Multiply connected wandering domains of meromorphic functions: internal dynamics and connectivity

We discuss how the nine-way classification scheme devised by Benini et al. for the dynamics of simply connected wandering domains of entire functions, based on the long-term behaviour of the hyperbolic distance between iterates of pairs of points and also the distance between orbits and the domains' boundaries, carries over to the general case of multiply connected wandering domains of meromorphic functions. Most strikingly, we see that not all pairs of points in such a wandering domain behave in the same way relative to the hyperbolic distance, and that the connectivity of the wandering domain greatly influences its possible internal dynamics. After illustrating our results with the well-studied case of Baker wandering domains, we further illustrate the diversity of multiply connected wandering domains in general by constructing a meromorphic function with a wandering domain without eventual connectivity. Finally, we show that an analogue of the"convergence to the boundary"classification of Benini et al. does hold in general, and add new information about how this convergence takes place.

As concerns Theorem A, we see that things are not the same for multiply connected wandering domains: it is not true that all points in a wandering domain behave in the same way relative to , and in fact even 'commonplace' examples can combine different long-term behaviours of the hyperbolic metric. That is not to say, however, that we are lost at sea; even these 'mixed-type' wandering domains can still exhibit interesting structures. We postpone a discussion of what this means until after Theorem 1.1; here, we make the following distinction.
Definition. Let be a multiply connected wandering domain of a transcendental meromorphic function . We say that is trimodal (respectively, bimodal) if it exhibits all (respectively, two out of three) possible behaviours described in Theorem A.
Another aspect of multiply connected wandering domains that helps us in classifying their internal dynamics is their connectivity ( ), defined as the number of connected components of ℂ ⧵ (which, of course, was a non-issue in the simply connected case). Recall that, as defined by Kisaka and Shishikura [19], the eventual connectivity of a wandering domain (if it exists) is the number ∈ ℕ ∪ {+∞} such that ( ) = for all large . Note that it often does exist -Kisaka and Shishikura also showed that multiply connected wandering domains of entire functions always have an eventual connectivity, and it is either two or infinity. Our first result (proved in Section 2) shows that a wandering domain's geometry and eventual connectivity severely restrict its possible internal dynamics. (i) if ⩾ 3, is eventually isometric; (ii) if = 2 and deg | is finite for all large , either (a) mod is constant † and is eventually isometric, or (b) mod → +∞ and is trimodal and admits contracting and eventually isometric transversal laminations.
The laminations in the theorem above encode the aforementioned 'structure' in mixed-type domains. A precise definition is the following.
Definition. Let be a wandering domain of the transcendental meromorphic function .
• We say that admits a contracting lamination if there exists a lamination  of such that ( ( ), ( )) → 0 for all and on the same leaf of . † For a definition of the modulus of , see Subsection 2.1.
• We say that admits an eventually isometric lamination if there exists a lamination  of such that, for all large , ( ( ), ( )) = ( , ) > 0 for every and on the same leaf of .
We will see that, if admits both a contracting and an eventually isometric lamination, then points , ∈ that are not on the same leaf for either lamination behave 'semi-contractingly'that is, ( ( ), ( )) ↘ ( , ) > 0 as → +∞.
In other words, a wandering domain admitting both laminations is automatically trimodal (see Section 2 and Figure 1). Theorem 1.1 can be thought of as a 'silhouette theorem': just by knowing the topology of the iterates of a wandering domain, we can (in some cases) predict their internal dynamics. The hypothesis of finite eventual connectivity is necessary, since infinite connectivity offers, in general, sufficient flexibility for many kinds of behaviours (compare, for instance, [21,Example 1] and [17,Theorem (iii)]). However, in a well-studied case, both finite and infinite connectivity are associated to particular internal dynamics.
