This is a preprint of: “Classifying simply connected wandering domains”, Anna Miriam Benini, Vasiliki Evdoridou, Nuria Fagella, P. J. Rippon, Gwyneth M. Stallard, Math. Ann., vol. 383, 1127–1178, 2022. DOI: [10.1007/s00208-021-02252-0] Classifying simply connected wandering domains Anna Miriam Benini∗1 , Vasiliki Evdoridou †2 , Núria Fagella‡3 , Philip J. Rippon †2 , and Gwyneth M. Stallard †2 1 Dep. of Mathematical, Physical and Computer Sciences, Università di Parma, Italy. 2 School of Mathematics and Statistics, The Open University, Milton Keynes, UK. 3 Dep. de Matemàtiques i Informàtica, Universitat de Barcelona, Catalonia. Emails: ambenini@gmail.com, vasiliki.evdoridou@open.ac.uk, fagella.nuria@gmail.com, phil.rippon@open.ac.uk, gwyneth.stallard@open.ac.uk Communicating author: Philip J. Rippon Dedicated to Misha Lyubich on his 60th birthday Abstract While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in sim- ply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic dis- tances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifica- tions are indeed realizable. 1 Introduction We consider dynamical systems defined by the iteration of holomorphic maps f : C → C on the complex plane, and particularly transcendental ones, that is, those with an essential singularity at infinity. The complex plane, seen as the phase space of the system, splits into two completely invariant subsets: the Fatou set, or those points in a neighbourhood of which the iterates {f n } form a normal family, and its complement, the Julia set. The Fatou set is open and consists typically of infinitely many connected components called Fatou components. ∗ This project has received funding from the European Unions Horizon 2020 research and innovation pro- gramme under the Marie Sklodowska-Curie Grant Agreement No. 703269 COTRADY. † Supported by the EPSRC grant EP/R010560/1. ‡ Partially supported by the Spanish grant MTM2017-86795-C3-3-P, the Maria de Maeztu Excellence Grant MDM-2014-0445 of the BGSMath and the Catalan grant 2017SGR1374. 1 Fatou components map from one to another and this leads to dynamics on the set of these components. In this setting, periodic Fatou components were completely classified a century ago by Fa- tou, in terms of the possible limit functions of the family of iterates; see, for example, [Ber93]. Indeed, if U is a periodic Fatou component of period p ≥ 1, then U can only be one of the following: a domain on which the iterates {f pn |U }n converge to an attracting or parabolic fixed point of f p (known as an attracting or parabolic component, respectively); or a domain on which the iterates {f pn |U }n converge to infinity locally uniformly (known as a Baker do- main); or a topological disk on which f p is conjugate to a rigid irrational rotation (known as a Siegel disk). If a Fatou component U is neither periodic, nor preperiodic (that is, eventually periodic), then f i (U ) ∩ f j (U ) = ∅ for all i, j ≥ 0, i 6= j and U is called a wandering domain. On a wandering domain all limit functions must be constant [Fat20]. Those for which the only limit function is the point at infinity are called escaping, while the rest are either oscillating (if infinity is a limit function and some other finite value also) or dynamically bounded (if all limit functions are points in the plane). A major open problem in transcendental dynamics is whether dynamically bounded wandering domains exist at all. We believe that any progress towards solving this problem will require a deeper knowledge of the dynamics inside (and around) wandering domains, our main motivation for the work in this paper. An essential role in the theory of holomorphic dynamics is played by the singular values, that is, those points for which not all inverse branches are locally well defined. In transcendental dynamics, these can be critical values (images under f of zeros of f 0 ), asymptotic values or accumulations thereof. For a wide class of functions known as finite type maps (those maps with a finite number of singular values), every Fatou component is periodic or preperiodic. Indeed, the absence of wandering domains for polynomials (actually for rational maps) [Sul85] and for transcendental entire functions of finite type [GK86], [EL92] was a major breakthrough in the theory of complex dynamics, and meant that the possible types of dynamical behaviours of all such maps within the Fatou set was fully classified. The result about the absence of wandering domains for the class of transcendental maps of finite type was particularly striking because in the 1970’s Baker [Bak76] had constructed a transcendental entire function which had a nested sequence of multiply connected Fatou components, each mapping to the next and whose orbits escaped to infinity, showing that wandering domains can indeed exist. While the wandering domains in Baker’s example were multiply connected, since then a wide variety of examples of simply connected wandering domains have been given; see, for example, [Her84, p. 106], [Sul85, p. 414], [Bak84, p. 564, p. 567], [Dev90, p. 222], [EL92, Examples 1 and 2] and [FH08, Sect. 4.3.]. But it is only more recently that wandering domains have emerged as a major focus of attention, as the least understood of all the different types of Fatou components. Indeed, several important advances have been made in recent years. For example, (oscillat- ing) wandering domains have been constructed for functions in the Eremenko-Lyubich class B (those maps with a bounded sets of singular values) [Bis15, MPS20, FJL18], a landmark result because escaping wandering domains have been shown not to exist for maps in this class [EL92]. We also mention the recent construction by Bishop [Bis18] of an entire function 2 with Julia set of Hausdorff dimension 1, solving a long standing problem in transcendental dynamics. This function has multiply connected wandering domains and also exhibits other remarkable properties; for example, all the boundary components of its wandering domains are Jordan curves. Moreover, a detailed description of the dynamics of entire functions within multiply con- nected wandering domains was obtained in [BRS13]. Perhaps surprisingly it turns out that in these wandering domains all orbits behave in essentially the same manner, eventually landing in and remaining in a sequence of very large nested round annuli. This detailed description has proved crucial in establishing results about classes of commuting transcendental entire functions [BRS16]. Surprisingly, however, very little is known about the full range of possible behaviours of the orbits inside simply connected wandering domains, relative to the components themselves. This behaviour is connected to the relation between the postsingular set P (f ) (that is, the union of the forward orbits of the singular values) and the wandering domains, another major open problem in the subject. Indeed, recent results in [BFJK19, MBRG13] establish that if U is a wandering domain and Un is the Fatou component containing f n (U ) for n ∈ N, then for every z ∈ U , there exists a sequence pn ∈ P (f ) such that dist(pn , Un )/ dist(f n (z), ∂Un ) → 0 as n → ∞. Hence the understanding of the possible dynamics of orbits inside wandering domains may throw some light on the possible relations between the postsingular set and the wandering domains, both issues being potentially relevant in any future attempt to eliminate dynamically bounded wandering domains. One of the challenges is that several different types of behaviour are known to exist. Let us elaborate a bit further on this observation. Consider any holomorphic self-map of C \ {0}, or an entire map F : C → C for which z = 0 is either an omitted value or has itself as its only preimage; for example, Fλ (z) = λz d exp(z) with d ∈ N, λ ∈ C \ {0}. Such a map F can be lifted by the exponential map to a transcendental entire function f : C → C satisfying exp(f (z)) = F (exp(z)). Observe that f is not uniquely defined, since any map of the form fk (z) = f (z) + 2kπi for k ∈ Z will have the same property. Now notice that if F had, say, an attracting component U (not containing z = 0), then any logarithm of U , say U e , would be a wandering domain for fk (for an appropriate choice of k). Nevertheless, the orbits of points in Ue would still “remember” that they were lifted from an attracting component, in the sense that the iterates of any given point would be successively closer to the orbit of pe := log p ∈ Ue, where p is the fixed point of F in U . Likewise, if U had been, for example, a Siegel disk, the iterates of points in the successive images of U e would “rotate” around a centre point (actually orbit), again the iterates of pe. See Figure 1, and also Figure 3 in Section 3.3 for a lift of a parabolic component. With this lifting procedure, one can construct examples of simply connected escaping wan- dering domains exhibiting the three different types of internal dynamics that correspond to the possible dynamics inside a periodic component: attracting, parabolic or rotation-like. Thus we already have a contrast with multiply connected wandering domains, where only one type of dynamical behaviour is possible, as noted above. These observations suggest a very natural question: How special are the three examples above in the general world of wan- dering domains; in other words, is there a classification of wandering domains in the spirit of Fatou’s classification of periodic Fatou components or is any orbit behaviour realizable? Let 3 √ Figure 1: Left: Dynamical plane of F (w) = λw2 e−w with λ = e2−ρ /(2 − ρ) and ρ = eπi(1− 5) . There is a (bounded) super-attracting component centred at w = 0 (white) and a Siegel disk centred at w0 = 2 − λ (gray). Right: Dynamical plane of f (z) = 2z − ez + log λ satisfying exp(f (z)) = F (exp(z)). The super-attracting component lifts to a Baker domain (white), while the Siegel disk lifts to infinitely many orbits of wandering domains on which f is univalent (gray). See [Ber95a, FH08, FG03] for details. The range is [−9, 9] × [−9, 9]. us note that, due to the lack of periodicity, the dynamics of f on a sequence of wandering domains can be thought of as a non-autonomous system (at every iterate we apply a “differ- ent” map), and such systems are a priori difficult to study because they may exhibit a wide range of behaviours. This might be an indication that such a classification may not exist. On the other hand, the successful description of the dynamics in multiply connected wandering domains obtained in [BRS13] is encouraging. The dynamics of points which belong to wandering domains can be seen from two perspec- tives. While points have to move together with the wandering domain which contains them (in the way that passengers on a cruise ship must follow the ship’s trajectory), on the one hand they may or may not cluster together as they move along (as happens when lifting an attracting component but not when lifting a Siegel disk), and on the other hand orbits may stay away from the boundaries of their domains (as happens when lifting an attracting basin but not when lifting a parabolic basin). Our results will address both of those points of view. More precisely we give a complete and precise description of the possible dynamics of orbits inside the wandering components in terms of both the contraction properties with respect to the hyperbolic metric, and the distance of orbits to the boundary of the wandering