On the connectivity of the escaping set in the punctured plane

We consider the dynamics of transcendental self-maps of the punctured plane, C∗=C\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^*=\mathbb {C}{\setminus } \{0\}$$\end{document}. We prove that the escaping set I(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(f)$$\end{document} is either connected, or has infinitely many components. We also show that I(f)∪{0,∞}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(f)\cup \{0,\infty \}$$\end{document} is either connected, or has exactly two components, one containing 0 and the other ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}. This gives a trichotomy regarding the connectivity of the sets I(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(f)$$\end{document} and I(f)∪{0,∞}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(f)\cup \{0,\infty \}$$\end{document}, and we give examples of functions for which each case arises. Finally, whereas Baker domains of transcendental entire functions are simply connected, we show that Baker domains can be doubly connected in C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^*$$\end{document} by constructing the first such example. We also prove that if f has a doubly connected Baker domain, then its closure contains both 0 and ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}, and hence I(f)∪{0,∞}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(f)\cup \{0,\infty \}$$\end{document} is connected.


Introduction
Let S be the complex plane, ℂ , or the punctured plane, ℂ * ∶= ℂ⧵{0} , and suppose that f ∶ S → S is a holomorphic function such that Ĉ ⧵S consists of essential singularities of f, where Ĉ ∶= ℂ ∪ {∞} is the Riemann sphere. When S = ℂ , f is a transcendental entire function, and when S = ℂ * , we say that f is a transcendental self-map of ℂ * . This paper concerns the iteration of this second class of functions, first studied by Rådström [27]. We define the Fatou set of f by and we define the Julia set of f as its complement in S, that is, J(f ) ∶= S⧵F(f ) . We use the term Fatou component to refer to each connected component of F(f ) . For more background and definitions, we refer to [5].
For a transcendental entire function f, the escaping set of f is defined by Eremenko [12] was the first to study this set in full generality. He showed that I(f ) ≠ � , J(f ) = I(f ) , and that the components of I(f ) are all unbounded. He conjectured that, in fact, all the components of I(f ) are unbounded. Although significant progress has been made on this important conjecture, it remains open, and has motivated much research on transcendental dynamics in recent years. It is straightforward to see that Eremenko's conjecture holds whenever I(f ) is connected. Because of this property and the relation between I(f ) and J(f ) discussed above, it is natural to study the connectivity of this set. Rippon and Stallard [31, Corollary 5.1 (a)] (see also [33,Theorem 1.3]) showed that either I(f ) is connected, or has infinitely many components. There exist several examples of transcendental entire functions with a connected escaping set; for example, this is the case for the exponential function [29]. Furthermore, for many functions I(f ) is a spider's web, that is, a connected set that separates every point of ℂ from ∞ ; see, for example, [13]. Rippon and Stallard [31] also proved that I(f ) ∪ {∞} is a connected subset of Ĉ ; note that this does not rule out the possibility that I(f ) has a bounded component.
Now, suppose that f is a transcendental self-map of ℂ * . Many authors have studied the dynamics of these maps, and shown that there are many similarities with the dynamics of transcendental entire functions, though also striking differences. In line with these studies, our principal goal in this paper is to generalise the results mentioned above to the escaping set of f, which in this setting is defined as where (z, f ) ∶= ⋂ n∈ℕ {f k (z) ∶ k ⩾ n} , and the closure is taken in Ĉ . This set was studied extensively in [20] where, in analogy with Eremenko's results, it was shown that I(f ) ≠ � , J(f ) = I(f ) and all the components of I(f ) are unbounded in ℂ * ; in other words, their closure in Ĉ meets {0, ∞}.
Note that, unlike the escaping set of a transcendental entire function, the escaping set of a transcendental self-map of ℂ * can be partitioned in a natural way into uncountably many non-empty disjoint sets that are completely invariant; recall that a set X is completely invariant when z ∈ X if and only if f (z) ∈ X . Set ℕ 0 ∶= ℕ ∪ {0} . For every z ∈ I(f ) , we define the essential itinerary of z as the sequence e = (e n ) n∈ℕ 0 ∈ {0, ∞} ℕ 0 given by, for n ∈ ℕ 0 , F(f ) ∶= {z ∈ S ∶ {f n } n∈ℕ is a normal family in an open neighbourhood of z}, For each e ∈ {0, ∞} ℕ 0 , the set of escaping points whose essential itinerary is eventually a shift k (e) of e (here is the Bernoulli shift that removes the first symbol of a sequence and moves all the other symbols one position to the left) is We call I e (f ) the little escaping set with essential itinerary e, using the terminology from [22]. In the particular cases where e is the constant sequence 0 and ∞ , we denote the set I e (f ) by I 0 (f ) and I ∞ (f ) , respectively.