To understand what this particular case is, we recall the definition of a Baker wandering domain. A multiply connected wandering domain is said to be a Baker wandering domain if, for all large , surrounds the origin and → ∞ as → +∞ (see [22] for a discussion of this and other closely related types of wandering domains). If the function in question is entire, every multiply connected wandering domain is of this kind, and this remains true if we allow the function to have finitely many poles (see [21]). An even larger class of functions with Baker wandering domains with similar properties is the class of transcendental meromorphic functions with a direct tract (see Section 3 for a definition). As we will see in Section 3, all Baker wandering domains in this larger class have asymptotic behaviour similar to that of Baker wandering domains of entire functions, and -as our next 'silhouette theorem' shows -have very rigid internal dynamics. Since all entire functions have a direct tract, Theorem 1.2(ii) can be seen as a recontextualisation of Bergweiler, Rippon, and Stallard's results in [8]. It does not offer an alternative way of proving them (quite the opposite: the proof of Theorem 1.2 relies explicitly on their tools and findings), but rather bridges the gap between the ideas of [8] and the hyperbolic-metric approach of [6] and this paper. Now, an interesting question regarding Theorem 1.1 is whether the hypothesis that has an eventual connectivity is necessary; every previously known example of a multiply connected wan-dering domain does. However, it turns out that this is not always the case: In Section 4, we will invoke Arakelyan's theorem to obtain a sequence of meromorphic functions approximating handpicked functions, and will use it to construct the following example. Theorem 1.3. There exists a transcendental meromorphic function g with a wandering domain such that: (i) each 4 , ⩾ 0, is unbounded and simply connected; (ii) each 4 +1 , ⩾ 0, is bounded and doubly connected; (iii) each 4 +2 , ⩾ 0, is bounded and simply connected; and (iv) each 4 +3 , ⩾ 0, is unbounded and simply connected.
After such strange and wild behaviour from multiply connected wandering domains, it is encouraging to know that not everything about Benini et al.'s classification scheme is overturned. As our final theorem (proved in Section 5) shows, it remains (mostly) true that all orbits behave the same way regarding convergence to the boundary. However, we need to restrict our attention to a particular part of the boundary -the 'outer' boundary, defined for our purposes as the boundary of˜, the topological convex hull of , which is the union of and its bounded complementary components. Note that˜is not necessarily connected or bounded; take, for example, the outer boundary of { ∶ |ℑ | < 1}.  The final assertion saying that all orbits converge to 'the same parts' of the boundary (when they do so at all) is new even for simply connected wandering domains, but still uses the original techniques of [6].

THE HYPERBOLIC METRIC AND MULTIPLY CONNECTED WANDERING DOMAINS
This section is devoted to the proof of Theorem 1.1. It is divided in two parts: first, we study a 'toy' model of composing power maps between annuli; then, we justify the attention given to such a simple model by showing that it is in many ways equivalent to wandering domains of eventual connectivity two (the proof of Theorem 1.1(i) is simple, and we do not dwell extensively on it).
Before that, however, we must clarify what we mean by a lamination, since this concept is central to transferring our annulus-based knowledge to a general setting. We would like to point out that different texts use slightly different definitions; ours is in the spirit of [18] and [20], and is tailored to the kinds of manifolds we will meet here.
Definition. Let be a Riemann surface, and let ⊆ be a subset such that ⧵ is at most countable. A lamination of is a partition { } ∈ of into injectively immersed real submanifolds of (real) dimension one such that: • ∩ = ∅ whenever ≠ . • For every ∈ , there exists a neighbourhood of and a conformal isomorphism ℎ ∶ → ℎ( ) ⊂ ℂ such that ℎ( ∩ ) is either empty or of the form { ∈ ℎ( ) ∶ ℑ = } for some constant = ( ) (that is, is 'straightened' onto a line segment, and different indices lead to different parallel line segments).
The submanifolds { } ∈ are called the leaves of the lamination. If = , the lamination is called a foliation of .

The annulus model
Now, let us consider the composition of power mappings between annuli; first, we must understand our domain. For any > 1, we define the modulus of ( ) is given by † mod ( ) ∶= log −1 = 2 log .
It is well known that every doubly connected domain onĈ is conformally isomorphic to either ℂ * , * , or ( ) for some > 1, that these model spaces are all incompatible with each other, and that mod ( ) is a conformal invariant defining equivalence classes of doubly connected domains with non-degenerate complementary components (see, for instance, [1, Section 6.5]). With that in mind, let us take a closer look at ( ); particularly important subsets are the circles ∶= { ∈ ( ) ∶ | | = }, ∈ (1∕ , ), and the ray segments In the following two lemmas, we gather some facts about the hyperbolic metric in ( ); these facts are either 'clear' from an explicit universal covering of the annulus, or can be found in [4], [12], or [10, Chapter 1].