components (Theorems A, B and C). This provides a wandering version of the Fatou Classification Theorem of periodic components, a cornerstone of holomorphic dynamics. Moreover, we show that all of the possible cases exist by proving a new general and potentially very useful tool (Theorem D) to establish the existence of wandering components for a given map. Finally, we prove that under certain 4 conditions the wandering domains have Jordan curve boundaries. Our proof includes a novel result (Theorem 7.5) on the Euclidean lengths of vertical geodesics of annuli, using a new technique involving the classical Féjer-Riesz Inequality, which is of independent interest. Statement of results The most natural intrinsic quantity that we have to hand – intrinsic in that it does not depend on the embedding of the wandering domains in the plane – are the hyperbolic distances between pairs of corresponding points of two orbits, and so our approach will be to evaluate how hyperbolic distances between such pairs of points evolve under iteration. Let us recall that a domain U ⊂ C is hyperbolic if its boundary (in C) contains at least two points. For a hyperbolic domain U , let ρU (z) denote the hyperbolic density at z ∈ U and for z, z 0 ∈ U let distU (z, z 0 ) denote the hyperbolic distance in U between z and z 0 . Also recall that if U, V are hyperbolic domains, and f : U → V is a holomorphic map, then the Schwarz-Pick Lemma ensures that f is a contraction for the hyperbolic distance. Hence, if U ⊂ C is a wandering domain of a transcendental entire function f and we define Un , as above, to be the Fatou component containing f n (U ), for n ∈ N, we have that, given any two points z, z 0 ∈ U , the sequence distUn (f n (z), f n (z 0 )) is decreasing and therefore converges to a value that we denote by c(z, z 0 ) = cU (z, z 0 ) := lim distUn (f n (z), f n (z 0 )) ≥ 0. n→∞ Our first classification result shows that whether or not c(z, z 0 ) is zero does not actually depend on the chosen pair (z, z 0 ), provided that the two points have distinct orbits. We also give a criterion to discriminate between these cases based on the concept of hyperbolic distortion [BM07, Sect. 5,11]. Definition 1.1 (Hyperbolic distortion). If f : U → V is a holomorphic map between two hyperbolic domains U and V , then the hyperbolic distortion of f at z is distV (f (z 0 ), f (z)) kDf (z)kVU := lim , 0 z →z distU (z 0 , z) and it equals the modulus of the hyperbolic derivative. Theorem A (First classification theorem). Let U be a simply connected wandering domain of a transcendental entire function f and let Un be the Fatou component containing f n (U ), for n ≥ 0. Define the countable set of pairs E = {(z, z 0 ) ∈ U × U : f k (z) = f k (z 0 ) for some k ∈ N}. Then, exactly one of the following holds. (1) distUn (f n (z), f n (z 0 )) −→ c(z, z 0 ) = 0 for all z, z 0 ∈ U , and we say that U is (hyperbol- n→∞ ically) contracting; 5 (2) distUn (f n (z), f n (z 0 )) −→ c(z, z 0 ) > 0 and distUn (f n (z), f n (z 0 )) 6= c(z, z 0 ) for all (z, z 0 ) ∈ n→∞ (U × U ) \ E, n ∈ N, and we say that U is (hyperbolically) semi-contracting; or (3) there exists N > 0 such that for all n ≥ N , distUn (f n (z), f n (z 0 )) = c(z, z 0 ) > 0 for all (z, z 0 ) ∈ (U × U ) \ E, and we say that U is (hyperbolically) eventually isometric. Moreover for z ∈ U and n ∈ N let λn (z) be the hyperbolic distortion kDf (f n−1 (z))kU n Un−1 . Then P • U is contracting if and only if ∞ n=1 (1 − λn (z)) = ∞; • U is eventually isometric if and only if λn (z) = 1, for n sufficently large. Note that, by the Schwarz-Pick Lemma, U is eventually isometric if and only if f : Un → Un+1 is univalent for large n and so a wandering domain obtained by lifting a Siegel disk is always eventually isometric. In contrast, we show that lifting an attracting or parabolic component results in a contracting wandering domain. To distinguish between these two cases, we refine the classification of contracting wandering domains according to the rate of contraction. Definition 1.2 (Rate of contraction). Let U be a simply connected wandering domain of a transcendental entire function f and let Un be the Fatou component containing f n (U ), for n ≥ 0. We say that U is strongly contracting if there exists c ∈ (0, 1) such that distUn (f n (z), f n (z 0 )) = O(cn ), for z, z 0 ∈ U. We say that U is super-contracting if it satisfies the stronger condition that lim (distUn (f n (z), f n (z 0 )))1/n = 0, for z, z 0 ∈ U. n→∞ It is easy to see that the lift of an attracting component is strongly contracting, and we prove in Section 3 that the lift of a parabolic component is contracting but not strongly contracting. We do this by a careful analysis of the behaviour of the hyperbolic distance between pairs of points in two orbits in any parabolic component; see Theorem 3.4. A special case of super-contracting wandering domains is given by wandering domains which contain an orbit consisting of critical points. An example of such a super-contracting domain which does not arise from a lifting procedure is given in Theorem F. Next, we give sufficient criteria for a wandering domain to be strongly contracting or super- contracting in terms of the long term average values of the hyperbolic distortion along the orbit of a point z0 ∈ U . We also show that this quantity is independent of the point z0 . Theorem B. Let U be a simply connected wandering domain of a transcendental entire function f and let Un be the Fatou component containing f n (U ), for n ≥ 0. Fix a point z0 ∈ U and, for z ∈ U and n ∈ N, let λn (z) = kDf (f n−1 (z))kU n Un−1 . Then the following hold: P (a) If lim supn→∞ n1 nk=1 λk (z0 ) < 1, then U is strongly contracting. P (b) If limn→∞ n1 nk=1 λk (z0 ) = 0, then U is super-contracting. 6 1 Pn 1 Pn (c) If z ∈ U , then lim supn→∞ n k=1 λk (z) = lim supn→∞ n k=1 λk (z0 ). Once the behaviour of orbits in relation to each other within simply connected wandering domains is well understood, we turn to the question of how these orbits interact with the boundaries of the wandering domains. The concept of orbits ‘approaching the boundary’ is in itself delicate to define since it depends on the shape of the sets Un , which may become highly distorted (think for example of the ratio of the diameter of the domains to their conformal radius, which may tend to infinity). There are alternative candidates for the definition of convergence to the boundary (see Section 4), but in this paper we use the following definition. Definition 1.3 (Boundary convergence). Let U be a simply connected wandering domain of a transcendental entire function f and let Un be the Fatou component containing f n (U ), for n ≥ 0. We say that the orbit of z ∈ U converges to the boundary (of Un ) if and only if dist(f n (z), ∂Un ) → 0 as n → ∞. We show that, with this definition, the following trichotomy holds. Theorem C (Second classification theorem). Let U be a simply connected wandering domain of a transcendental entire function f and let Un be the Fatou component containing f n (U ), for n ≥ 0. Then exactly one of the following holds: (a) lim inf n→∞ dist(f n (z), ∂Un ) > 0 for all z ∈ U , that is, all orbits stay away from the boundary; (b) there exists a subsequence nk → ∞ for which dist(f nk (z), ∂Unk ) → 0 for all z ∈ U , while for a different subsequence mk → ∞ we have that lim inf dist(f mk (z), ∂Umk ) > 0, for z ∈ U ; k→∞ (c) dist(f n (z), ∂Un ) → 0 for all z ∈ U , that is, all orbits converge to the boundary. We remark that we actually prove a stronger version of Theorem C (see Theorem 4.2), which takes into account different definitions of converging to the boundary. Construction of examples Theorems A and C combine to give nine different dynamical types of simply connected wandering domains. A natural question to ask is whether all of these can be realized. As far as we know, the existing examples of simply connected wandering domains in the literature belong to one of the following three cases: contracting and converging to the boundary (e.g. lifts of parabolic components); contracting and staying away from the boundary (e.g. lifts of attracting components); isometric and staying away from the boundary (e.g. lifts of Siegel disks). We will use approximation theory (see Section 5) to construct examples of each of the nine possibilities. In fact, we present a new general technique to construct bounded simply connected wandering domains (see Theorem 5.3) which allows us to keep good control on the internal dynamics, as well as on the degree of the resulting maps from one Fatou component to the next. As a key step we prove the following general result to show the existence of bounded simply connected wandering domains. Its statement uses the following terminology. 7 Definition 1.4. We say that a curve σ surrounds a set B if and only if B is contained in a bounded complementary component of σ. Also, for a Jordan curve η we denote by int η the bounded component of C \ η and by ext η the unbounded component of C \ η. We can now state our result. Theorem D (Existence criteria for wandering domains). Let f be a transcendental entire function and suppose that there exist Jordan curves γn and Γn , n ≥ 0, a bounded domain D, a subsequence nk → ∞ and compact sets Lk (associated with Γnk ) such that (a) Γn surrounds γn , for n ≥ 0; (b) for every k, n, m ≥ 0, m 6= n the sets Lk , D, Γm are in ext Γn ; (c) γn+1 surrounds f (γn ), for n ≥ 0; (d) f (Γn ) surrounds Γn+1 , for n ≥ 0; S (e) f (D ∪ k≥0 Lk ) ⊂ D; (f) max{dist(z, Lk ) : z ∈ Γnk } = o(dist(γnk , Γnk )) as k → ∞. Then there exists an orbit of simply connected wandering domains Un such that int γn ⊂ Un ⊂ int Γn , for n ≥ 0. Moreover, if there exists zn ∈ int γn such that both f (γn ) and f (Γn ) wind dn times around f (zn ), then f : Un → Un+1 has degree dn , for n ≥ 0. We use Theorem D to construct examples of each of the nine possible types, and also to construct simply connected wandering domains that contain any prescribed (finite) number of orbits consisting of critical points. A wandering domain U will be called k-super-attracting if there exist critical points z1 , . . . , zk ∈ U , such that f n (z1 ), . . . , f n (zk ) are critical points of f , for all n ∈ N. Theorem E (All types are realizable). (a) For each of the nine possible types of simply connected wandering domains arising from Theorems A and C, there exists a transcendental entire function with a bounded, simply connected escaping wandering domain of that type. (b) For each k ∈ N, there exists a transcendental entire function f having a bounded, simply connected escaping wandering domain U which is k-super-attracting. Note that our examples in part (b) of Theorem E are super-contracting wandering domains that are not lifts of super-attracting components. The bounded, simply connected wandering domains, (Un ) say, constructed in Theorem E all have a shape that tends to the shape of a Euclidean disk as n → ∞, but in fact the construction can easily be modified to give wandering domains with different limiting shapes. Also, in forthcoming work we combine our new ideas with techniques introduced by Eremenko and Lyubich [EL87] to construct examples of oscillating simply connected wandering domains that are bounded and have various types of internal dynamics. Finally, we show that our methods can be adapted to construct simply connected wandering domains bounded by Jordan curves. Theorem E is proved by obtaining entire functions that 8 approximate sequences of translates of Blaschke products associated with sequences of Jordan curves with the properties given in Theorem D, and we show that, if these Blaschke products are in a certain sense uniformly expanding and have uniformly bounded degree, then the resulting wandering domains have Jordan curve boundaries. Structure of the paper The first part of the paper (Sections 2, 3 and 4) is devoted to studying the possible behaviours of orbits in simply connected wandering domains, proving Theorems A, B and C. We begin in Section 2 by setting up related non-autonomous dynamical systems of self maps of the unit disk. We prove several results in this general setting which may be of wider interest. In Section 3 we use our results from Section 2 to prove Theorems A and B. We prove Theorem C in Section 4. The second part of the paper (Sections 5, 6 and 7) is devoted to the construction of exam- ples. In Section 5 we give the proof of Theorem D and develop a new general technique for constructing bounded wandering domains. In Section 6 we use this technique to construct examples of every possible behaviour classified in the first part of the paper, proving The- orem E. Finally, in Section 7 we show that, under certain conditions, our new construction technique gives simply connected wandering domains that are Jordan domains. Acknowledgments We are grateful to the Universitat de Barcelona, to the IMUB and to the Open University for hosting parts of this research. We would like to thank Chris Bishop, Xavier Jarque, Misha Lyubich, Lasse Rempe-Gillen and Dave Sixsmith for inspiring discussions. 