Martí-Pete [20] showed that for each e ∈ {0, ∞} ℕ 0 , we have I e (f ) ≠ � , J(f ) = I e (f ) and all components of I e (f ) are unbounded. Note that, although there are uncountably many non-empty disjoint subsets I e (f ) ⊆ I(f ) , several components of different sets I e (f ) may lie in the same component of I(f ) (this is the case, for example, when I(f ) is connected).
Our first result concerns the connectivity of these sets.
Remark Note that it is easy to deduce the same connectedness properties for the fast escaping set A(f ) and the little fast escaping sets A e (f ) , for e ∈ {0, ∞} ℕ 0 , instead of I(f ) and I e (f ) , respectively, where f is a transcendental self-map of ℂ * . However, the definitions of these sets are complicated, and so we refer to [20, Definition 1.2] for more details.
In Sect. 4, we give several examples to show that all three cases are attained, as well as to illustrate different properties of these sets; some of the examples have appeared before in the literature, but others are new. In [11], we proved that the function from [21, Example 3.3] has the property that its escaping set is a ℂ * -spider's web, that is, a connected set which separates every point of ℂ * from {0, ∞} , and hence is an example of type (I1) (see Example 1). When I(f ) is disconnected, the set I(f ) ∪ {0, ∞} can be connected or disconnected. We give a function f such that ℝ⧵{0} ⊆ I(f ) and iℝ⧵{0} ⊆ ℂ * ⧵I(f ) , and hence f is of type (I2) (see Example 2 and Fig. 1). On the other hand, to show that I(f ) ∪ {0, ∞} is disconnected, that is, f is a function of type (I3), it suffices to find a continuum in ℂ * ⧵I(f ) that separates 0 from ∞ . We discuss two different examples of situations in which this happens. First, observe that this is the case when f has a doubly connected Fatou component in ℂ * ⧵I(f ) ; the function in Example 3 has a doubly connected basin of attraction. Another situation in which I(f ) ∪ {0, ∞} is disconnected is when f has an invariant curve around the origin; the functions in Example 4 all satisfy that the unit circle is invariant. Finally, we give an example of a function f for which I ∞ (f ) is connected, but not a spider's web (see Example 5). We emphasise that although I(f ) and J(f ) satisfy similar properties, the connectivity of I(f ) is independent of that of J(f ) . Recall that if f is a transcendental entire function, then J(f ) is either connected or has uncountably many components [4] and J(f ) ∪ {∞} is connected if and only if f has no multiply connected Fatou components [19]. The function f (z) = sin z is an example for which J(f ) is connected [9] (and, in fact, a spider's web [26]) and I(f ) is disconnected as ℝ ⊆ ℂ⧵I(f ) . On the other hand, for Fatou's function f (z) = z + 1 + e −z we know that J(f ) is disconnected (it is an uncountable union of disjoint curves), but I(f ) is connected (in fact, it is a spider's web [13]).

Remarks
(1) Many authors [23,25,35] have studied the connectivity of other dynamically meaningful sets, such as the sets of bounded or unbounded orbits, for transcendental entire functions. It would be interesting to study the connectivity of these sets for transcendental self-maps of ℂ * . (2) In [22] it was shown that many properties of the dynamics of transcendental self-maps of ℂ * carry over to quasiregular maps of punctured Euclidean space. It is natural to ask if the connectivity results of this paper can also be transferred into this wider setting.
Let f be a transcendental entire function or a transcendental self-map of ℂ * . Suppose that U is a Fatou component of f, and let U n be the Fatou component containing f n (U) for n ∈ ℕ 0 . If U ⊆ U p for some minimal p ∈ ℕ , then we say that U is periodic of period p. If U is not periodic, but U k is periodic for some k ∈ ℕ , then we say that U is preperiodic. Otherwise we say that U is a wandering domain. If U is periodic and meets I(f ) , in which case U is contained in I(f ) , then we say that U is a Baker domain. In order to prove Theorem 1.2, we need a result concerning escaping points on the boundaries of Fatou components, which may be of independent interest. Observe first that Baker domains and escaping wandering domains are the only two types of Fatou components that lie in I(f ) . Recall that for entire functions, Rippon and Stallard [31, Theorem 1.1] proved that the boundaries of escaping wandering domains always contain escaping points; the problem of whether the boundaries of Baker domains always contain escaping points remains open (see [7,33]). Suppose that f is a transcendental self-map of ℂ * . We consider the following subset of I e (f ), which has the property that if a Fatou component U meets Ĩ e (f ) , then U ⊆ � I e (f ) (see Lemma 3.1). We call Ĩ e (f ) the immediate little escaping set with essential itinerary e, in analogy to the term immediate basin of attraction. Observe that, in general, these sets are not completely invariant. Moreover, I e (f ) =Ĩ e (f ) if and only if the sequence e only has one symbol. The result we need to prove Theorem 1.2 is the following.