Lemma 2.1. For any > 1,
(i) For any ∈ [0, 2 ), is a hyperbolic geodesic of ( ). Geodesics arcs in are distanceminimising for any two points on the same , and are also unique in their homotopy class. † We are using Beardon and Minda's definition [4]; other authors normalise it by a factor of 2 .   (2 ), ∈ [0, 1], of 1 traversed ∈ ℤ ⧵ {0} times is the only closed geodesic in its free homotopy class in ( ). Its hyperbolic length is given by If is the winding number of • relative to zero, then with equality if and only if = | | and ( ) = for some ∈ ℝ.
With these in place, let us begin. We fix some > 1, take a sequence ( ) ∈ℕ of natural numbers, and define 0 = 1 and, for ⩾ 1, We assume for the sake of convenience that → +∞ as → +∞; otherwise, we would have = 1 for all large , which does not lead to interesting behaviour. Now, take the sequence ∶ ( −1 ) → ( ), ⩾ 1, given by ( ) = ; it is clear that we can compose these maps, obtaining Without further ado, let us describe the long-term behaviour of the hyperbolic metric relative to .
Theorem 2.1. For any choice of > 1 and sequence ( ) ∈ℕ as above, any pair , ∈ ( ) satisfies exactly one of the following.
Proof of Theorem 2.1. The first part of our claim is clear; any pair , ∈ ( ) is either on the same circle, on the same ray, or neither. We are left to associate each case to its dynamical consequences, which we will do on a case-by-case basis.
First, assume that and are on the same circle ⊂ ( ), which we parametrise as where (here and throughout) Ω denotes the hyperbolic length of a curve; we also know that ( ) = , so that • ( ) = exp (2 ). Thus, • is a parametrisation of the circle ⊂ ( ), traversed times with constant speed. It follows that the length ( ) ( ) of traversed only once is ( ) ( )∕ ; connecting ( ) to ( ) with a simple arc on , we see that and it is clear that the right-hand side goes to zero as → +∞. Now, assume that | | ≠ | |, but and are both on the same ray ⊂ ( ). We take a distanceminimising geodesic ⊂ ( ) connecting to , and we know by Lemma 2.1(i) that ⊂ . Again by the fact that is locally isometric, we have ( ) ( • ) = ( ) ( ) = ( ) ( , ). Unlike in the previous case, however, we know that actually takes to in one-to-one fashion, meaning that • is a distance-minimising geodesic arc in and ( ) ( • ) = ( ) ( ( ), ( )). Finally, assume that and are neither on the same circle nor on the same ray. There exists, however, some * ∈ ( ) such that | * | = | | and arg * = arg , meaning that * belongs to the same circle as and to the same ray as . By the reverse triangle inequality, we have items (i) and (ii) of this theorem now tell us that ( ) ( ( ), ( * )) → 0 as → +∞, while

Finite eventual connectivity
Now, we consider a transcendental meromorphic function ∶ ℂ →Ĉ with a wandering domain , which we will assume has finite eventual connectivity . First, in order to deal with case (ii) of Theorem 1.1, we want to show that, in the case = 2, we can 'conjugate' the dynamics of | to the model discussed in Subsection 2.1. For that, we will need the following amalgamation of Theorems 1 and 3 from [9], proved using Ahlfors's theory of covering surfaces. With this in mind, let us begin.