2 Non-autonomous dynamical systems of self maps of the unit disk In this section we prove several results in the general setting of non-autonomous forward dynamical systems of holomorphic self maps of the unit disk fixing the origin. These results may be of wider interest with applications outside holomorphic dynamics. In the next section, we apply them to the case of transcendental entire functions with simply connected wandering domains in order to prove Theorem A and Theorem B. Our proofs are based on hyperbolic distances in the unit disk and we make frequent use of the fact that Z |w|   2 dt 1 + |w| distD (w, 0) = = log , for w ∈ D. (2.1) 0 1 − t2 1 − |w| In our first result, we characterize when the limits of such systems of holomorphic self maps of the unit disk are identically equal to zero, in terms of the values of the derivatives of the maps at 0. In particular, unless |gn0 (0)| → 1 as n → ∞, the limit of the maps Gn is always zero. 9 Theorem 2.1 (Criterion for converging to zero). For each n ∈ N, let gn : D → D be holomorphic with gn (0) = 0 and |gn0 (0)| = λn , and let Gn = gn ◦ · · · ◦ g1 . P (a) If ∞ n=1 (1 − λn ) = ∞, then Gn (w) → 0 as n → ∞, for all w ∈ D. P (b) If ∞ n=1 (1 − λn ) < ∞, then Gn (w) 9 0 as n → ∞, for all w ∈ D for which Gn (w) 6= 0 for all n ∈ N. Proof. We begin with ideas used by Beardon and Carne [BC92]. First, it follows from the hyperbolic triangle inequality and hyperbolic contraction that, if ψ : D → D is holomorphic, then for all w ∈ D we have distD (0, ψ(w)) ≤ distD (0, ψ(0)) + distD (ψ(0), ψ(w)) ≤ distD (0, ψ(0)) + distD (0, w), (2.2) and, similarly, distD (0, ψ(0)) ≤ distD (0, ψ(w)) + distD (0, w), for all w ∈ D. (2.3) We also use the fact that ∞ X (1 − λn ) = ∞ ⇐⇒ λm+n · · · λm+1 → 0 as n → ∞, for all m ∈ N. (2.4) n=1 In the case when λn 6= 0, for all n, and the right-hand side is λn · · · λ1 → 0 as n → ∞, this statement is a standard property of infinite products proved by taking logarithms. Here it is possible that some or all of the terms λn are zero, so the right-hand side of (2.4) takes account of these possibilities. Now take w0 ∈ D and, for simplicity, denote Gn (w0 ) by wn , for n ∈ N. To prove part (a), we assume that λm+n · · · λm+1 → 0 as n → ∞, for all m ∈ N, and deduce that wn → 0 as n → ∞. Suppose that wn 9 0 as n → ∞. Since wn = gn (wn−1 ), we deduce by Schwarz’s Lemma that |wn | ≤ |wn−1 |, and hence that |wn | decreases to some d > 0 as n → ∞. First choose m ∈ N so large that |wm | is sufficiently close to d to ensure that distD (0, wn+m /wm ) > distD (0, wm ), for n ∈ N. (2.5) Next we fix n ∈ N and define the holomorphic map ψ(w) = (gm+n ◦ · · · ◦ gm+1 (w))/w, for w ∈ D \ {0}, with ψ(0) = (gm+n ◦ · · · ◦ gm+1 )0 (0) = λm+n · · · λm+1 . Applying (2.2) to the function ψ at the point w = wm gives distD (0, wn+m /wm ) = distD (0, ψ(wm )) ≤ distD (0, ψ(0)) + distD (0, wm ) ≤ distD (0, λm+n · · · λm+1 ) + distD (0, wm ). 10 Since we have assumed that λm+n · · · λm+1 → 0 as n → ∞, it follows that distD (0, wn+m /wm ) ≤ distD (0, wm ), for m sufficiently large. This, however, contradicts (2.5), showing that wn → 0 as n → ∞. To prove part (b), we assume that, for some m0 ∈ N, λm0 +n · · · λm0 +1 → λ > 0 as n → ∞, and deduce that whenever wn 6= 0, for all n ∈ N, we have wn 9 0 as n → ∞. Suppose that wn → 0 as n → ∞. First choose m so large that m ≥ m0 and distD (0, wm ) < distD (0, λ), (2.6) and note that, for such m, λm+n · · · λm+1 ≥ λm+n · · · λm0 +1 = λm0 +(m−m0 )+n · · · λm0 +1 ≥ λ, for n ∈ N. (2.7) Next we fix n ∈ N and apply (2.3) with ψ defined as earlier and w = wm to give distD (0, λm+n · · · λm+1 ) ≤ distD (0, wm+n /wm ) + distD (0, wm ). Letting n → ∞, we obtain a contradiction to (2.6) in view of (2.7) and hence to the suppo- sition that wn → 0 as n → ∞. This completes the proof. The following corollary to Theorem 2.1 shows that if the hyperbolic distance between two distinct orbits converges to zero, then the same occurs for every pair of orbits. Corollary 2.2. For n ∈ N, let gn : D → D be holomorphic and let Gn = gn ◦ · · · ◦ g1 . If there exist w0 , w00 ∈ D such that Gn (w00 ) 6= Gn (w0 ) for all n ∈ N and distD (Gn (w00 ), Gn (w0 )) → 0 as n → ∞, then distD (Gn (w), Gn (w0 )) → 0 as n → ∞, for all w ∈ D. Proof. For each n ∈ N, let wn = gn (wn−1 ) and, for n ≥ 0, let Mn : D → D be a Möbius map −1 satisfying Mn (wn ) = 0. Then, for each n ∈ N, the map hn = Mn ◦ gn ◦ Mn−1 is a holomorphic self map of the unit disk and hn (0) = 0. For n ∈ N, let Hn := hn ◦ · · · ◦ h1 and notice that Hn (0) = 0. Since Möbius maps are isometries and Hn = Mn ◦ Gn ◦ M0−1 , for n ∈ N, we have distD (0, Hn (M0 (w00 ))) = distD (Hn (0), Hn (M0 (w00 ))) = distD (Mn ◦ Gn ◦ M0−1 (0), Mn ◦ Gn ◦ M0−1 ◦ M0 (w00 )) = distD (Mn ◦ Gn (w0 ), Mn ◦ Gn (w00 )) = distD (Gn (w0 ), Gn (w00 )) → 0 as n → ∞, and hence Hn (M0 (w00 )) → 0 as n → ∞. Since Hn (M0 (w00 )) = Mn (Gn (w00 )) 6= 0, for each n ∈ N, it follows from Theorem 2.1 that Hn (w0 ) → 0 as n → ∞ for all w0 ∈ D. The result now follows since distD (Gn (w), Gn (w0 )) = distD (Hn (M0 (w)), Hn (0)) = distD (Hn (M0 (w)), 0), for w ∈ D. 11 Theorem 2.1 and Corollary 2.2 will be used in the proof of Theorem A (see Section 3.1). We now prove several results giving estimates for the rate at which limits tend to zero in the case when the limit in Theorem 2.1 is identically equal to zero. The results proven in the remainder of this section will be used in Section 3.2 to prove Theorem B, that is, the subclassification of contracting wandering domains. We use the following result which includes a generalization of Schwarz’s Lemma. Lemma 2.3 (Variation of Schwarz’s Lemma). Let ψ : D → D be holomorphic. Then |ψ(0)| − |w| |ψ(0)| + |w| ≤ |ψ(w)| ≤ , for w ∈ D. 1 − |ψ(0)||w| 1 + |ψ(0)||w| Proof. The right-hand inequality arises from (2.2) and is given in [BC92, p.217]. We prove the left-hand inequality using similar methods. First note that it follows from (2.3) that distD (0, ψ(w)) ≥ distD (0, ψ(0)) − distD (0, w), that is, 1 + |ψ(w)| 1 + |ψ(0)| 1 + |w| log ≥ log − log . 1 − |ψ(w)| 1 − |ψ(0)| 1 − |w| By the monotonicity of the logarithm, this is equivalent to the following inequality:    1 + |ψ(w)| 1 + |ψ(0)| 1 − |w| ≥ , 1 − |ψ(w)| 1 − |ψ(0)| 1 + |w| which gives    1+|ψ(0)| 1−|w| 1−|ψ(0)| 1+|w| −1 |ψ(0)| − |w| |ψ(w)| ≥    = , 1+|ψ(0)| 1−|w| +1 1 − |ψ(0)||w| 1−|ψ(0)| 1+|w| as claimed. We make frequent use of the following corollary of Lemma 2.3. Corollary 2.4. Let g : D → D be holomorphic with g(0) = 0 and |g 0 (0)| = λ. Then, for all w ∈ D,     λ − |w| λ + |w| |w| ≤ |g(w)| ≤ |w| 1 − λ|w| 1 + λ|w| Proof. The result follows by applying Lemma 2.3 to the holomorphic map ψ : D → D defined by ψ(w) = g(w)/w, for w ∈ D \ {0}, with ψ(0) = g 0 (0). We first use Corollary 2.4 to prove the following result giving rather precise upper and lower estimates of the rate at which the sequences |Gn (w)| in Theorem 2.1 decrease, expressed in terms of the derivatives |gn0 (0)|. This result can be used to give a more direct proof of Theorem 2.1; see the remark after the proof of Theorem 2.5. 12 Theorem 2.5. For each n ∈ N, let gn : D → D be holomorphic with gn (0) = 0 and |gn0 (0)| = λn = 1 − µn , and let Gn = gn ◦ · · · ◦ g1 . If w ∈ D and wn = Gn (w), n ∈ N, then (a) n Y |wn | ≤ |w| (1 − cw µk ), where cw = (1 − |w|)/2; (2.8) k=1 (b) if |w| ≤ λk , for 1 ≤ k ≤ n, then n Y 1 + |w| |wn | ≥ |w| (1 − dw µk ), where dw = . (2.9) 1 − |w| k=1 Proof. Set w0 = w. We begin the proof of part (a) by noting that it follows from Corollary 2.4 that, for k ≥ 0 and w ∈ D,   λk+1 + |wk | |wk+1 | = |gk+1 (wk )| ≤ |wk | 1 + λk+1 |wk |   µk+1 (1 − |wk |) = |wk | 1 − 1 + λk+1 |wk | µk+1 (1 − |wk |) ≤ |wk |(1 − )) 2 ≤ |wk |(1 − cw µk+1 ), where the third inequality follows because λk+1 |Gk (w)| < 1 and the last inequality follows because |wk | = |Gk (w)| ≤ |w| by Schwarz’s Lemma. The result of (2.8) now follows and this completes the proof of part (a). We now prove part (b). Using Corollary 2.4 again,   λk+1 − |wk | |wk+1 | = |gk+1 (wk )| ≥ |wk | . (2.10) 1 − λk+1 |wk | Now we use the elementary calculus estimate that   λ−r 1+r ≥1− (1 − λ), for 0 < r < λ ≤ 1, 1 − λr 1−r to deduce from (2.10) that, for k ≥ 0, if |w| ≤ λk+1 , then         1 + |wk | 1 + |w| |wk+1 | ≥ |wk | 1 − µk+1 ≥ |wk | 1 − µk+1 , 1 − |wk | 1 − |w| using the fact that |wk | = |Gk (w)| ≤ |w| again. The result of (2.9) now follows and this completes the proof of part (b). Remark. Theorem 2.5 can be used to give a proof of Theorem 2.1. To do so, it is first necessary to use Hurwitz’ Theorem in order to show that either Gn (w) → 0 as n → ∞ for all w ∈ D or Gn (w) → 0 as n → ∞ only for those points w ∈ D for which Gn (w) = 0 eventually. 13 We now prove another result giving upper estimates for the rate at which the sequences |Gn (w)| decrease, this time expressed in terms of the average of the derivatives |gn0 (0)|. The proof of this result is also based on Corollary 2.4. Theorem 2.6. For each n ∈ N, let gn : D → D be holomorphic with gn (0) = 0 and |gn0 (0)| = λn = 1 − µn , and let Gn = gn ◦ · · · ◦ g1 . Then, for all n ∈ N, if w0 ∈ D and wn = Gn (w0 ), for n ∈ N, !n n n−1 1X 1X |wn | ≤ λk + |wk | . (2.11) n n k=1 k=0 Hence (a) if n 1X lim sup λk = a < 1, n→∞ n k=1 then |Gn (w)| = O(cn ) as n → ∞, for w ∈ D, where c ∈ (a, 1); (b) if n 1X lim λk = 0, n→∞ n k=1 then |Gn (w)|1/n → 0 as n → ∞, for w ∈ D. Proof. By using Corollary 2.4, and then applying the fact that the geometric mean of n positive numbers is at most equal to their arithmetic mean, we see that, for w0 ∈ D and n ∈ N,     λn + |wn−1 | λ1 + |w0 | |wn | ≤ |w0 | ... 1 + λn |wn−1 | 1 + λ1 |w0 | ≤ |w0 |((λn + |wn−1 |) . . . (λ1 + |w0 |))  n 1 ≤ |w0 | ((λn + |wn−1 |) + · · · + (λ1 + |w0 |)) n n n−1 !n 1X 1X = |w0 | λk + |wk | . n n k=1 k=0 This proves (2.11). P P Next, if lim supn→∞ n1 nk=1 λk = a < 1, then ∞ (1 − λk ) = ∞ and so it follows from k=1P 1 n−1 Theorem 2.1 that wn → 0 as n → ∞ and hence that n k=0 |wk | → 0 as n → ∞. So, in this case, it follows from (2.11) that |Gn (w0 )|1/n = |wn |1/n ≤ a + o(1) as n → ∞. The results of parts (a) and (b) now follow. 