function , the escaping set of is defined by
If is a polynomial, then infinity is an attracting fixed point and the escaping set consists of the basin of attraction of infinity, , which is part of the Fatou set and is connected. Moreover, , so the escaping set has a close relationship to the Julia set.
For transcendental entire functions, the escaping set also plays an important role. It was first studied by Eremenko [eremenko89] who used WimanValiron theory to show that, for a transcendental entire function ,

;

;

the components of are unbounded.
Furthermore, in [eremenkolyubich92] the socalled EremenkoLyubich class
was introduced (the set consists of the critical values and asymptotic values of ) and the authors showed that

if , then .
In his article [eremenko89] Eremenko conjectured that all the components of the escaping set are also unbounded. In 2011 it was proved that this is true for functions of finite order in class [rrrs11], but for general transcendental entire functions this remains an open question.
Key progress on Eremenko’s conjecture was obtained by studying the fast escaping set defined by
where and is chosen to be sufficiently large so that as . Here and throughout .
The set , which consists of the points that escape about as fast as possible, was introduced by Bergweiler and Hinkkanen, and it shares some properties with , for example, and , see [bergweilerhinkkanen99] and [ripponstallard05]. But it also has some much nicer properties. Rippon and Stallard showed that all the components of are unbounded, and hence has at least one unbounded component. The paper [ripponstallard12] gives a compilation of results about .
For transcendental selfmaps of we define the escaping set by
where . The set contains points that escape to as well as to ,
and also contains points that escape from by jumping infinitely many times between a neighbourhood of and a neighbourhood of . The sets and were studied by Fang [liping98] who proved that
by using WimanValiron theory in the way that Eremenko did for the entire case, but the full set of escaping points has not previously been studied.
To classify the various types of escaping orbits we introduce the followingconcept.
Definition 1.1 (Essential itinerary).
Let be a transcendental selfmap of . We define the essential itinerary of a point to be the symbol sequence such that
for all .
For each , the set of escaping points whose essential itinerary is eventually a shift of is
We also have a notion of fast escaping points related to an essential itinerary , defined using the iterates of the maximum and minimum modulus functions
Definition 1.2 (Fast escaping set).
Let be a transcendental selfmap of . Let and let be sufficiently large (or small) so that the sequence defined for by

, if ,

, if ,
accumulates to . We say that a point is fast escaping if there are such that

, if ,

, if ,
for all . We denote the set of all fast escaping points by and the set of fast escaping points with essential itinerary by .
Observe that if is of the form (1.1), then the behaviour of in a neighbourhood of depends mainly on that of the entire function while the behaviour near depends mainly on that of .
We begin by proving an analogue of property (I1), namely that and indeed are nonempty for any essential itinerary . We follow the approach of Rippon and Stallard in [ripponstallard13] where they proved the existence of points escaping to infinity at different rates by constructing points with different annular itineraries.
Theorem 1.1.
Let be a transcendental selfmap of . For each , and hence .
Our notation for annular itineraries is as follows. Let and be respectively large enough and small enough such that, for all , and, for all , we have . Then define
and the following sequences of annuli:
Each point has an associated annular itinerary with respect to the partition such that for all . We prove a covering result (see Theorem 4.2) which allows us to construct orbits with certain annular itineraries including the ones listed in Theorem 1.2 below.
Remark 1.1.
In this article we deal with two kinds of itineraries for escaping points that should not be confused: essential itineraries that describe how an escaping point accumulates to the two essential singularities, and annular itineraries that depend on the partition . For large values of , and correspond respectively to negative and positive terms in the annular itinerary.
Theorem 1.2.
Let be a transcendental selfmap of . Given an annular partition defined as above with sufficiently large, we can construct points with the following itineraries:

fast escaping itineraries;

periodic itineraries;

bounded itineraries (uncountably many);

unbounded nonescaping itineraries (uncountably many);