Proof. We proceed inductively; let us start with = 1. Then, = 0 is conformally isomorphic to a unique symmetric annulus ( ), and the isomorphism 0 ∶ 0 → ( ) is unique up to rotation and inversion. The next iterate 1 is also isomorphic to some unique ( ), but in this case we might have to post-compose the isomorphism 1 ∶ 1 → ( ) with a rotation or inversion. For now, this induces a holomorphic map g 1 ∶ ( ) → ( ) by and we know that deg g 1 = deg | 0 = 1 is finite -we exercise our freedom of choice over 1 here, choosing it so that 1 is also positive. Thus, by Lemma 2.2, we want to show that equality holds, whence (also by Lemma 2.2) g 1 ( ) = 1 , whereupon we exercise our freedom of choice again to rotate 1 and ensure that = 1. To this end, note that, by the Riemann-Hurwitz formula (Lemma 2.3(i)), | 0 (and hence g 1 ) has no critical points, and is therefore an unbranched covering map. This means that g 1 ∶ ( ) → ( ) is a local hyperbolic isometry; it takes the closed geodesic ( ) = exp(2 ), ∈ [0, 1], to a closed geodesic of ( ). By Lemma 2.1(ii), g 1 • must be another (monotonic) parametrisation of the unit circle -but one that traverses it 1 times (or, in other words, g 1 • winds 1 times around the origin). Since ( ) ( ) = ( ) (g 1 • ), Lemma 2.1(ii) also tells us that whence mod ( ) = 1 ⋅ mod ( ) and we are done. The rest of the sequences ( ) and (g ) can be built in a similar fashion, with +1 being rotated and inverted as necessary to accommodate (which is already fixed at the th stage). □ An immediate consequence of Lemma 2.4 is that, by induction, where ∶= g • ⋯ •g 0 , each g is a power map, and the are hyperbolic isometries. If  ′′ and  ′′ are the foliations given by Theorem 2.1 for , we can pull them back to obtain transversal foliations  ′ ∶= ( 0 ) *  ′′ = { −1 0 ( ) ∶ ∈ } and  ′ ∶= ( 0 ) *  ′′ = { −1 0 ( ) ∶ ∈ } on , and -since all are hyperbolic isometries -they encode the same dynamical information for and that we had for ( ) and .
This takes care of the case of constant connectivity two; next, we complete the proof of Theorem 1.1.
Proof of Theorem 1.1. Assume first that has eventual connectivity 3 ⩽ < +∞. We claim that ∶ → +1 is a proper map for all large ; indeed, the only way for it not to be proper is for = deg | to be infinite (since maps onto +1 , it follows from Ahlfors's first fundamental theorem that ∶ → +1 either is proper and satisfies the Riemann-Hurwitz formula, or has infinite degree; see, for example, [ Thus, we can apply the Riemann-Hurwitz formula to ∶ → +1 , which tells us that − 2 = ⋅ ( − 2) + ( ).
Since and ( ) are both non-negative integers and ⩾ 3, the only solution is ( ) = 0 and = 1. It follows that ∶ → +1 is a conformal isomorphism for all large , and thus that the hyperbolic metric is preserved by | .
For = 2, we take a sufficiently large that ( ) = 2 for all ⩾ , and apply Lemma 2.4. If mod is a constant sequence, then | is conjugated to rigid rotations of the same ( ) for some > 1, and so the hyperbolic metric is preserved. If mod is not constant, it is (by Lemma 2.4) an increasing sequence diverging to infinity, and we obtain transversal foliations  ′ and  ′ on that encode its dynamics. In order to transfer this knowledge to , we pull the foliations back as  ∶= ( ) *  ′ = { − ( ) ∩ ∶ ∈  ′ } and  ∶= ( ) *  ′ = { − ( ) ∩ ∶ ∈  ′ }. This process 'breaks down' at the critical points Crit( | ) of , which form a discrete set, but works conformally everywhere else. It follows that  and  are transversal laminations of that fail to be foliations at critical points of . By their definitions and Theorem 2.1, it also follows that  is a contracting lamination, and  is an eventually isometric one.

BAKER WANDERING DOMAINS
In this section, we will examine the internal dynamics of Baker wandering domains in direct tracts. Of course, in order to do so, we should first define the latter: an unbounded domain ⊂ ℂ with piecewise smooth boundary is said to be a direct tract for the meromorphic function if ℂ ⧵ is unbounded (but not necessarily connected: ℂ ⧵ can consist of infinitely many bounded components), has no poles in , and there exists > 0 such that | ( )| = for ∈ while | ( )| > for ∈ . Every meromorphic function with finitely many poles has a direct tract, and so do many with infinitely many poles (such as Euler's gamma function, for instance); this gives us a substantially larger class to study than entire functions, while still leaving us with plenty of machinery to do so.