14 Theorem 2.5 (a) and Theorem 2.6 Pgive uniform upper estimates on the rate that |Gn (w)| tends to 0, in the situation where ∞ n=1 (1 − λn ) = ∞. It is natural to ask whether we can demonstrate such a uniform rate if we know the rate at which |Gn (w)| tends to 0 on some subset of D. It is clear that we cannot deduce any uniform rate at which Gn (w) → 0 from information about the behaviour of Gn at a single point w0 ∈ D. However, if we have an upper bound for |Gn (w)| on some circle {w : |w| = r0 }, where 0 < r0 < 1, then we can obtain an upper estimate for |Gn (w)| for all w ∈ D by applying the following simple proposition. Proposition 2.7 (Hadamard convexity). Let f : D → D be holomorphic and satisfy |f (w)| ≤ a, for |w| ≤ r0 , where 0 < a ≤ r0 < 1. Then, log r |f (w)| ≤ a log r0 for |w| ≤ r, for all r such that r0 ≤ r < 1. Proof. For 0 ≤ r < 1, let M (r) = M (r, f ) := sup |f (z)| |z|=r denote the maximum modulus function and put ϕ(t) = log M (et ), for −∞ < t < 0. Then ϕ is convex by Hadamard’s Three Circles Theorem [Tit39, page 172], negative and increasing, and by hypothesis ϕ(log r0 ) ≤ log a. Hence   log a ϕ(t) ≤ t, for − ∞ < t < 0; log r0 that is,   log a log M (r) ≤ log r, for r0 ≤ r < 1, log r0 and hence log r M (r) ≤ a log r0 , for r0 ≤ r < 1, as required. Remark. In Proposition 2.7, the circle {w : |w| = r0 } can be replaced by any subset of D of positive logarithmic capacity, using a more delicate argument involving Green potentials in D. We omit the details. 15 3 Contraction trichotomy: Proof of Theorems A and B This section is devoted to a classification of simply connected wandering domains based on hyperbolic distances between orbits of points. More precisely we prove Theorems A and B and we also show that lifts of parabolic components are contracting yet not strongly contracting; (see Theorem 3.4). The proofs are based on the results from Section 2 concerning self maps of the unit disk. We first show how the hyperbolic distances between orbits of points in the wandering do- main compare with the distances between related orbits of points in the unit disk. We also compare the hyperbolic distortion along an orbit of a point in the wandering domain with the derivatives of the related maps of the unit disk. Let f be a transcendental entire function with a simply connected wandering domain U and let Un be the Fatou component containing f n (U0 ), for n ≥ 0. Note that each of the domains Un is simply connected; indeed, if some Un is multiply connected, then by [Bak84, Theorem 3.1], all the Fatou components are bounded, so f is a proper map between Fatou components and the claim follows from the Riemann–Hurwitz formula. Although Un = f n (U0 ) if U0 is bounded, this is not necessarily true in the case that U0 is unbounded when Un \ f n (U ) may contain one point; see for example [Her98]. We prove Theorem A and Theorem B by considering a sequence (gn ) of holomorphic self maps of the unit disk associated to f and Un in the following way. Fix a point z0 ∈ U0 and, for each n ≥ 0, choose ϕn : Un → D to be a Riemann map such that ϕn (f n (z0 )) = 0. Then, for n ∈ N, consider the holomorphic maps gn : D → D defined as gn = ϕn ◦ f ◦ ϕ−1 n−1 , and the composite maps Gn : D → D defined as Gn = gn ◦ · · · ◦ g1 = ϕn ◦ f n ◦ ϕ−1 0 . Because of the choice of normalization for the Riemann maps we have that gn (0) = Gn (0) = 0. This set up is illustrated in Figure 3. Each of the maps gn and Gn is an inner function, but we do not use this fact in this paper. Before stating the next theorem, we recall that if f : U → V is a holomorphic map between two hyperbolic domains U and V , then the hyperbolic distortion of f at z is defined to be distV (f (z 0 ), f (z)) kDf (z)kVU := lim . 0 z →z distU (z 0 , z) Lemma 3.1. Let U be a simply connected wandering domain of a transcendental entire function f and let Un be the Fatou component containing f n (U ), for n ≥ 0. Let z0 ∈ U0 and let gn , Gn be as defined above. (a) If z ∈ U and ϕ0 (z) = w, then   n n 1 + |Gn (w)| distUn (f (z), f (z0 )) = log , for n ∈ N. 1 − |Gn (w)| 16 U0 U1 Un−1 Un zn−1 z0 z1 zn ... f f f f ϕ0 ϕ1 ϕn−1 ϕn g1 g2 gn−1 gn 0 0 ... 0 0 D D D D Gn Figure 2: Self maps of the unit disk arising from an orbit of wandering domains (b) For each n ∈ N, |gn0 (0)| = λn (z0 ) := kDf (f n−1 (z0 ))kU n Un−1 . Proof. (a) Let n ∈ N. Since Gn = ϕn ◦f n ◦ϕ−1 0 and ϕn is conformal, if z ∈ U and ϕ0 (z) = w then   n n 1 + |Gn (w)| distUn (f (z), f (z0 )) = distD (Gn (w), Gn (0)) = distD (Gn (w), 0) = log , 1 − |Gn (w)| where the last equality follows from (2.1). (b) Let n ∈ N. Since gn = ϕn ◦ f ◦ ϕ−1 n−1 and ϕn is conformal we have kDgn (0)kD D = kDf (f n−1 (z0 ))kU n Un−1 = λn (z0 ). Since gn (0) = 0, it follows from (2.1) that distD (gn (w), gn (0)) distD (gn (w), 0) kDgn (0)kD D = lim = lim w→0 dist (w, 0) w→0 distD (w, 0)  D  log 1+|g n (w)| 1−|gn (w)| = lim   w→0 1+|w| log 1−|w| 2|gn (w)| = lim = |gn0 (0)|, w→0 2|w| by using the Taylor expansion for the logarithm. 17 3.1 Proof of Theorem A We now use the results of Section 2 together with Lemma 3.1 to prove Theorem A, that is, the classification of simply connected wandering domains according to the behaviour of the hyperbolic distances between orbits of points. Let U be a simply connected wandering domain of a transcendental entire function f and let Un be the Fatou component containing f n (U ), for n ≥ 0. Also, let E = {(z, z 0 ) ∈ U × U : f k (z) = f k (z 0 ) for some k ∈ N}. Let z0 ∈ U0 and let ϕn , gn , Gn be as defined at the beginning of this section. First we suppose that there exists z00 ∈ U0 with (z00 , z0 ) ∈ /E and distUn (f n (z00 ), f n (z0 )) −→ 0. (3.1) n→∞ Let w00 = ϕ0 (z00 ). By (3.1) and Lemma 3.1 (a), we have that Gn (w00 ) −→ 0 and Gn (w00 ) 6= 0 P n→∞ for all n ∈ N. Hence by Theorem 2.1 (b) we have that ∞ n=1 (1−λ n ) = ∞, where λn = |gn0 (0)|, and therefore that Gn (w) −→ 0, for all w ∈ D, by Theorem 2.1 (a). By Lemma 3.1 (a) again, n→∞ distUn (f n (z), f n (z0 )) −→ 0, for all z ∈ U0 . We conclude that distUn (f n (z), f n (z 0 )) −→ 0, n→∞ n→∞ for all z, z 0 ∈ U0 , by the triangle inequality, which is case (1). P We have shown that (3.1) implies that ∞ n=1 (1 − Pλn ) = ∞ and that this implies that U0 is contracting. Thus U0 is contracting if and only if ∞ n=1 (1 − λn ) = ∞, where λn = |gn0 (0)| = kDf (f n−1 (z0 ))kU Un−1 = λn (z0 ), for n ∈ N, n by Lemma 3.1 (b). Now suppose that there exist z, z 0 ∈ U0 and N ∈ N with distUn (f n (z), f n (z 0 )) = c(z, z 0 ) > 0, for all n ≥ N. (3.2) Then, by the Schwarz-Pick Lemma, f : Un → Un+1 is an isometry, for all n ≥ N , and so for every pair z, z 0 ∈ U0 we have that distUn (f n (z), f n (z 0 )) = distUN (f N (z), f N (z 0 )), for n ≥ N. Thus, if (z, z 0 ) ∈ (U × U ) \ E we have that distUn (f n (z), f n (z 0 )) = c(z, z 0 ) > 0 for all n ≥ N and that U0 is eventually isometric, which is case (3). In this case, λn (z) = 1 for all z ∈ U0 and for n ≥ N , by the Schwarz-Pick Lemma, as required. Finally, we show that case (2) is the only other possibility. It follows from the above proof that, if there exists z00 ∈ U0 for which neither (3.1) nor (3.2) holds, then the only possibility is that neither of these conditions hold for any z ∈ U0 ; that is, U0 is semi-contracting, which is case (2). This completes the proof of Theorem A. 3.2 Subclassification of contracting wandering domains: Proof of Theo- rem B In this subsection we prove Theorem B, which gives sufficient conditions for a simply con- nected wandering domain to be strongly contracting or super-contracting. We prove parts (a) and (b) by using the results of Section 2 together with Lemma 3.1. 18 Let U be a simply connected wandering domain of a transcendental entire function f and let Un denote the Fatou component containing f n (U ), for n ≥ 0. Let z0 ∈ U0 and let gn , Gn be as defined in the beginning of this section. Also, for n ∈ N, we let λn = λn (z0 ) = kDf (f n−1 (z0 ))kU n Un−1 and note from Lemma 3.1 (b) that λn = |gn0 (0)|. P To prove part (a), observe that if lim supn→∞ n1 nk=1 λk = a < 1, then it follows from Theorem 2.6 (a) that |Gn (w)| = O(cn ) as n → ∞, for w ∈ D, c ∈ (a, 1). So, by Lemma 3.1 (a), if we take z ∈ U0 and put w = ϕ0 (z), then distUn (f n (z), f n (z0 )) = O(cn ) as n → ∞, for c ∈ (a, 1). This proves part (a) of Theorem B. 1 Pn To prove part (b), we note that, if limn→∞ n k=1 λk = 0, then, from Theorem 2.6 (b), (distUn (f n (z), f n (z0 )))1/n → 0 as n → ∞, and hence U0 is super-contracting. To prove part (c) we need to show that, for n ∈ N, z ∈ U0 , n n 1X 1X lim sup λk (z) = lim sup λk , for z ∈ U0 . (3.3) n→∞ n n→∞ n k=1 k=1 P (Recall that λk = λk (z0 ).) We begin by supposing that lim supn→∞ n1 nk=1 λk = a < 1 and fix c ∈ (a, 1) and z ∈ U0 . From part (a) above, there exists C > 0 such that distUk (f k (z), f k (z0 )) ≤ Cck , for k ∈ N. (3.4) We now use the following result of Beardon and Minda to obtain a bound on the difference between λk (z) and λk . Lemma 3.2 ([BM07, Theorem 11.2]). Let U, V be hyperbolic domains and let f : U → V be holomorphic. Then distD (kDf (z)kVU , kDf (w)kVU ) ≤ 2 distU (z, w), for all z, w ∈ U. It follows from Lemma 3.2 together with (3.4) that, under our supposition, distD (λk (z), λk ) ≤ 2Cck , for k ∈ N. Since Z Z λk (z) λk (z) 2 dt distD (λk (z), λk ) = ≥ 2 dt = 2|λk (z) − λk |, λk 1 − t2 λk it follows that |λk (z) − λk | ≤ Cck , for k ∈ N. 19 1 Pn So, if lim supn→∞ n k=1 λk = a < 1, then n n n 1X 1X CX k lim sup λk (z) ≤ lim sup λk + lim sup c n→∞ n n→∞ n n→∞ n k=1 k=1 k=1 n 1X = lim sup λk = a. n→∞ n k=1 Since the rolesPof z0 and z are interchangeable, we have shown that (3.3) holds whenever lim supn→∞ n1 nk=1 λk < 1. The only remaining case is that n n 1X 1X lim sup λk (z) = 1 = lim sup λk . n→∞ n n→∞ n k=1 k=1 This completes the proof of Theorem B. 3.3 Rate of contraction in parabolic components It is clear that if a wandering domain U is the lift of an attracting component V , then U is strongly contracting and, if V is super-attracting, then U is super-contracting. We end this section by showing that if a wandering domain U occurs as a lift of a parabolic component, then U is contracting but not strongly contracting. We need the following lemma; see [Sha93, p. 157], for example. Lemma 3.3. If G is a simply connected domain, not the whole complex plane, then for z, w ∈ G,   1 |z − w| distG (z, w) ≥ log 1 + . 