Theorem 1.3 Let f be a transcendental self-map of ℂ * , and let U be a wandering domain
Moreover, the set U⧵Ĩ e (f ) has zero harmonic measure relative to U. In particular, the set U⧵I e (f ) has zero harmonic measure relative to U.
One of the striking differences between the iteration of transcendental entire functions and that of transcendental self-maps of ℂ * lies in the nature of their multiply connected Fatou components. Baker [2] proved that all Fatou components of transcendental self-maps of ℂ * are either simply or doubly connected, and that there is at most one doubly connected Fatou component. Baker and Domínguez [3] showed that if U is a doubly connected periodic Fatou component that is bounded away from 0 and ∞ , then U must be a Herman ring, that is, a doubly connected domain on which the function is conjugated to an irrational rotation. However, there is no such restriction if U is a doubly connected periodic Fatou component that is unbounded in ℂ * . The first example of a doubly connected Fatou component in ℂ * was given by Baker [2, Theorem 1.2], and was the basin of attraction of an attracting fixed point (see also Example 3).
For transcendental entire functions, Baker [1, Theorem 3.1] proved that Baker domains are all simply connected. We construct a transcendental self-map of ℂ * with a doubly connected Baker domain; we are not aware of any previous such example.

Theorem 1.4 There exists a transcendental self-map of ℂ * that has a doubly connected Baker domain.
Observe that every Baker domain U in ℂ * contains a simply connected absorbing set H, that is, f (H) ⊆ H and for every compact set K ⊆ U , there exists n ∈ ℕ 0 such that f n (K) ⊆ H (see Lemma 5.2).

3
Finally, we prove a connection between the fact that a transcendental self-map f of ℂ * has a doubly connected Baker domain and the connectivity of I(f ) ∪ {0, ∞} . Namely, if f has a doubly connected Baker domain, then f cannot be of type (I3).

Notation
We denote the open ball centred at a ∈ ℂ and of radius r > 0 by If X ⊆ ℂ * , we denote by X and X the closure of X in ℂ * and Ĉ , respectively. We always use X to refer to the boundary of X in ℂ * . Recall that we say that X is unbounded in ℂ * if X ∩ {0, ∞} ≠ �.

Connectivity of the escaping set
In this section we prove Theorem 1.1. We begin by giving a more general result, which is a version of [31, Theorem 5.2].

is not connected, then it has infinitely many components.
Remark Note that in ℂ * we only have two cases for the connectivity of the set E above, whereas for entire functions [31,Theorem 5.2] there is the possibility that E has two components, in which case one of the components must be a singleton consisting of the only possible exceptional point. Picard's theorem implies that holomorphic self-maps of ℂ * do not have any exceptional points.
The proof of Theorem 2.1 is based on the following key property of the Julia set, which is known as the blowing-up property (see [27,Theorem 4.1]).