Of course, when talking about the dynamics of Baker wandering domains, one must talk about the results of Bergweiler, Rippon, and Stallard [8]. Although they deal with entire functions, they remark that their results can be generalised to Baker wandering domains of meromorphic functions with direct tracts. The one key step not directly related to their techniques is to generalise a theorem -originally proved by Zheng [25] for functions with finitely many poles -saying that the iterates of a Baker wandering domain contain large annuli { ∶ < | | < } with ∕ → +∞. We now give a brief outline of how to do this. Many of the results and tools used here were also introduced by Bergweiler, Rippon, and Stallard in a different paper [7]. Firstly, they showed that if is a Baker wandering domain of and has a direct tract , then ⊂ for all large , so that is in fact the only direct tract of and all components of ℂ ⧵ are bounded. Secondly, they introduced the subharmonic function ∶ ℂ → [0, +∞), defined as where is a direct tract of where is the positive constant in the definition of the tract . Combined, these two things have key consequences for the distribution of zeros and poles of . since all complementary components of are bounded, there are infinitely many thereof, and = ( ) → ∞ as → +∞, we also have → +∞ and the conclusion follows. □ The function was also used by Bergweiler, Rippon, and Stallard to prove an analogue of the Wiman-Valiron theorem for meromorphic functions with direct tracts (see [7,Theorems 2.2 and 2.3]). This, in turn, implies that if is a meromorphic function with a direct tract , then the image of any sufficiently large annulus lying sufficiently far away from the origin contains another large annulus lying even farther away. This fact coupled with Lemma 3.1 shows that, as concerns Baker wandering domains, functions with a direct tract behave similarly to functions with finitely many poles.
As promised, Lemma 3.1 and the generalised Wiman-Valiron theorem (the latter to substitute for Bohr's theorem) allow us to extend Zheng's theorem to functions with a direct tract. This, in turn, serves as the starting point to generalise Bergweiler, Rippon, and Stallard's results [8] concerning the harmonic function where 0 is any point in a Baker wandering domain of an entire function, and so to generalise their approach to describing the dynamics of Baker wandering domains -as was stated above; we refer to [8] and [7] for the remaining details. Thus, given a transcendental meromorphic function with a Baker wandering domain and a direct tract , and a point 0 ∈ , Equation (3.1) defines a positive, non-constant harmonic function, and the properties of ℎ tell us many things about the internal dynamics of .
Naturally, the reason we did this was so we could apply the results of [8] to our setting. However, we will have to change our notation slightly; since ℎ can be defined taking as starting points any point in any domain on the orbit, we will use ℎ( ; 0 , ) to mean the function defined in by equation (3.1) using the base point 0 ∈ . With that in mind, it is clear that ℎ is -invariant in the sense that ℎ( ( ); ( 0 ), 1 ) = ℎ( ; 0 , ); (3.2) note that, since every Baker wandering domain is bounded and preserves the Fatou and Julia sets, ( ) = 1 in this case. The level sets of ℎ form a lamination of , and this lamination was already studied by Sixsmith [23] in relation to the fast escaping set of entire functions. Here, we are interested in level sets of ℎ for another reason.

Lemma 3.2. Let be a transcendental meromorphic function with a direct tract and a Baker wandering domain .
Choose 0 ∈ , and define ℎ ∶ → (0, +∞) according to equation (3.1). Then, the level sets of ℎ form a contracting lamination of . Before proving Lemma 3.2, we want to convince ourselves that the level curves of ℎ are 'well behaved'. Of course, being level sets of a harmonic function, we know that they are made of analytic curves, but we want more than that.

Lemma 3.3. In the setting of Lemma 3.2, every level curve of ℎ is closed.