2 min{dist(z, ∂G), dist(w, ∂G)} We have the following general result about the contraction rate in a parabolic component. The estimates in this result may be known but we are not aware of a reference. Theorem 3.4. Let V be an invariant parabolic component of a transcendental entire func- tion f . Then, for all z0 , z00 ∈ V , either f m (z0 ) = f m (z00 ) for some m ∈ N or there exist positive constants k and K depending on z0 , z00 and p, the number of petals, such that k K ≤ distV (f n (z0 ), f n (z00 )) ≤ , for n ∈ N. (3.5) n n Proof. Without loss of generality we assume that 0 is the parabolic fixed point in ∂V and let p be the number of petals of f at 0. The following proof uses detailed estimates from the dis- cussion of Abel’s functional equation in [Bea91, pages 110–122] and we start by summarising this discussion, mainly using the notation from [Bea91]. First, the function f is conformally conjugate near 0 to a function of the form F (z) = z − z p+1 + O(z 2p+1 ) as z → 0. Substituting w = z −p , z = w−1/p , where w−1/p denotes the principal root, we obtain  p g(w) = 1/ F (w−1/p ) = w + p + A/w + O(1/w1+1/p ) as w → ∞, 20 for some constant A, from which it follows that there exists a parabola-shaped domain of the form Π = {u + iv : v 2 > 4K(K − u)}, K > 0, that is forward invariant under g. For w ∈ Π, we have A g n (w) = np + log n + un (w), for n ∈ N, (3.6) p where the functions un are holomorphic in Π and converge locally uniformly on Π to a univalent function u, which satisfies u(g(w)) = u(w)+p, a form of Abel’s functional equation. −1/p Now suppose that w0 , w00 ∈ Π are distinct points and put z0 = w0 and z00 = (w00 )−1/p . Then z0 and z00 lie in the invariant petal-shaped domain for F at 0, which corresponds to Π under the mapping w 7→ w−1/p and which is symmetric with respect to the positive real axis, subtending an angle of 2π/p at 0. It follows from the above properties of the functions un and the univalence of u that g n (w00 ) − g n (w0 ) = un (w00 ) − un (w0 ) → u(w00 ) − u(w0 ) 6= 0 as n → ∞. (3.7) On substituting z = w−1/p we find that zn = F n (z0 ) and zn0 = F n (z00 ) both approach 0 tangentially to the positive real axis through the petal-shaped domain mentioned above. Moreover, by (3.6), 1 1 |zn | = |g n (w0 )|−1/p = A ∼ as n → ∞, (3.8) |np + p log n + un (w0 )|1/p (np)1/p and 1 1 |zn0 | = |g n (w00 )|−1/p = A 0 ∼ as n → ∞. (3.9) |np + p log n + un (w0 )|1/p (np)1/p Also, by (3.6), (3.7) and a short calculation, |u(w0 ) − u(w00 )| |zn − zn0 | = |g n (w0 )−1/p − g n (w00 )−1/p | ∼ as n → ∞. (3.10) p(np)1+1/p Since F is conformally conjugate to f near 0, we deduce that these estimates for zn and zn0 hold if zn = f n (z0 ) and zn0 = f n (z00 ), for n ∈ N, where z0 and z00 are redefined to be the corresponding points in the invariant parabolic component V for f . Also, note that V is one of p distinct invariant parabolic components for f at 0, each containing an invariant petal-shaped domain subtending an angle of 2π/p at 0. Therefore, by Lemma 3.3, (3.8), (3.9) and (3.10), together with the fact that dist(zn , ∂V ) ≤ |zn |, we have   0 1 |zn − zn0 | distV (zn , zn ) ≥ log 1 + 2 min{dist(zn , ∂V ), dist(zn0 , ∂V )} 1/n1+1/p k ≥ k 1/p = , for n ∈ N, 1/n n for some positive constant k depending on z0 , z00 and p. Finally, for n ∈ N, let γn denote the line segment joining zn to zn0 . Then, in view of the fact that zn and zn0 approach 0 tangentially to the positive real axis, the line segment γn , 21 for n sufficiently large, lies in the invariant petal-shaped domain in V , which is conformally equivalent near 0 to the petal-shaped domain obtained from Π. Also, for n sufficiently large, we have 1 min{dist(zn , ∂V ), dist(zn0 , ∂V )} ≥ sin(π/p)|zn |. 2 Therefore, by the standard hyperbolic density estimate in a simply connected domain (see, for example, [CG93, page 13]), (3.8), (3.9), and the triangle inequality, we have Z 0 distV (zn , zn ) ≤ ρV (z) |dz| γ Zn 2 ≤ |dz| γn dist(z, ∂V ) 2|zn − zn0 | ≤ min{dist(zn , ∂V ), dist(zn0 , ∂V )} − 12 |zn − zn0 | 1/n1+1/p K ≤ K = , 1/n1/p n for some positive constant K depending on z0 , z00 and p, and n sufficiently large. Finally, we note that, for all pairs of points z0 , z00 ∈ V with disjoint orbits, we have f (z0 ), f n (z00 ) ∈ V for n sufficiently large. This completes the proof. n Remark. Using a more careful analysis of the size of the hyperbolic density in V near the points zn and zn0 we can show that the estimate (3.5) in Theorem 3.4 can be replaced by c distV (f n (z0 ), f n (z00 )) ∼ as n → ∞, n for some positive constant c depending on z0 , z00 and p. The proof uses results about the behaviour of a Riemann map from a sector of angle 2π/p onto V which maps 0 to 0, justified by using standard results about angular derivatives of conformal mappings at boundary points. By conformality and Definition 1.2, we have the following corollary of Theorem 3.4. Corollary 3.5. Let U be a simply connected wandering domain that is the lift of an invariant parabolic component V and let Un be the Fatou component containing f n (U ), for n ≥ 0. Then, for all z0 , z00 ∈ U , either f m (z0 ) = f m (z00 ) for some m ∈ N or there exist positive constants k and K depending on z0 and z00 such that k K ≤ distUn (zn , zn0 ) ≤ , for n ∈ N. n n In particular, U is contracting but not strongly contracting. Here are two examples of simply connected wandering domains, obtained by lifting parabolic components, which are contracting but not strongly contracting. Example 1. Consider the entire functions f (z) = z + e−z + 2πi, g(z) = z + e−z and F (w) = we−w . 22 Then both f and g are obtained by lifting F under the exponential function w = e−z . Since F has an invariant parabolic component associated with the fixed point at 0, the function g has congruent unbounded invariant Baker domains Un , n ∈ Z, such that Un ⊂ {z : (2n − 1)π < Im(z) < (2n + 1)π}; see [Dom98, FH06]. Since J(f ) = J(g), by [Ber95b], the components Un form a sequence of simply connected wandering domains which, by Corollary 3.5, are contracting but not strongly contracting. Figure 3: Left: Dynamical plane of F in Example 2. The super-attracting basin of w = 0 is shown in light blue, while in gray we see the parabolic basin of w = 1. Right: Dynamical plane of f . In blue the Baker domain (lift of the superattracting basin). In black the parabolic invariant basin at z = 0. In gray the wandering domains. The range is [−9, 9] × [−9, 9]. Example 2. As another example, consider f (z) = 2z + 1 − ez and F (w) = e w2 e−w , which belongs to the same family as the Example in Figure 1, both closely related to an example of Bergweiler [Ber95a]. In this case, f is a lift of F under w = ez , and F has an invariant parabolic component associated with the fixed point at 1 which lifts to congruent, bounded, simply connected Fatou components, Vn , n ∈ Z, say, of f such that 0 ∈ ∂V0 and Vn = V0 + 2nπi, for n ∈ Z, and f (Vn ) = V2n , for n ∈ Z. From this it follows that V2n , n ≥ 1, is a sequence of bounded, escaping, simply connected wandering domains which, by Corollary 3.5, are contracting but not strongly contracting. 4 Convergence to the boundary: Proof of Theorem C In this section we give the proof of Theorem C, the classification of simply connected wander- ing domains in terms of whether orbits of points converge to the boundary. Recall that the 23 Euclidean distance of a point z from the boundary of a hyperbolic domain U is closely related to the hyperbolic density ρU (z) of the point in the domain. Indeed, if U is a simply con- nected wandering domain of a transcendental entire function f , Un is the Fatou component containing f n (U ), for n ≥ 0, and z ∈ U0 , then by standard estimates [CG93, page 13] distEucl (f n (z), ∂Un ) → 0 ⇐⇒ ρUn (f n (z)) → ∞. We prove Theorem C by considering the hyperbolic densities ρUn (f n (z)). In fact, we show that a trichotomy as in Theorem C occurs if we consider the quantities an ρUn (f n (z)), for any sequence an and not just for an = 1. As we mentioned in the introduction, the issue of convergence to the boundary is somehow delicate in that it is tightly connected to the shape of the wandering domains, and there may be situations where it is more appropriate to use an alternative definition involving different sequences an . For example, if the domains Un are shrinking then it may make sense to say that zn converges to the boundary if an ρUn (f n (z)) → ∞ as n → ∞ where an = sup{diam D : D is a disk contained in Un }. D In order to prove Theorem C we need the following lemma, which can be thought of as a Har- nack inequality for hyperbolic density in a simply connected domain; see [BC08, Lemma 6.2] for a similar type of result (with a different proof) for hyperbolic density in the unit disk. Lemma 4.1 (Estimate of hyperbolic quantities). Let U ⊂ C be a simply connected domain. Then, for all z, z 0 ∈ U , ρU (z 0 ) exp(−2 distU (z, z 0 )) ≤ ≤ exp(2 distU (z, z 0 )). ρU (z) Proof. Let z, z 0 ∈ U and let ϕ : D → U be a Riemann map with ϕ(0) = z and ϕ(r) = z 0 , for some r ∈ [0, 1). By conformal invariance of the hyperbolic metric, together with (2.1), 1+r distU (z, z 0 ) = distD (0, r) = log , 1−r and, by the definition of the hyperbolic density on U , ρU (z) = ρD (0)/|ϕ0 (0)| = 2/|ϕ0 (0)| 2  0 ρU (z 0 ) = ρD (r)/|ϕ0 (r)| = |ϕ (r)| . 1 − r2 Also, by a standard distortion theorem for conformal maps [Pom92, p. 9], 1−r |ϕ0 (r)| 1+r 3 ≤ 0 ≤ . (1 + r) |ϕ (0)| (1 − r)3 Putting everything together we obtain the lower bound, ρU (z 0 ) 1 |ϕ0 (0)| (1 − r)2 = ≥ = exp(−2 distU (z, z 0 )), ρU (z) 1 − r2 |ϕ0 (r)| (1 + r)2 and the upper bound follows by symmetry. 24 Remark It is easy to check that the inequalities in Lemma 4.1 are sharp in the case when U is C \ (−∞, 0] and the points z, z 0 lie on the positive real axis. We now prove the main result of this section. Theorem 4.2. Let U be a simply connected wandering domain of a transcendental entire function f , let Un be the Fatou component containing f n (U ), for n ≥ 0, and let (an ) be a real positive sequence. (a) If there is a subsequence nk → ∞ and a point z ∈ U0 such that ank ρUnk (f nk (z)) → ∞, then the same is true for all other points in U0 . (b) If there is a subsequence mk → ∞ and a point z ∈ U0 such that amk ρUmk (f mk (z)) is bounded, then the same is true for all other points in U0 . Proof. (a) Suppose that ank ρUnk (f nk (z)) → ∞ as k → ∞ and let z 0 ∈ U0 with z 0 6= z. By the contraction property of the hyperbolic metric, we have that distUn (f n (z), f n (z 0 )) ≤ distU0 (z, z 0 ) =: C, for n ∈ N. By Lemma 4.1, ρUn (f n (z 0 )) ≥ e−2C ρUn (f n (z)), for n ∈ N. Hence ank ρUnk (f nk (z 0 )) ≥ e−2C ank ρUnk (f nk (z)) → ∞ as k → ∞ (b) Now suppose that amk ρUmk (f mk (z)) ≤ M , for k ∈ N, and let z 0 ∈ U0 with z 0 6= z. Again, by the contraction property of the hyperbolic metric, we have that distUmk (f mk (z), f mk (z 0 )) ≤ distU0 (z, z 0 ) =: C. Now, applying Lemma 4.1 and interchanging z and z 0 , we obtain that ρUmk (f mk (z 0 )) ≤ e2C ρUmk (f mk (z)), which implies that amk ρUmk (f mk (z 0 )) ≤ e2C amk ρUmk (f mk (z)) ≤ M e2C , so amk ρUmk (f mk (z 0 )) is bounded, for k ∈ N. The result of Theorem C follows from Theorem 4.2, by taking an = 1, for n ≥ 0. 5 Constructing wandering domains We begin this section with the proof of Theorem D, which we then use together with an extension of Runge’s Approximation Theorem to prove Theorem 5.3. This result enables us to construct bounded simply connected wandering domains in which various different dynamical behaviours can be specified and is the main tool that we use to construct examples in Section 6. 