arbitrarily slowly escaping itineraries.
Note that our proof uses a different annular covering lemma to those used in [ripponstallard13] and, in this setting we are able to avoid the exceptional sets which feature in [ripponstallard13, Theorem 1.1 and Theorem 1.2].
We now state a result in the spirit of property (I2) but for any essential itinerary . For the special cases of and this is due to Fang and it also follows from the results in [bakerdominguezherring01].
Theorem 1.3.
Let be a transcendental selfmap of . For each , . Also .
Since there are uncountably many different essential itineraries, in particular this means that there is an uncountable collection of disjoint sets, each of which has the Julia set as its boundary.
We can also prove the analogue of property (I3) for any essential itinerary. When we say that a set is unbounded in we mean that .
Theorem 1.4.
Let be a transcendental selfmap of . For each , the connected components of are unbounded, and hence the connected components of are unbounded.
Finally we show that, as for transcendental entire functions, the components of are all unbounded.
Theorem 1.5.
Let be a transcendental selfmap of . For each , the connected components of are unbounded, and hence the connected components of are unbounded.
In [fagellamartipete] we study the escaping set of holomorphic selfmaps of of bounded type,
In that paper, we prove the analogue of property (I4): if then . Following the ideas of [rrrs11], we show that for functions in class of finite order every escaping point can be connected to either or by a curve of points escaping uniformly.
Structure of the paper. In Section 2 we prove the basic properties of and that we are going to need later. The discussion about the notions of essential itinerary and the fast escaping set is in Section 3. Section 4 is devoted to the construction of the annular itineraries and the proof of Theorem 1.2. The main result in this section, Theorem 4.2, in fact allows you to construct many more types of orbits than the ones listed in the statement of Theorem 1.2. Theorem 1.1 is proved in Section LABEL:sec:fastescapingset and we prove Theorems 1.3, 1.4 and 1.5 in Section LABEL:sec:therest. In doing so we also show that if a Fatou component intersects the fast escaping set then (see Theorem LABEL:thm:fastescapingFatoucomponents).
Notation. In this paper , denotes the open disc of radius centered at , and if ,
Acknowledgments. The author would like to thank his supervisors Phil Rippon and Gwyneth Stallard for all their support and patient guidance in the preparation of this article, as well as Núria Fagella for useful discussions and suggesting such an interesting topic for my Ph.D. thesis.
2. Properties of and
Before proving the annular covering results we need some basic properties of the maximum and minimum modulus functions. Note that we will not usually make explicit the dependence on and we will just write and . As a consequence of the maximum modulus principle, both and are unimodal functions. In the following lemma we summarise their main properties. Throughout this section we will only prove the statements for when , and the other three statements can be deduced from these by using the facts that if then
Lemma 2.1.
Let be a transcendental selfmap of . The functions and satisfy the following properties:

, as , and
, as ; 
and are convex functions of ;

such that
and such that

for
Proof.

This property follows from the analogous result for transcendental entire functions using the fact that , where and are nonconstant entire functions (see (1.1)), so
where .

This means that is a convex function of and the property is usually referred to as the Hadamard three circles theorem, see [ahlfors53]. Observe that in the hypothesis of that theorem you only need that the function is analytic in an annulus and it therefore applies to holomorphic selfmaps of .

See [ripponstallard09, Lemma 2.2] or [bergweilerripponstallard13, Theorem 2.2] for the analogous result for transcendental entire functions. We reproduce the proof here for completeness.
Let . By property (i), as , so we can take large enough that
Let denote the right derivative of . Then, by property (ii) and the previous inequality,
Hence is an increasing function for . Thus, if , then
for . Taking exponentials on both sides we get the result, with .

For every value of we can write , where
By property (iii), for large enough,
and then, using property (i),
so
∎
The following result compares the iterates of and with those of their ‘relaxed’ versions and , where . The analogous property for entire functions was used by Rippon and Stallard in [ripponstallard12, Theorem 2.9].
Lemma 2.2.
Let be a transcendental selfmap of , and let and , where . Then there exists such that, for ,
and, for ,
Proof.
Let be large enough that for all . By property (iv) in Lemma 2.1, with , there is such that,
and therefore
Hence, , for all and . If is the constant required for the corresponding inequality with and , then we define. If is the constant such that the second pair of inequalities hold for , then we put . ∎
Finally let us prove a property of and that will be used later in the construction of the annular itineraries.
Lemma 2.3.
Let be a transcendental selfmap of , and let and , where . Then there exists such that, for ,
and, for ,
Proof.
Consider and let be the constant defined in Lemma 2.2. Then
for all and . Now let be large enough that for all . Then, applying to both sides of , we get
for . Hence,
for all and . If is the constant required for the corresponding inequality with and , then the required result holds with. ∎
3. The escaping and fast escaping sets
In this section we discuss some basic properties of the escaping and fast escaping sets of transcendental selfmaps of . Recall that in Definition 1.1 we defined the essential itinerary of an escaping point to be the symbol sequence such that
and denotes the set of points whose essential itinerary is eventually a shift of , that is,
Remark 3.1.