Proof. Assume that this is not the case; that is, that ℎ has a level curve with ℎ( ) = ∈ ℝ + that is not closed. Since ℎ is harmonic, must escape to the boundary of , and by [8, Theorem 1.6(b)] (which tells us that ℎ has a continuous extension to ⧵˜, and is constant there) and the maximum principle it must escape to the outer boundary † of . Note that, since every is bounded and the Julia set is completely invariant, ∶ → is a proper map for every (see [21,Lemma 4]), and therefore ( ) will always be a level curve of ℎ( (⋅); ( 0 ), ) that escapes to the outer boundary of .
Take now another level set Γ of ℎ for which ℎ(Γ) = ′ > max{ , 1} (by the maximum principle, such a level set must be non-empty for an appropriate choice of ′ ), and any point ∈ . The definition of ℎ implies that | ( 0 )| − < | ( )| < | ( 0 )| + , where ↘ 0; in other words, ( ) lies in some definite annulus . At the same time, [8, Theorem 7.1] says that (Γ) has (for large enough ) a connected component Γ ⊂ that is a Jordan curve surrounding the origin and lying in the annulus where ′ and ′′ are positive sequences going to zero with → +∞. Most importantly, we see that if is large enough the annuli and ′ are disjoint, with ′ surrounding . Finally, recall that ( ) was supposed to escape to the outer boundary of . It follows from the Jordan curve theorem that ( ) intercepts Γ , which is a contradiction since both are supposed to be level curves of ℎ( (⋅); ( 0 ), ) corresponding to different levels (by equation (3.2)). □ Thus pacified, we can prove Lemma 3.2.
Proof of Lemma 3.2. Let ⊂ be a simple closed level curve of ℎ, and define = • ⊂ . Note that is another closed level curve of ℎ by equation (3.2), and it is clear from the maximum principle that every closed subcurve of surrounds at least one bounded component of ℂ ⧵ . If is sufficiently large, then is (by [8, Theorem 1.3]) contained in a large annulus ⊂ centred at 0, with its iterates • being contained in similar annuli + ⊂ + . It follows that, for such , is a simple closed curve, which we will take to be traversed once. Assume now that has eventual connectivity two (we will postpone proving that has eventual connectivity either two or infinity to the end of this section). Then (assuming that is large enough), surrounds the only bounded complementary component of , and by the argument principle • winds around the origin = − times (we are using the notation of Lemma 3.1). We see that • is a simple closed curve traversed times; let +1 stand for the same curve traversed only once. Since +1 ( • ) ⩽ ( ) by the Schwarz-Pick lemma, we have that † Bergweiler, Rippon, and Stallard use a different definition of the outer boundary, but -fortunately -it is equivalent to ours on bounded domains.
and by induction and the -invariance of ℎ we conclude that Therefore, if and are any two points of , we clearly have + ( ( ), ( )) ⩽ + ( + ) → 0 as → +∞.
This concludes the proof for eventual connectivity two; if ( ) = +∞, we must deal with the simple closed level curves ⊂ of ℎ that do not surround the component of ℂ ⧵ containing the origin (by Lemma 3.3, we have no non-closed level curves to deal with). However, by [21,Lemma 2], any closed curve ⊂ that is not null-homotopic in must have an iterate • that surrounds a pole of , and therefore a bounded complementary component of . In that case, by Lemma 3.1 and the argument principle, +1 • surrounds the origin, and the previous arguments apply.
Finally, we complete the proof of Theorem 1.2.