25 5.1 Proof of Theorem D Let f be a transcendental entire function and let γn , Γn , nk , Lk , and D be as in Theorem D; see Figure 4. It follows from properties (a) and (b) of Theorem D that for each n, m ∈ N with n 6= m the curve γn is in ext γm and so, by property (c) and Montel’s theorem, there exist Fatou components Un such that int γn ⊂ Un , for n ≥ 0. (5.1) Notice that, a priori, the components Un need not be different from each other. One of our goals is to show that they are indeed different, by proving that Un ⊂ int Γn , for n ≥ 0. L0 Γ0 Lk Γ1 Γnk Γnk +1 Cnk +1 γ0 γ1 f f f γnk f C0 Cnk C1 γnk +1 D Figure 4: Sketch of the setup of the proof of Theorem D. By property (e), the domain D must contain an attracting fixed point and so it is contained in an attracting Fatou component, say V . It then follows by property (e) that for all k ≥ 0 the set Lk is contained in a union of Fatou components, Vk say, that maps into V . As above, notice that the Vk ’s may all be part of the same component. Since for every n we have that D ⊂S S γn ⊂ int Γn we deduce that U0 is not in the grand orbit of V and hence ext Γn while that n≥0 Un ∩ k≥0 Vk = ∅. Therefore dist(z 0 , ∂Uk ) < δk := max{dist(z, Lk ) : z ∈ Γnk }, for all z 0 ∈ Γnk ∩ Uk . (5.2) Note that Un is simply connected for n ≥ 0. Indeed, if Un is multiply connected for some n ≥ 0, then it is a wandering domain and by [BRS13, Theorem 1.2] there exists N > 0 such that f k (int γn ) contains an annulus A(rk , Rk ) for all k ≥ N with Rk /rk → ∞ as k → ∞. It follows by property (c) that A(rk , Rk ) is contained in int γn+k and this contradicts property (b). So Un must be simply connected for n ≥ 0. We now show that Un ⊂ int Γn , for n ≥ 0, using proof by contradiction. If there exists m ≥ 0 for which Um is not a subset of int Γm , then it follows from (5.1) and property (a) 26 that Um ∩ Γm 6= ∅ and so we can take zm ∈ int γm and zm 0 ∈ U ∩ Γ , and join them by a m m compact curve Cm ⊂ (Um ∩ int Γm ). Then, by properties (c) and (d), we can choose simple curves Cn , n ≥ m, such that Cn ⊂ f n−m (Cm ) ⊂ (Un ∩ int Γn ) and also Cn joins zn := f n−m (zm )∈ int γn to a point zn0 ∈ Γn ∩ f n−m (Cm ) ⊂ Un , while Cn lies in int Γn . Such a curve Cn must also intersect γn . Then, on the one hand, since Cn ⊂ f n−m (Cm ) and f n−m : Um → Un is a hyperbolic contraction, we have that lengthUn Cn ≤ lengthUn f n−m (Cm ) ≤ lengthUm Cm < ∞, (5.3) for all n ≥ m. On the other hand, by Lemma 3.3 and (5.2), for nk ≥ m, we have lengthUn Cnk ≥ distUnk (znk , zn0 k ) k   1 |znk − zn0 k | ≥ log 1 + 2 min{dist(znk , ∂Unk ), dist(zn0 k , ∂Unk )}   1 |znk − zn0 k | ≥ log 1 + 2 dist(zn0 k , ∂Unk )   1 dist(γnk , Γnk ) ≥ log 1 + . 2 δk By property (f), this quantity tends to infinity as k → ∞, which contradicts (5.3), so Um ⊂ int Γm and hence Um is a bounded wandering domain by property (b). Finally, suppose that, for some n ≥ 0, there exists zn ∈ int γn such that both f (γn ) and f (Γn ) wind dn times round f (zn ). Since f (Γn ) winds dn times around f (zn ), we deduce that f takes the value f (zn ) exactly dn times in int Γn . Similarly, f takes the value f (zn ) exactly dn times in int γn . Hence f takes the value f (zn ) exactly dn times in Un . Since Un is a bounded Fatou component, f : Un → Un+1 is a proper map; since the above argument holds for a neighbourhood of f (zn ), we deduce that the degree of f on Un is equal to dn . 5.2 Main construction result In the proof of our main construction result, Theorem 5.3 below, we use the following ex- tension of the main lemma in [EL87], which is a strong version of the well-known Runge’s Approximation Theorem. Lemma 5.1 (Approximating on infinitely many compact sets). Let (En ) be a sequence of compact subsets of C with the following properties: (i) C \ En is connected, for n ≥ 0; (ii) En ∩ Em = ∅, for n 6= m; (iii) min{|z| : z ∈ En } → ∞ as n → ∞. S Suppose ψ is holomorphic on E = ∞ n=0 En and j ∈ N. For n ≥ 0, let εn > 0 and let zn,i ∈ En , 1 ≤ i ≤ j. Then there exists an entire function f satisfying, for n ≥ 0, |f (z) − ψ(z)| < εn , for z ∈ En ; (5.4) 0 0 f (zn,i ) = ψ(zn,i ), f (zn,i ) = ψ (zn,i ), for 1 ≤ i ≤ j. (5.5) 27 The main lemma in [EL87] allows for one point zn in every compact set at which f and f 0 can be specified, but its proof can easily be modified to hold for finitely many points in every En , as stated above. The following lemma will also be used in the proof of Theorem 5.3. Lemma 5.2 (Hyperbolic distance on disks). Suppose that 0 < s < r < 1 < R and set 1 − s2 c(s, R) = , Dr = D(0, r) and DR = D(0, R). R − s2 /R If |z|, |w| ≤ s, then distDR (z, w) = distD (z/R, w/R) ≥ c(s, R) distD (z, w), (5.6) and 1 distDr (z, w) = distD (z/r, w/r) ≤ distD (z, w). (5.7) c(s/r, 1/r) Also, 0 < c(s, R) < 1 and if the variables s, r and R satisfy in addition 1 − r = o(1 − s) as s → 1 and R − 1 = O(1 − r) as r → 1, (5.8) then c(s, R) → 1 as s → 1, (5.9) and c (s/r, 1/r) → 1 as s → 1. (5.10) Proof. Suppose that 0 < s < r < 1 < R and take z, w ∈ D with |z|, |w| ≤ s. Let γ be the hyperbolic geodesic in D joining z/R to w/R. Then Z 2 |dt| distD (z/R, w/R) = 2 . γ 1 − |t| Now substitute ζ = Rt, t ∈ D, so |dζ| = R|dt|. Also let Rγ := {Rz : z ∈ γ}. Since R > 1, we have Z Z 1 2 |dζ| 2 |dζ| distD (z/R, w/R) = 2 2 =R 2 2 . R Rγ 1 − |ζ| /R Rγ R − |ζ| Now for ζ ∈ Rγ we have |ζ| ≤ s, so R2 − |ζ|2 R2 − s2 ≤ , for ζ ∈ Rγ. 1 − |ζ|2 1 − s2 Hence Z Z 2 |dζ| R(1 − s2 ) 2 |dζ| distD (z/R, w/R) = R ≥ ≥ c(s, R) distD (z, w), Rγ R2 − |ζ|2 R 2 − s2 Rγ 1 − |ζ|2 since Z  2 |dζ| 0 distD (z, w) = min : for all paths γ joining z to w in D . γ0 1 − |ζ|2 28 This proves (5.6). Next, z w s , distDr (z, w) = distD (z/r, w/r) and ≤ < 1. r r r Hence, by (5.6), with r and R replaced by s/r and 1/r, and z, w replaced by z/r and w/r, we obtain 1 1 distD (z/r, w/r) ≤ distD1/r (z/r, w/r) = distD (z, w). c(s/r, 1/r) c(s/r, 1/r) This proves (5.7). It is clear that 0 < c(s, R) < 1 since 0 < s < 1 < R. Finally, suppose that (5.8) holds. Then R − 1 = O(1 − r) = o(1 − s) as s → 1 and hence R(1 − s)(1 + s) R(1 − s) 1+s c(s, R) = = → 1 as s → 1, (R − s)(R + s) 1 − s + o(1 − s) R + s and (r − s)(1 + s/r) 1 − s + o(1 − s) 1 + s/r c(s/r, 1/r) = = → 1 as s → 1, (1 − s)(1 + s) 1−s 1+s which give (5.9) and (5.10). We now give our main construction result, which we use in Section 6 to construct examples. In these examples, we shall prescribe the orbits of at most two points z1 , z2 ∈ D(0, r0 ), although the result below allows us to prescribe the orbits of any finite number of points in D(0, r0 ). Theorem 5.3 (Main construction). Let bn , n ∈ N, be a sequence of Blaschke products of corresponding degrees dn ≥ 1 and let Tn , n ≥ 0, be the sequence of translations z 7→ z + 4n and Dn , n ≥ 0, be the sequence of disks Dn = {z : |z − 4n| < 1}. Suppose also that j ∈ N and zi ∈ D0 , 1 ≤ i ≤ j. Then there exists a transcendental entire function f having an orbit of bounded, simply connected, escaping, wandering domains Un such that, for n ≥ 0, (i) ∆0n := D(4n, rn ) ⊂ Un ⊂ D(4n, Rn ) := ∆n , where 0 < rn < 1 < Rn and rn , Rn → 1 as n → ∞; (ii) fn+1 := Tn+1 ◦ bn+1 ◦ Tn−1 is holomorphic on ∆n , and |f (z) − fn+1 (z)| → 0 uniformly on ∆n as n → ∞; (iii) f n (zi ) = Fn (zi ) and f 0 ((f n )(zi )) = fn+1 0 (Fn (zi )), 1 ≤ i ≤ j, where Fn = fn ◦ · · · ◦ f1 ; (iv) f : Un → Un+1 has degree dn+1 . Finally, if z, z 0 ∈ D(0, r0 ), then we have kn distDn (f n (z), f n (z 0 )) ≤ distUn (f n (z), f n (z 0 )) ≤ Kn distDn (f n (z), f n (z 0 )), (5.11) where 0 < kn < 1 < Kn and kn , Kn → 1 as n → ∞. 29 Fn L0 L1 Ln−1 ∆n−1 ∆0 ∆1 Ln ∆n f1 f2 fn−1 ∆0n−1 fn ∆0n ∆00 ∆01 0 4 4(n − 1) 4n Figure 5: Sketch of the setup of Theorem 5.3. In green, the disks Dn centred at 4n. In blue, the boundaries of the disks of radii rn and Rn in between which lie the boundaries of the wandering domains. In red, the curves Ln introduced in the proof. Proof. For n ≥ 0, let dn Y iθn z + an,j bn (z) = e , 1 + an,j z j=1 where an,j ∈ D are not necessarily different from each other, and θn ∈ [0, 2π). We first define the increasing sequence (rn ) and the decreasing sequence (Rn ) inductively. These sequences determine the following circles which play a key role in the proof (see Fig- ure 5): γn = {z : |z − 4n| = rn } and Γn = {z : |z − 4n| = Rn }. (5.12) First, take R0 ∈ (1, 3/2) such that R0 < 1/maxj {|a1,j |}}, which ensures that b1 is holomorphic inside and in a neighborhood of Γ0 , and take r0 ∈ (1/2, 1) such that r0 > maxi |zi | and also such that b1 (z) = w has exactly d1 solutions in D(0, r0 ) for w ∈ D(0, 1/2). Now assume that rk , Rk have been chosen for k = 0, . . . , n − 1, for some n ∈ N. We choose rn and Rn so that the following statements all hold:   1 − rn−1 2 0 < 1 − rn ≤ min , dist(fn (γn−1 ), ∂Dn ) ; (5.13) 2 fn+1 (γn ) winds exactly dn+1 times round D(4n, 1/2); (5.14)   Rn−1 − 1 1 1 0 < Rn − 1 ≤ min , 1 − rn , dist(fn (Γn−1 ), ∂Dn ), − 1 . (5.15) 2 2 maxj {|an+1,j |} These properties prescribe the values rn and Rn , and hence the circles γn and Γn . In partic- ular, by (5.13) and (5.15), the sequence (rn ) increases to 1 and the sequence (Rn ) decreases to 1, and the maps fn+1 , n ≥ 0, defined in property (ii), satisfy γn+1 surrounds fn+1 (γn ), (5.16) fn+1 (Γn ) surrounds Γn+1 . (5.17) 30 Our aim is to use Lemma 5.1 to approximate all the maps fn by a single entire function f such that, for n ≥ 0, γn+1 surrounds f (γn ) and f (Γn ) surrounds Γn+1 . We first define δn = Rn − rn , for n ≥ 0, (5.18) and observe that δn → 0 as n → ∞. We then define Ln , n ≥ 0, to be the curve Ln := {z : |z − 4n| = Rn + δn2 /2, | arg(z)| ≤ π − δn2 }, (5.19) so max{dist(z, Ln ) : z ∈ Γn } ≤ 2δn2 , for n ≥ 0, (5.20) and define the error quantities   1 1 εn = min dist(fn (γn−1 ), ∂Dn ), dist(fn (Γn−1 ), ∂Dn ), δn /4 > 0, (5.21) 4 4 for n ≥ 1. Since 0 < εn ≤ δn /4, we have that εn < δ0 /4 < 1/4, n ≥ 1, and εn → 0 as n → ∞. We now apply Lemma 5.1 to the sets E0 = D(−4, 1) and E2k+1 = Lk , E2k+2 = ∆k , for k ≥ 0, with the function ψ defined by   −4, if z ∈ D(−4, 1), ψ(z) = −4, if z ∈ Ln , n ≥ 0,   fn+1 (z), if z ∈ ∆n , n ≥ 0. Lemma 5.1 allows us to choose finitely many points zn,i , 1 ≤ i ≤ j, in each set En where S we do the approximation. The choice of these points in D(−4, 1) ∪ ∞ k=0 Lk plays no role in our argument. In E2 = ∆0 we choose z2,i = zi ∈ D0 , 1 ≤ i ≤ j, and in E2k+2 = ∆k , k ≥ 1, we choose z2k+2,i = Fk (zi ), 1 ≤ i ≤ j, where Fk = fk ◦ · · · ◦ f1 . It then follows from Lemma 5.1 that there exists an entire function f such that, for n ≥ 0, |f (z) − fn+1 (z)| < εn+1 , for z ∈ ∆n ; (5.22) |f (z) + 4| ≤ 1/2, for z ∈ Ln ; (5.23) |f (z) + 4| ≤ 1/2, for z ∈ D(−4, 1); (5.24) n f (zi ) = Fn (zi ), for 1 ≤ i ≤ j; (5.25) f 0 ((f n )(zi )) = f 0 (Fn (zi )) = fn+1 0 (Fn (zi )), for 1 ≤ i ≤ j. (5.26) It follows from (5.13), (5.15), (5.22) and (5.21) that, for n ≥ 0, γn+1 surrounds f (γn ) (which surrounds the point 4(n + 1)); (5.27) f (Γn ) surrounds Γn+1 . (5.28) We now apply Theorem D to the Jordan curves γn , Γn , n ≥ 0, the compact curves Ln , n ≥ 0, and the bounded domain D = D(−4, 1) ⊂ E0 , noting that these sets satisfy the required 31 hypotheses. Indeed, the hypotheses (a) and (b) are clearly true, (c) follows from (5.27), (d) follows from (5.28), (e) holds by (5.23) and (5.24), and (f) holds by (5.18) and (5.20). Part (i) of our result now follows from Theorem D, part (ii) is true by construction, and part (iii) follows from (5.25) and (5.26). We now show that part (iv) holds. By (5.21) and (5.22), we can write f (z) = fn+1 (z) + en+1 (z) for some holomorphic map en (z) which satisfies |en+1 (z)| < 1/4, for z ∈ ∆n . By (5.14), we have |fn+1 (z) − 4(n + 1)| ≥ 1/2, for z ∈ γn = ∂∆0n . It follows from this together with the fact that |en (z)| < 1/4, for z ∈ ∆n and (5.14) that |f (z) − 4(n + 1)| ≥ 1/4, for z ∈ γn , (5.29) and f (γn ) winds exactly dn+1 times around 4(n + 1), so f takes the value 4(n + 1) exactly dn+1 times in ∆0n . Similarly, by (5.14), (5.21) and (5.22), f (Γn ) winds exactly dn+1 times around 4(n + 1), so f takes the value 4(n + 1) exactly dn+1 times in ∆n . Therefore, by the final statement of Theorem D, f : Un → Un+1 has degree dn+1 . It remains to prove the double inequality (5.11), which compares the hyperbolic distances in Un between points of two orbits under f with the corresponding hyperbolic distances in the disks Dn . To do this, we let sn := 1 − 34 dist(fn (γn−1 ), ∂Dn ), for n ≥ 1, and note that, if z, z 0 ∈ D(0, r0 ), then f n (z), f n (z 0 ) ∈ D(4n, sn ) ⊂ ∆0n , for n ∈ N, by (5.16), (5.21) and (5.22). Now 1 − rn = o(1 − sn ) as n → ∞, by (5.13), and Rn − 1 ≤ 1 − rn , by (5.15), so the properties (5.8) hold for the sequences (sn ), (rn ) and (Rn ). Also, dist∆n (f n (z), f n (z 0 )) ≤ distUn (f n (z), f n (z 0 )) ≤ dist∆0n (f n (z), f n (z 0 )), since ∆0n ⊂ Dn ⊂ ∆n . Therefore, we deduce from Lemma 5.2 (translated to the disks Dn ) that 1 c(sn , Rn ) distDn (f n (z), f n (z 0 )) ≤ distUn (f n (z), f n (z 0 )) ≤ distDn (f n (z), f n (z 0 )) c(sn /rn , 1/rn ) and c(sn , Rn ) → 1 as n → ∞, c(sn /rn , 1/rn ) → 1 as n → ∞, which gives (5.11). 