Lemma 2.2 Let f be a transcendental self-map of
We now prove Theorem 2.1. We suppose that E ⊆ ℂ * is a completely invariant and disconnected set with the property that J(f ) = E ∩ J(f ) , and we need to show that E has infinitely many components.
Proof of Theorem 2.1 Suppose, by way of contradiction, that the set E is not connected but consists of finitely many components E 1 , The fact that E has finitely many components implies that there exists a positive number r sufficiently small that

Connectivity of the escaping set union zero and infinity
To match the notation used in [20], observe that, for e ∈ {0, ∞} ℕ 0 , the little escaping set with essential itinerary e, I e (f ) , can be written as the union where, for ∈ ℕ 0 and k ∈ ℕ 0 , Then, the set � I e (f ) ⊆ I e (f ) from the introduction, which consists of the points whose essential itinerary eventually coincides with e, can be written as .
Fatou component U satisfies that U ∩ I e (f ) ≠ � , then U ⊆ I e (f ) , but the following lemma gives more precise information. Proof Choose z ∈ U and let e ∈ {0, ∞} ℕ 0 be such that z ∈Ĩ e (f ) . Suppose to the contrary that U ∩ (ℂ * ⧵Ĩ e (f )) ≠ � . Then, we can find a point z � ∈ U ∩ Ĩ e (f ) . It is easy to see that the family of iterates of f is not equicontinuous on any neigbhourhood of z ′ , contradicting the Arzelà-Ascoli theorem; this proves the lemma. ◻ Next, we prove Theorem 1.3, which says that if f is a transcendental self-map of ℂ * and U is a wandering domain of f such that U ⊆ � I e (f ) , then U⧵Ĩ e (f ) has zero harmonic measure relative to U. To that end, we require the following lemma (see [24,Lemma 4.1]), which is a generalisation of [31, Theorem 1.1]. Here d(z, w) denotes the spherical distance between two points z, w ∈Ĉ . If G ⊆ ℂ * is a domain and E ⊆ G is a Borel set, then (z, E, G) denotes the harmonic measure of E relative to G at a point z ∈ G (see [18] for a precise definition). If (z, E, G) = 0 for some z ∈ G and hence all z ∈ G , then we say that E has harmonic measure zero relative to G. Lemma 3.2 Let (G n ) n∈ℕ 0 be a sequence of disjoint simply connected domains in Ĉ . Suppose that, for each n ∈ ℕ , g n ∶ G n−1 → G n is analytic in G n−1 , continuous in G n−1 , and satisfies g n ( G n−1 ) ⊆ G n . Set Suppose that there exist ∈Ĉ , ∈ (0, 1) and z 0 ∈ G 0 such that Suppose finally that c > 1 , and let Then H has harmonic measure zero relative to G 0 .
We now prove Theorem 1.3.

Proof of Theorem 1.3
Suppose that f is a transcendental self-map of ℂ * with a wandering domain U ⊆ � I e (f ) for some e ∈ {0, ∞} ℕ 0 . It suffices to prove that U⧵Ĩ e (f ) has zero harmonic measure relative to U. For n ∈ ℕ 0 , let U n be the Fatou component containing f n (U).
By a result of Baker [2], there is at most one value N ∈ ℕ 0 such that U N is doubly connected, and U n is simply connected for all n ≠ N . It follows from [28,Theorem 4.3.8] that if n ∈ ℕ and E ⊆ U n has zero harmonic measure relative to U n , then f −n (E) ∩ U has zero harmonic measure relative to U, regardless of whether U is simply or doubly connected. We can assume, therefore, that U n is simply connected for n ∈ ℕ 0 .
Suppose that there is a sequence (n k ) k∈ℕ ⊆ ℕ such that e n k = ∞ for all k ∈ ℕ ; otherwise the sequence e is eventually the constant sequence 0 and the argument below can be applied with = 0 . We apply Lemma 3.2 with = ∞ , G 0 = U and, for k ∈ ℕ , G k = U n k and g k = f n k −n k−1 , where n 0 = 0 , so that h k = f n k . We obtain a subset H ⊆ U , of harmonic h n ∶= g n • ⋯ •g 2 •g 1 , for n ∈ ℕ. H ∶= {z ∈ G 0 ∶ d(h n (z), ) ⩾ c for infinitely many values of n ∈ ℕ}. measure zero relative to U, such that U⧵ � I e (f ) ⊆ H ; recall that if the essential itineraries of two points disagree on an infinite sequence, then the two points cannot lie in the same set Ĩ e (f ) for e ∈ {0, ∞} ℕ 0 . In particular, the set U⧵Ĩ e (f ) , and hence also the subset U⧵I e (f ) , has zero harmonic measure relative to U as required. implies that the set V intersects I(f ) , but Theorem 1.3 gives more precise information, namely that V contains points z such that |Ref n (z)| → +∞ as n → ∞ ; see the following corollary.   G ∩ I e (f ) ≠ � for all e ∈ {0, ∞} ℕ 0 , as required. The last part of Lemma 3.5 holds because there are uncountably many disjoint sets A e (f ) for e ∈ {0, ∞} ℕ 0 . Note that this follows from Lemma 3.4(1) and the fact that the set of sequences e ∈ {0, ∞} ℕ 0 for which the sets I e (f ) are disjoint is uncountable (see [20,Remark 3