First, we choose a real constant ∈ (2, ), and a positive < 1∕ (as we proceed, we will impose further restriction on ). Next, consider this rational function of degree two has critical points ±1, which are also fixed points. More importantly for us, the vertical strip { ∶ |ℜ | < 1} has a simply connected pre-image component (of degree one) under in , and this component touches the unit disc exactly at ±1 and ± . Since < , we know that −1 ({ ∶ |ℜ | < log − }) (of course, we take < log ) has a simply connected component ⊂ that touches exactly at ± . Now, we want to choose a constant ∈ ℂ so that the (re-scaled) Joukowski mapping maps the annulus { ∶ 1∕ < | | < } into † { ∈ ∶ dist( , ) > }, and then choose > such that ({ ∶ 1∕ < | | < }) ⊃ { ∶ 1 + }, and some > ; we will also require to satisfy other constraints to be specified ahead, but is relatively 'free'. For now, we make a few observations about the Joukowski mapping ↦ + 1∕ : it is a rational function of degree two, with critical points at ±1 and roots at ± ; it maps the unit circle in two-to-one fashion onto the closed interval [−2, 2], and is symmetric under ↦ 1∕ . Consequently, maps annuli of the form { ∶ 1∕ < | | < }, > 1, properly and with degree two onto simply connected ellipses containing the origin. With these preliminaries in place, we are ready to define our approximating sets. We will need sequences and , ⩾ 0, constructed as follows. Start with 0 = 1, and then choose 0 such that 0 − log(2 ′ ) > 0 , where the criteria for selecting ′ > will be explained further ahead. The next value to be chosen is 1 , taken to satisfy 1 > 0 + log(2 ′ ). To choose 1 , we choose a different criterion, namely 1 − > 1 and then for 2 we want 2 > 1 + . The next constant 2 is chosen so that 2 − 1 − > 2 . For 3 , we pick a number so that 3 > 2 + 1 + , and for 3 we want 3 > 3 + log + 1. Finally, the last 'different' step in this construction is 4 , which is taken so that 4 > 3 + log + 1. From now on, the construction can be carried out recursively with the same basic structure as above (we start by treating 4 as 0 ), cycling between the rules with 'period' four.
With these sequences ready, we define our sets. We have ∶= { ∶ ℜ ⩽ }, and the more complicated and finally ∶= ∪ . Let us make a few remarks about the sets : first, all of them are closed, and all of them contain a left half-plane and exactly one other component. If mod 4 = 0 or mod 4 = 3, these are vertical † In particular, since ⊂ , satisfies | | < 1∕2.  Figure 3 for the most 'relevant' parts of these sets, and how they relate to each other.

F I G U R E 3
The 'main parts' of the sets involved in this construction. Poles of the g , which are also the poles of g, are marked in red (colour online). In orange, we see some of the ⊂ , and, in light grey, the lines We will also need vertical lines ∶= { ∶ ℜ = }. These are set up so that < < +1 (except for mod 4 = 2, in which case we have +1 < < +1 ) and so that ∩ = ∅ for any and . We will reserve the right to fine-tune the position of some of these lines later.
We are ready to start our approximations. First, we must map 0 into 1 while simultaneously mapping 0 into itself. To this end, we apply Arakelyan's theorem (see, for instance, [14, Section IV.C]) to the closed set 0 ∪ 0 , obtaining an entire function g 0 satisfying where 0 is the first term in a sequence of positive numbers such that < ∕10 . Our next step is to approximate an appropriately translated version of in 1 . However, since has a pole, a simple application of Arakelyan's theorem will not suffice.

(4.2)
Proof. We apply Arakelyan's theorem to find an auxiliary entire function ℎ 1 satisfying ⎧ ⎪ ⎨ ⎪ ⎩ |ℎ 1 ( ) + ( − 1 ) + 2 | < 3 1 , ∈ 1 |g 0 ( ) + ℎ 1 ( )| < 3 1 , ∈ 1 |g 0 ( ) + ℎ 1 ( ) + ( − 1 ) + 2 | < 3 1 , ∈ 1 , and define g 1 as g 1 ( ) = ℎ 1 ( ) + ( − 1 ) + 2 . It follows easily from the definition that g 1 has a single pole, which is at 1 , and satisfies the desired inequalities. □ whenever a choice is made regarding , , or ′ , we can continue to shrink without affecting this choice. It is in this sense that we say that they are 'independent of '. First, we ask that log + 2 < 1. We now fix ∈ ℂ as explained before: it is such that ({ ∶ 1∕ < | | < }) ⊂ { ∈ ∶ dist( , ) > }. Now, notice that (since | | < 1∕2) −1 ({ ∶ | | < 1 + }) is a doubly connected domain surrounding