6 Examples: Proof of Theorem E In this section we construct the examples described in Theorem E. In every case we use Theorem 5.3 and the notation there. Hence (bn ) denotes the sequence of Blaschke products of degree dn ≥ 1; (Tn ) the sequence of real translations z 7→ z + 4n; and (Dn ) the sequence of disks Dn = {z : |z − 4n| < 1}, n ≥ 0. Moreover, for n ∈ N, we set Bn = bn ◦ · · · ◦ b1 , −1 fn = Tn ◦ bn ◦ Tn−1 , and Fn = fn ◦ · · · ◦ f1 , so Fn = Tn ◦ Bn ; see Figure 6. 32 Fn f1 f2 fn−1 fn D0 D1 Dn−1 Dn Bn Tn Figure 6: The maps fn , Bn , Tn 6.1 Preliminary lemmas We first prove two lemmas that will be used in the constructions. Lemma 6.1. Let f be a transcendental entire function with an orbit of bounded, simply connected, wandering domains Un , n ≥ 0, arising from Theorem 5.3, with Blaschke products bn and associated functions Bn and Fn such that f n (0) = Fn (0), for n ∈ N. Then, we have the following cases. (a) If Bn (0) → 0 as n → ∞, then, for all z ∈ U0 , lim inf dist(f n (z), ∂Un ) > 0, n→∞ that is, all orbits stay away from the boundary. (b) If there exists a subsequence nk → ∞ with Bnk (0) → 1 and a different subsequence mk → ∞ with Bmk (0) → 0, then dist(f nk (z), ∂Unk ) → 0 for all z ∈ U0 , while lim inf dist(f mk (z), ∂Umk ) > 0, for all z ∈ U0 . k→∞ (c) If Bn (0) → 1 as n → ∞, then dist(f n (z), ∂Un ) → 0 for all z ∈ U0 , that is, all orbits converge to the boundary. Proof. It follows from Theorem C that all points in a simply connected wandering domain have the same limiting behaviour in relation to the boundary and so, in each case, it is sufficient to find just one point whose orbit behaves as required. We choose this point to be 0 ∈ U0 . If Bn (0) → 0 as n → ∞ then f n (0) − 4n = Fn (0) − 4n → 0 as n → ∞ 33 and so, by Theorem 5.3 part (i), we have lim inf dist(f n (0), ∂Un ) = 1 > 0, n→∞ which is sufficient to prove part (a). If Bn (0) → 1 as n → ∞ then f n (0) − (4n + 1) = Fn (0) − (4n + 1) → 0 as n → ∞ and so, by Theorem 5.3 part (i), we have dist(f n (0), ∂Un ) → 0 as n → ∞, which is sufficient to prove part (c). The proof of part (b) follows in a similar way. In some of our constructions we use the following properties about a specific family of Blaschke products of degree 2.  2 z+a Lemma 6.2. Let b(z) = 1+az , where 1/3 ≤ a < 1, and let 0 < r < s < 1. Then (a) the function b has a fixed point at 1, which is attracting if a > 1/3 and parabolic if a = 1/3, and bn (r) → 1 as n → ∞ for all a ≥ 1/3; (b) distD (bn (r), bn (s)) 9 0 as n → ∞ if a > 1/3; (c) distD (bn (r), bn+1 (r)) = O(1/n) as n → ∞ if a = 1/3. Proof. The proof of part (a) is straightforward. For part (b) note first that Z bn (s) Z bn (s) n n 2 dt dt 1 − bn (r) distD (b (r), b (s)) = ≥ = log . (6.1) bn (r) 1 − t2 bn (r) 1−t 1 − bn (s) Also, since 1 is an attracting fixed point of b when a > 1/3, there exist λ ∈ (0, 1) and d > c > 0 such that 1 − bn (r) ∼ dλn and 1 − bn (s) ∼ cλn as n → ∞. Hence, by (6.1), d lim distD ((bn (r), bn (s)) ≥ log > 0. n→∞ c For part (c) we use a similar approach. First note that Z bn+1 (r) n n+1 2 dt 1 − bn (r) distD (b (r), b (r)) ≤ = 2 log . (6.2) bn (r) 1−t 1 − bn+1 (r) When a = 1/3, we have b(1) = 1, b0 (1) = 1, b00 (1) = 0 and b000 (1) 6= 0, so 1 − bn (r) ∼ c/n1/2 as n → ∞, where c > 0; see (3.8). We deduce that   1 − bn (r) (n + 1)1/2 1 1/2 1 n+1 ∼ 1/2 = 1+ =1+ + O(1/n2 ) as n → ∞. 1−b (r) n n 2n The result now follows by putting this estimate into (6.2). 34 6.2 The nine types of simply connected wandering domains We now prove part (a) of Theorem E by constructing examples corresponding to each of the nine cases given in Theorems A and C. The following maps will play key roles in the constructions:   z+a 2 b(z) = , for 1/3 ≤ a < 1, 1 + az z + an z − a2n µn (z) = en (z) = and µ , for n ∈ N, (6.3) 1 + an z 1 − a2n z where an ∈ (0, 1) is an arbitrary sequence satisfying an → 1 as n → ∞. Examples 1, 2 and 3, which follow, correspond to the three cases of Theorem A. Within each of them we give three functions, corresponding to the three cases of Theorem C. Example 1 (Three contracting wandering domains). For each of the cases (a), (b) and (c) of Theorem C, there exists a transcendental entire function f having an orbit of bounded, simply connected, escaping contracting wandering domains Un , n ≥ 0, with the stated behaviour: (a) for all z ∈ U0 , lim inf dist(f n (z), ∂Un ) > 0, n→∞ that is, all orbits stay away from the boundary; (b) there exists a subsequence nk → ∞ for which dist(f nk (z), ∂Unk ) → 0 for all z ∈ U , while for a different subsequence mk → ∞ we have that lim inf dist(f mk (z), ∂Umk ) > 0, for z ∈ U0 ; k→∞ (c) dist(f n (z), ∂Un ) → 0 for all z ∈ U0 , that is, all orbits converge to the boundary. Proof. (a) Let bn (z) = z 2 , for n ∈ N, and apply Theorem 5.3 with the points z1 = 0 and z2 = 1/2. For n ∈ N, we have distDn (f n (0), f n (1/2)) = distD (Fn (0), Fn (1/2)) = distD (Bn (0), Bn (1/2)) n = distD (0, 1/22 ) → 0 as n → ∞. It follows from (5.11) that distUn (f n (0), f n (1/2)) → 0 as n → ∞. By Theorem A, this is sufficient to show that U0 is contracting. Since Bn (0) = 0, for n ∈ N, the result now follows from case (a) of Lemma 6.1. (b) In this case, for n ∈ N, we let 35   2 if n = 3k − 2, k ≥ 1, z bn (z) = µk (z), if n = 3k − 1, k ≥ 1,   −1 µk (z), if n = 3k, k ≥ 1, where µk is as defined in (6.3). As in case (a), we apply Theorem 5.3 with z1 = 0 and z2 = 1/2. For k ∈ N, we have distD3k (f 3k (0), f 3k (1/2)) = distD (F3k (0), F3k (1/2)) = distD (B3k (0), B3k (1/2)) k = distD (0, 1/22 ) → 0 as k → ∞. As in case (a), this is sufficient to show that U0 is contracting. Since B3k (0) = 0, for k ∈ N, and B3k−1 (0) = a3k−1 → 1 as k → ∞, the conclusion now follows from case (b) of Lemma 6.1.  2 z+1/3 (c) In this case we let bn (z) = b(z) = 1+z/3 , for n ∈ N, and we apply Theorem 5.3 with z1 = 0 and z2 = b(0). For n ∈ N, we have distDn (f n (0), f n (b(0))) = distD (Fn (0), Fn (b(0))) = distD (Bn (0), Bn (b(0))) = distD (bn (0), bn+1 (0)) → 0 as n → ∞, by Lemma 6.2(c). As before, this is sufficient to show that U0 is contracting. It also follows from Lemma 6.2(a) that Bn (0) = bn (0) → 1 as n → ∞ and the result now follows from case (c) of Lemma 6.1 Remark. All three cases of Example 1 are in fact super-contracting (see Definition 1.2). Example 2 (Three semi-contracting wandering domains). For each of the cases (a), (b) and (c) of Theorem C, there exists a transcendental entire function f having an orbit of bounded, simply connected, escaping, semi-contracting, wandering domains Un , n ≥ 0, with the stated behaviour. en ((µn (z))2 ), for n ∈ N, where µn and µ Proof. (a) In this case we let bn (z) = µ en are as defined in (6.3). We apply Theorem 5.3 with the points z1 = 0 and z2 = 1/2 2an A calculation shows that, for n ∈ N, we have bn (0) = 0 and b0n (0) = 1+a 2 → 1 as n → ∞. n Hence we can choose (an ) so that, in addition, ∞ X (1 − b0n (0)) < ∞. (6.4) n=1 It follows from Theorem 2.1(b) that Bn (1/2) 9 0 as n → ∞. Thus distDn (f n (0), f n (1/2)) = distD (Fn (0), Fn (1/2)) = distD (Bn (0), Bn (1/2)) = distD (0, Bn (1/2)) 9 0 as n → ∞. 36 It follows from (5.11) that distUn (f n (0), f n (1/2)) 9 0 as n → ∞, and so U0 is not contracting. Also, for n ∈ N, the Blaschke product bn has degree 2 and so, by Theorem 5.3 part (iv), f : Un−1 → Un has degree 2. Thus U0 is not eventually isometric and so it follows from Theorem A that U0 is semi-contracting. Since Bn (0) = 0, for n ∈ N, the result now follows from case (a) of Lemma 6.1. (b) In this case, for n ∈ N, we let   µk (z), if n = 3k − 2, k ≥ 1, bn (z) = z 2 , if n = 3k − 1, k ≥ 1,   µek (z), if n = 3k, k ≥ 1, where µk and µek are as defined in (6.3). Note the similarity to case (a), where each Blaschke product bn was defined to be the composite of the three maps above. As in case (a), we apply Theorem 5.3 with z1 = 0 and z2 = 1/2. Using similar arguments to those used in part (a), we can choose (an ) such that distU3k (f 3k (0), f 3k (1/2)) 9 0 as k → ∞, and so U0 is not contracting. Also, for k ∈ N, the Blaschke product b3k−1 has degree 2 and so, by Theorem 5.3 part (iv), f : U3k−2 → U3k−1 has degree 2. Thus U0 is not eventually isometric and so it follows from Theorem A that U0 is semi-contracting. Since B3k (0) = 0, for k ∈ N, and B3k−2 (0) = a3k−2 → 1 as k → ∞, the conclusion now follows from case (b) of Lemma 6.1.  2 z+a (c) In this case we choose a > 1/3 and let bn (z) = b(z) = 1+az , for n ∈ N. As in Example 1(c), we apply Theorem 5.3 with z1 = 0 and z2 = b(0). For n ∈ N, we have distDn (f n (0), f n (b(0))) = distD (Fn (0), Fn (b(0))) = distD (Bn (0), Bn (b(0))) = distD (bn (0), bn+1 (0)) 9 0 as n → ∞, by Lemma 6.2(b). Arguing as above, this is sufficient to show that U0 is not contracting. Also, for n ∈ N, the Blaschke product bn has degree 2 and so, as above, it follows that U0 is not eventually isometric. Hence, by Theorem A, it is semi-contracting. It also follows from Lemma 6.2(a) that Bn (0) = bn (0) → 1 as n → ∞ and the result now follows from case (c) of Lemma 6.1 Example 3 (Three eventually isometric wandering domains). For each of the cases (a), (b) and (c) of Theorem C, there exists a transcendental entire function f having an orbit of bounded, simply connected, escaping, eventually isometric, wandering domains Un , n ≥ 0, with the stated behaviour. 37 Proof. (a) In this case we let bn (z) = z, for n ∈ N, and apply Theorem 5.3 with z1 = 0. For n ∈ N, the map bn is univalent and so, by Theorem 5.3 part (iv), f : Un−1 → Un is also univalent. Thus U0 is eventually isometric. Since Bn (0) = 0, for n ∈ N, the result now follows from case (a) of Lemma 6.1. (b) In this case, for n ∈ N, we let ( µk (z), if n = 2k − 1, k ≥ 1, bn (z) = −1 µk (z), if n = 2k, k ≥ 1, where µk is as defined in (6.3). We apply Theorem 5.3 with z1 = 0. For n ∈ N, the map bn is univalent and so, as in case (a), U0 is eventually isometric. Since B2k (0) = 0, for k ∈ N, and B2k−1 (0) = a2k−1 → 1 as k → ∞, the conclusion now follows from case (b) of Lemma 6.1. (c) In this case, for n ∈ N, we let bn (z) = z+1/2 1+z/2 . As in cases (a) and (b), we apply Theorem 5.3 with z1 = 0 and, since the map bn is univalent, for n ∈ N, we deduce that U0 is eventually isometric. Since bn (x) > x, for x ∈ [0, 1), we deduce that Bn (0) → 1 as n → ∞. The conclusion now follows from case (c) of Lemma 6.1. 