.1(2)]). ◻
We now give two corollaries of Theorem 1.3, and prove Theorem 1.2. These three results are the analogues in ℂ * of [31,Theorem 4.1]. However, note that they concern I e (f ) instead of the whole of I(f ) , so in some sense they are more precise, as a component of I(f ) may comprise components of several I e (f ).

Corollary 3.7 Let f be a transcendental self-map of
Proof Suppose, by way of contradiction, that G ⊆ ℂ * is a domain that is bounded in ℂ * , In the first example, we give a transcendental self-map of ℂ * of type (I1), that is, a function f for which I(f ) is connected, and hence also I(f ) ∪ {0, ∞} is connected.

Example 1 In [21, Example 3.3]
, it was proved that for sufficiently large values of > 0 , the function has an invariant Baker domain in which points escape to ∞ . This was the first explicit example of a transcendental self-map of ℂ * with a Baker domain. Later, in [11, Theorem 1.5], we showed that for this family of functions, provided that > 0 is large enough, I(f ) has the structure of a ℂ * -spider's web; recall that a connected set X ⊆ ℂ * is called a ℂ * -spider's web if it separates every point of the punctured plane from {0, ∞} . △ The second example we give is a transcendental self-map f of ℂ * that satisfies property (I2), that is, with a disconnected I(f ) , but I(f ) ∪ {0, ∞} being connected.

Example 2 Consider the function
Note that if z ∈ ℝ⧵{0} , then the term in the main exponential is positive and so the main exponential term is greater than one. Hence ℝ⧵{0} ⊆ I(f ) and, by Theorem 1.2, the set I(f ) ∪ {0, ∞} is connected (see Fig. 1).
It remains to show that I(f ) is not connected for this map. We consider the dynamics on the imaginary axis (minus the origin). Observe that so we need to consider the action of f on the real line (see Fig. 2).
Note that f has a unique attracting fixed point on the positive real line at a point y 0 ≈ 1.087 , and it is bounded on the real line. We can deduce that f has two attracting fixed points at ±y 0 i , and that {0 + iy ∶ y > 0} and {0 + iy ∶ y < 0} each lie in an attracting Fatou component. Hence the imaginary axis separates I(f ) , and the result follows. △ f (z) = z n exp(g(z) + h(1∕z)), Next, we give examples of transcendental self-maps f of ℂ * satisfying property (I3). Recall that, for such maps, I(f ) ∪ {0, ∞} is disconnected, and hence also I(f ) is disconnected. We give two different situations in which this happens.
First, observe that if f has a doubly connected Fatou component in ℂ * ⧵I(f ) , then f is of type (I3). One class of functions with a doubly connected Fatou component is that of the so-called functions of disjoint type. We say that a transcendental self-map f of ℂ * is of disjoint type if F(f ) is connected and consists of the immediate basin of attraction of a fixed point, which is doubly connected in ℂ * (see [15,Definition 3.10]). In the next example, we give a disjoint-type transcendental self-map of ℂ * , which satisfies the additional property that each component of I(f ) is contained in a single set I e (f ) for some e ∈ {0, ∞} ℕ 0 .

Example 3 The function defined by
is a transcendental self-map of ℂ * of disjoint type (see [15,Example 3.12]) and hence satisfies (I3). Indeed, it can be shown that there exists a round annulus A separating 0 from ∞ that maps compactly inside itself, and this implies that f has an attracting fixed point ∈ A (see Fig. 3).
Since f has finite order as a transcendental self-map of ℂ * (see [ Next, we give a different situation in which I(f ) ∪ {0, ∞} is disconnected, and hence f is of type (I3). This is the case when f has a forward invariant closed curve around the origin. Note that a transcendental self-map of ℂ * can only have one such curve. In the next example we study a well-known family with this property.