the origin and bounded away from both zero and infinity. Therefore, there exists an annulus ∶= {1∕ < | | < } such that | ( )| > 1 + for every ∈ , and we can decrease without affecting the choice of as promised. We shrink so that 1∕ > , and then pick ′ such that 1∕ ′ + < 1∕ and ′ − > . After all that trouble, it follows that † , if is a curve in an -neighbourhood of 4 +3 , ⩾ 0, we can 'pull it back' by g at least three times while remaining within the sets 4 + , = 0, 1, 2 (see Figure 4). Hence, if the values of 4 +2 and 4 +3 are such that 4 +3 − 4 +2 < log + and 4 +3 − 4 +3 < log + , then 4 +2 and 4 +3 will be pulled back as shown in Figure 4. Since every belongs to the orbit of the attracting domain , all these pullbacks will be in the same attracting grand orbit, and thus cannot intersect any of the . The conclusion follows. □ F I G U R E 4 With (mostly) the same colour scheme as Figure 3, we have added pre-images of 2 and 3 in dashed blue ( 2 and 3 are highlighted in solid blue). If the sets and constants are chosen as specified in Claim 4.2, then these pre-images will persist even after the approximation This suffices for showing the boundedness or unboundedness of the wandering domains ; next, we tackle the problem of connectivity. Proof. We will derive the argument for 2 , which is the case = 0; any other follows an analogous argument. First, it is clear from the proof of Claim 4.2 that g −1 ( 2 ∪ 3 ) consists of two analytic arcs that connect the poles of g at 2 ± , surrounding 2 = + 2 and hence 2 . Therefore, no closed curve ⊂ 2 can surround the poles of g; nevertheless, by [21,Lemma 2], any such that is not null-homotopic in 2 must have an iterate g ( ) that surrounds a pole of g. The only hope for , then, is to have an iterate in one of the Fatou components 4 +1 for some ⩾ 1 that surrounds the pole of g at 4 +1 . However, such an iterate would require g −1 ( ) to be a closed curve in 4 , which must (since g is holomorphic on 4 ⊃ 4 ) surround a pre-image * ∈ 4 of 4 +1 . Still, by the triangle inequality (with ( ) = exp(( − 4 )∕2) + 4 +1 ) and the error bound for g on 4 , we have |g( ) − 4 +1 | ⩾ | ( ) − 4 +1 | − | ( ) − g( )| > 1∕ ′ − for every ∈ 4 .
If we choose such that < (2 ′ ) −1 , then the right-hand side of this inequality is greater than (2 ′ ) −1 , which means that no ∈ 4 can reach 4 +1 through g. We conclude that 2 cannot contain any closed curve that is not null-homotopic in 2 , and so 2 is simply connected. □ A similar argument shows that the domains 4 and 4 +3 , ⩾ 0, are also simply connected. Finally, the domains 4 +1 are at least doubly connected, for each surrounds a pole of g at 4 +1 ; if the connectivity is greater than two, then (by Lemma 2.3) 4 is infinitely connected, which is a contradiction. This concludes the proof of Theorem 1.3.
Remark. It remains to be seen if the construction above can be simplified to yield a connectivity sequence of period less than four. However (and this was brought to my attention by Reem Yassawi), it can be irregularly 'padded' with strips and translations to yield a connectivity sequence that is not periodic at all.

CONVERGENCE TO THE BOUNDARY IN MULTIPLY CONNECTED WANDERING DOMAINS
In this section, we shall prove Theorem 1.4. Our starting point will be the same as in [6]: namely, the fact that for any hyperbolic region , we have ( ) → +∞ ⇔ dist( , ) → 0.
Since we will be working with˜, which is simply connected, we will also be able to use their elegant 'Harnack-type' estimates [ Remark. The proof of Lemma 5.1 applies the Koebe distortion theorem to the Riemann map from to Ω. As such, similar estimates can be obtained for multiply connected domains by applying the distortion theorems in [26] to a universal covering map -provided we assume that Ω is uniformly perfect, and replace the universal factor of 2 by a constant depending on Ω. See also [5] for other properties of the hyperbolic metric of simply connected domains that generalise to multiply connected ones under the assumption of uniform perfectness and domain-dependent constants. Now, note that the first claim of Theorem 1.4 (that all orbits go to the boundary or stay away from it together) follows from the second one (that, if an orbit approaches a certain sequence in , so does every other orbit). As such, we can prove Theorem 1.4 by showing the following.