Observe that we used the unit circle to define the boundaries of neighbourhoods of zero and infinity but we could have used any circle with , because the orbits of escaping points are eventually as close as we want to the essential singularities.

These sets are not always disjoint. In fact, if and only if for some , where denotes the Bernoulli shift map. In this case we say that is equivalent to and write . However, it is easy to see that there are uncountably many nonequivalent essential itineraries.
Recall that in Definition 1.2 we said that a point is fast escaping if there are such that, for all ,
• , if ,
• , if ,
(3.1)
where is a sequence of positive numbers defined using the iterates of and according to the essential itinerary . We now define
which is a closed set. Note that , and since , the set
is a nested union of closed sets. Finally stands for all the fast escaping points regardless of their essential itinerary.
The following two lemmas show that, with the definition above, the set has appropriate properties for the fast escaping set.
Lemma 3.1.
Let be a transcendental selfmap of , and let . There is large enough such that if is as in Definition 1.2 and or , then as . Hence, .
Proof.
Since and grow faster than any power of (see Lemma 2.1 (i)),we can take large enough that and if ,and and if . Therefore if , then as , as required. ∎
Lemma 3.2.
Let be a transcendental selfmap of . For each , is completely invariant and independent of , where satisfies the assumptions in Definition 1.2. Hence, is completely invariant and independent of .
Proof.
The set is completely invariant by constuction, because if , then
We give the details that is independent of for the case where there exists a sequence such that as . The argument when as is similar.
Suppose that and let and be the sequences given by Definition 1.2 starting with and respectively. Then as . Since and are unimodal functions, for large enough values of they are respectively increasing and decreasing (and the opposite when is small enough). Therefore, and hence
In the other direction, we use the fact that we have assumed that there is a sequence such that as . Let be such that . Then
where is the symbol sequence preceded by the string . Since , we have
and therefore is independent of the value of . ∎
We will continue studying the dynamical and topological properties of in Section LABEL:sec:fastescapingset.
4. Annular itineraries
In this section we study annular itineraries for our class of functions. By Lemma 2.1, there exist respectively large and small enough such that and as . We define and
so that is a partition of . Each point has an associated annular itinerary such that . Note that this sequence depends on the values and used to define the partition. By construction, it follows from the maximum modulus principle that
To create escaping orbits with certain types of annular itineraries we will use the following version of a wellknown result.
Lemma 4.1.
Let , , be compact sets in and be a continuous function such that
Then such that , for .
In our construction, the compact sets will be compact annuli with some covering properties.
Theorem 4.2.
Let be a transcendental selfmap of . If is the set of annuli defined above, then there exists a sequence of closed annuli , , with the following covering properties:

if , such that for ,

if , such that for ,
and when and have the same sign.
We compare Theorem 4.2 with the corresponding result for entire functions [ripponstallard13, Theorem 1.1]. In that setting there is a subsequence such that for with at most one exception while in our case all cover the other with if . Also the proof in [ripponstallard13] is significantly more involved than ours due to the possible presence of zeros of the function and multiplyconnected Fatou components, and it requires the use of several new covering lemmas.
In order to prove Theorem 4.2 we use the following recent covering result due to Bergweiler, Rippon and Stallard, see [bergweilerripponstallard13, Theorem 3.3]. Here stands for the hyperbolic distance between and relative to , where is a hyperbolic domain, that is, has at least two boundary points.
Lemma 4.3 (Bergweiler, Rippon and Stallard 2013).
There exists an absolute constant such that if is analytic, where , then for all such that
we have
Now we prove Theorem 4.2.
Proof of Theorem 4.2.
If , , and , then the hyperbolic length of with respect to is
(see [beardonminda07, Example 12.1]). Since the hyperbolic length is invariant under conformal transformations, the hyperbolic length of the circle with respect to is also . We choose to be sufficiently small that
where is the absolute constant of Lemma 4.3.
Let and let be the constant in Lemma 2.3. Then we claim that there exists such that if , then
(4.1) 
for all . Indeed Lemma 2.3 ensures that the first inequality is satisfied. The two middle inequalities are clear because , and the last one is due to the fact that is increasing for large values of and .
Let and, for , define
Take such that and
This is possible by (4.1). Our choice of ensures that and, for large enough, the condition is trivially satisfied.
Finally observe that if is large enough we can make sure that for every , . Then Lemma 4.3 tells us that