Example 4
The complex Arnol'd standard family is given by and the iteration of this family of transcendental self-maps of ℂ * was originally studied by Fagella [14]. Since f , ( ) ⊆ , we have that ∩ I(f , ) = � . Thus, for any 0 ⩽ < 2 and ⩾ 0 , the function f , satisfies property (I3) (see Fig. 4). Note that for some parameters J(f , ) = ℂ * , but otherwise f , can have a Herman ring, or other types of Fatou components. △ f , (z) ∶= ze i e (z−1∕z)∕2 , for 0 ⩽ < 2 and ⩾ 0, It is natural to ask if the set I e (f ) can be connected for some e ∈ {0, ∞} ℕ 0 . Our goal in the next example is to answer this question in the affirmative by showing that there is a transcendental self-map f of ℂ * such that I ∞ (f ) is connected. We will also show that for this function, I(f ) is connected, and hence f is of type (I1). Observe that the set I e (f ) is never a ℂ * -spider's web for any e ∈ {0, ∞} ℕ 0 . Indeed, there are infinitely many sequences e � ∈ {0, ∞} ℕ 0 ⧵{e} for which A e � (f ) ∩ I e (f ) = � (as I e � (f ) ∩ I e (f ) ≠ � if and only if m (e � ) = n (e) for some m, n ∈ ℕ ) and, it follows from Lemma 3.4(3) that all the components of A e � (f ) are unbounded in ℂ * .
We first prove the following general proposition that will be used in the example; this is based on the proof in [32] that the escaping set of the map z ↦ cosh 2 z is connected. Recall that if f is a transcendental entire function or a transcendental self-map of ℂ * , we say that a set X is backward invariant under f if f −1 (X) ⊆ X.  backward invariant, this is in fact a subset of X. By repeated application of this argument, we deduce that T = ⋃ n⩾0 f −n (E) is a connected subset of X. Since J(f ) is the whole plane, and the backward orbit of any non-exceptional point is dense in J(f ) , it follows that T = ℂ . Hence T ⊆ X ⊆ T , and so X is connected. ◻ We now give the example.
Example 5 Set f (z) ∶= cosh z , which is a transcendental entire function that is a lift of the transcendental self-map of ℂ * given by Observe  + 1∕z)).
Suppose that U is a Baker domain of a transcendental meromorphic function or transcendental self-map f of ℂ * . We say that a set H ⊆ U is an absorbing set if f (H) ⊆ H and for every compact set K ⊆ U , there exists n ∈ ℕ such that f n (K) ∈ H . The existence of simply connected absorbing sets for Baker domains of transcendental entire functions was established by Cowen [8]. In [7], the authors study when a Baker domain of a transcendental meromorphic functions admits a simply connected absorbing domain. Even though Baker domains can be doubly connected in ℂ * as we have seen in Theorem 1.4, it is easy to show that they always contain a simply connected absorbing domain. is an absorbing set for U. Indeed, for every z ∈ U , there exists a point w ∈ exp −1 (z) that lies in V and if n ∈ ℕ is such that f n (w) ∈ H � , then f n (z) ∈ H , as we wanted to show. ◻ Observe that this means that we can transfer the classification of Baker domains for transcendental entire functions (see, for example, [30,Section 5]) to transcendental selfmaps of ℂ * by using the fact that f and any of its lifts f are conjugated on the absorbing set.
In the case that f has a doubly connected Baker domain, we can deduce some additional properties of f. Recall that given a transcendental self-map f of ℂ * , we define the index of f as the index (or winding number) of f ( ) with respect to 0, where ⊆ ℂ * is any positively oriented simple closed curve around 0. This quantity is a topological invariant of f. We prove that ind(f ) = 0 when f has a doubly connected Baker domain.  where H n ⊆ H n+1 for n ∈ ℕ . Since U is doubly connected, there exists n 0 ∈ ℕ such that H n is doubly connected and n 0 is minimal with this property; note that the union of an increasing sequence of open simply connected sets is simply connected. We claim that the closure Ĥ n 0 +1 necessarily contains both 0 and ∞ . Indeed, let ⊆ H n 0 be a curve of index 1 around 0. Each of the components of the preimage of in H n 0 +1 is a cuve ′ that is unbounded in ℂ * . Observe that each of the complementary components of H n 0 in U must contain a component of the preimage of , and so the closure of U in Ĉ contains {0, ∞} . This proves the claim and the theorem.