6.3 2−super–attracting wandering domains We prove part (b) of Theorem E by giving an example of a transcendental entire function with an orbit of wandering domains (Un ) containing two orbits consisting of critical points. The case of finitely many critical orbits is completely analogous. We choose the sequence of Blaschke products (bn ). We let z + a1 b1 (z) = z 2 , 1 + a1 z with a1 < 0 chosen so that 1/2 is a critical point, and, for n ∈ N define bn inductively by setting z + an+1 bn+1 (z) = z 2 , 1 + an+1 z with an+1 ∈ (−1, 1) chosen so that bn ◦ bn−1 ◦ · · · ◦ b1 (1/2) is a critical point of bn+1 . In this way we construct a sequence of Blaschke products (bn ) of degree 3 such that, for n ∈ N, we have bn (0) = 0 and the two critical points of bn are 0, Bn−1 (1/2). We now apply Theorem 5.3 with z1 = 0 and z2 = 1/2. We deduce that there exists a tran- scendental entire function f which has a sequence of bounded, simply connected, escaping, wandering domains (Un ) such that, for n ≥ 0, f n (0) = Fn (0) = 4n, f n (1/2) = Fn (1/2) and f 0 (f n (0)) = f 0 (f n (1/2)) = 0. Hence, there are two points in U0 , namely 0 and 1/2, whose orbits under f consist of critical points of f . Remark. It follows from Theorem 5.3 that distUn (f n (0), f n (1/2)) → 0 as n → ∞ and, in fact, one can check that these wandering domains are super-contracting. 38 7 Wandering domains whose boundaries are Jordan curves In this section we prove that, if the Blaschke products in Theorem 5.3 satisfy certain con- ditions, then the boundaries of the resulting wandering domains are Jordan curves. Apart from wandering domains arising from lifting constructions, as far as we are aware these are the first examples of simply connected wandering domains for which it is possible to obtain information concerning the boundary. Examples of multiply connected wandering domains for which it is known that connected components of the boundary are Jordan curves can be found in [Bis18] and in [Bau15]. In order for the boundaries of the resulting wandering domains to be Jordan curves, we need the Blaschke products in Theorem 5.3 to be uniformly expanding in the following precise sense. Definition 7.1 (Uniformly expanding Blaschke products). Let bn , n ∈ N, be a sequence of Blaschke products. We say that the Blaschke products in the sequence (bn ) are uniformly expanding if there exists ξ > 1 and an ε-neighborhood Uε of the unit circle such that 1. each bn is holomorphic in Uε , that is, bn has no poles in Uε ; 2. |b0n | ≥ ξ on Uε . Note that the second condition implies that the bn have no critical points in Uε . Our aim for this section is to prove the following theorem. Theorem 7.2. Let bn , n ∈ N, be a sequence of uniformly expanding Blaschke products such that maxn {deg bn } < ∞ and let Un , n ≥ 0, be the resulting orbit of wandering domains given by Theorem 5.3. Then, for n ≥ 0, the boundary of the wandering domain Un is a Jordan curve. The proof of Theorem 7.2 follows in outline the proof that the Julia sets of certain quadratic polynomials are Jordan curves (see [Bea91, Section 9.9], for example) but with very significant changes and novel arguments due to the fact that we are dealing with a lack of uniformity arising from the associated non-autonomous system of maps. The proof has three steps: 1. For each n ∈ N, we let An be the annulus bounded by the circles γn and Γn , which were defined in (5.12) and played a key role in the proof of Theorem 5.3. We consider the annulus Ăn lying in An between Γn and a component of f −1 (An+1 ) and show that the vertical geodesics of Ăn have uniformly bounded Euclidean length. 2. We then use pullbacks under f −n of these vertical geodesics together with the uniformly expanding property of the functions bn (and hence of f on the annuli An ) to induce a continuous map Σ from Γ0 to a closed curve Σ(Γ0 ), and a continuous map σ from γ0 to a closed curve σ(γ0 ). 3. Finally, we show that ∂U0 , which is squeezed between Σ(Γ0 ) and σ(γ0 ), is a Jordan curve. 39 The first step in the proof relies on a general geometric result of independent interest about the Euclidean lengths of vertical geodesics of annuli, which we prove using the Fejér-Riesz Inequality. We prove this and other preliminary geometric results in Section 7.1, and then give the proof of Theorem 7.2 in Section 7.2. 7.1 Preliminary results We begin with a result about pre-images of annuli bounded by Jordan curves. Related results appear in [Bis18, Lemma 11.1] and [RS11, Lemma 5]. In this lemma, we denote the inner boundary component of a topological annulus A by ∂Ainn and the outer boundary component by ∂Aout . As usual, when we say that f is holomorphic on A we mean that f is holomorphic on a neighborhood of A. Lemma 7.3. Let A and B be annuli with Jordan curve boundary components, both surround- ing 0, and let f be holomorphic on A, with f and f 0 non-zero on A. Suppose that ∂Binn surrounds f (∂Ainn ) and f (∂Aout ) surrounds ∂Bout . (7.1) Then A contains a unique component A b of f −1 (B), which is an annulus that surrounds 0 with Jordan curve boundary components that satisfy binn ) = ∂Binn f (∂ A and bout ) = ∂Bout . f (∂ A (7.2) Proof. Since f (∂A) ∩ B = ∅ we deduce that every connected component H of f −1 (B) which intersects A is in fact contained in A. Since f (∂A) intersects both components of B c , we deduce that f (A) ⊃ B and that there exists at least one component  of f −1 (B) that is contained in A. We now claim that  is doubly connected, surrounds 0, and that there are no other preimage components of B in A. Let H be any preimage component of B which is contained in A. By the Riemann–Hurwitz formula H is at least doubly connected. Let X be a bounded complementary component of H. Since f : H → B is proper, f (X) is the bounded complementary component of B, hence 0 ∈ f (X). If X does not contain 0, then X ⊂ A, on which f cannot take the value 0 by assumption, giving a contradiction. Hence every preimage component H of B in A is doubly connected and surrounds 0. Now suppose that there are two such components. Then one of them is contained in the bounded complementary component of the other, and hence maps to the bounded complementary component of B, again a contradiction since it is a preimage component of B. Since the boundary components of B are Jordan curves, the fact that f :  → B is proper, together with the fact that f 0 6= 0 on A, implies that the boundaries of  are also Jordan curves, and also implies (7.2). The main result in this subsection concerns the notion of a vertical foliation of an annulus which we now define. Definition 7.4 (Vertical foliations). Let A be an open annulus and consider the straight annulus Aρ = {z : ρ < |z| < 1} such that ϕ : Aρ → A is a biholomorphism. The vertical foliation FA of A consists of the image curves under ϕ of the radial segments connecting 40 the two circles which form the boundary of Aρ . Each of these image curves is a hyperbolic geodesic which we refer to as a vertical geodesic. We can now state the main result of this subsection, which concerns the Euclidean lengths of geodesics in vertical foliations of annuli. The proof uses the Fejér-Riesz Inequality (stated below); similar reasoning using instead the Gehring-Hayman Theorem (see [GH62] or [Pom92, Section 4]) is possible. Theorem 7.5. Let A be an annulus for which both boundary components are analytic Jordan curves with length at most S, and such that the bounded component of C \ A contains a disk of radius r > 0. Then there exists M = M (S, r) > 0 such that `Eucl (γ) ≤ M, for all γ ∈ FA . We prove Theorem 7.5 using the following technical lemma. Lemma 7.6. Let A be an annulus for which both boundary components are analytic Jordan curves with length at most S, and consider the straight annulus Aρ = {z : ρ < |z| < 1} such that ϕ : Aρ → A is a biholomorphism. For θ ∈ [0, 2π], let σθ := ϕ({reiθ : ρ < r < 1}) and `(θ) := `Eucl (σθ ). Then the Lebesgue measure of the set of θ such that `(θ) < 2S 1 ρ (1 − ρ) is at least 2π − 2 . Proof. Consider the integral Z 1Z 2π Z 1 Z 2π 0 I= |ϕ (re )| drdθ =iθ dr |ϕ0 (reiθ )| dθ. ρ 0 ρ 0 The function Z 2π I(r) := |ϕ0 (reiθ )| dθ, ρ < r < 1, 0 is a convex function of log r since |ϕ0 | can be extended to be subharmonic in a neighbourhood of Aρ , so I(r) ≤ max{I(ρ), I(1)}, for ρ ≤ r ≤ 1. Then Z 2π ρI(ρ) = ρ|ϕ0 (ρeiθ )| dθ = `Eucl (∂Ain ) ≤ S, 0 Z 2π I(1) = |ϕ0 (eiθ )| dθ = `Eucl (∂Aout ) ≤ S. 0 Hence I(r) ≤ S/ρ, for ρ ≤ r ≤ 1, so Z 1 S I= I(r) dr ≤ (1 − ρ). ρ ρ Changing the order of integration we obtain Z 2π Z 1  Z 2π 0 iθ S I= |ϕ (re | dr dθ = `(θ) dθ ≤ (1 − ρ). 0 ρ 0 ρ 2S Hence the Lebesgue measure of the set {θ : `(θ) > ρ (1 − ρ)} is at most 1/2. 41 In particular, Lemma 7.6 shows that the annulus A has many vertical geodesics whose Euclidean length is at most 2S(1 − ρ)/ρ. Next, we state the following classical result (see[FR21] and [Dur70, Theorem 3.13]) about the space H p , p > 0, of functions g holomorphic in D such that Z 2π  iθ p sup |g(re )| dθ < ∞. 0≤r<1 0 Lemma 7.7 (Fejér-Riesz Inequality). If g ∈ H p , then Z 1 Z 2π p 1 |g(x)| dx ≤ |g(eiθ )|p dθ. −1 2 0 We can now give a proof of Theorem 7.5. Proof of Theorem 7.5. Since the result remains true under a translation, we can assume that C \ A contains a disk of radius r centered at 0. We first claim that there exists L = L(S, r) and a vertical geodesic σ ∈ FA such that `Eucl (σ) ≤ L. Since C \ A contains a disk of radius r centered at 0, and the outer boundary has length at most S, the modulus of A is bounded from above by a constant depending only on S and r. The claim then follows by Lemma 7.6, since ρ is bounded from below by a positive constant depending only on S and r. Now let log A be a lift of A \ σ under the exponential map, using a suitable branch of the logarithm. Observe that log A is simply connected and that vertical geodesics in A lift to geodesic cross cuts in log A. For any vertical geodesic γ ∈ FA consider its lift log γ in log A. Let ψ : D → log A be a biholomorphism such that ψ({z : −1 < Re z < 1}) = log γ. This can be done by mapping 0 to a point in log γ, observing that geodesics are mapped to geodesics, and pre-composing with a rotation if necessary. By applying the Fejér-Riesz Inequality (Lemma 7.7) with p = 1 and g = ψ 0 , we obtain 1 `Eucl (log γ) ≤ `Eucl (∂ log A). (7.3) 2 So it remains to show that `Eucl (∂ log A) is bounded by a uniform constant and that the resulting bound on `Eucl (log γ) can be translated into a bound for `Eucl (γ). We do this by studying the distortion of lengths of curves under the lift via the exponential. Let t 7→ z(t) for t ∈ [0, 1] be a parametrization of a curve C in A and let log C be its lift in log A. Then t 7→ log(z(t)) for t ∈ [0, 1] is a parametrization of the curve log C, so Z 1 Z 1 z 0 (t) `Eucl (C) = |z 0 (t)| dt and `Eucl (log C) = dt. 0 0 z(t) Since A is contained in the straight annulus A(r, S/2) (this follows from considering the extremal case for ∂Aout ), we have that r ≤ |z(t)| ≤ S/2, so 2 1 `Eucl (C) ≤ `Eucl (log C) ≤ `Eucl (C). (7.4) S r 42 It follows that `Eucl (log σ) ≤ 1r `Eucl (σ) ≤ Lr and that if α = ∂Ainn and β = ∂Aout are the inner and outer boundary components, respectively, of A, then we have 1 2S `Eucl (log α) + `Eucl (log β) ≤ (`Eucl (α) + `Eucl (β)) ≤ . r r So 2L 2S `Eucl (∂ log A) = 2`Eucl (log σ) + `Eucl (log α) + `Eucl (log β) ≤ + . r r It now follows from (7.3) and (7.4) that, for any vertical geodesic γ ∈ FA , we have S S(L + S) `Eucl (γ) ≤ `Eucl (log γ) ≤ . 2 2r This concludes the proof of Theorem 7.5. 7.2 Proof of Theorem 7.2 Let bn , n ∈ N, be a sequence of uniformly expanding Blaschke products of degree at most d and let f be the transcendental entire function with an associated orbit of wandering domains (Un ) arising from Theorem 5.3. We will show that the boundary of U0 is a Jordan curve. For each n ≥ 0, we let An be the annulus bounded by the circles γn and Γn which were defined in (5.12) in the proof of Theorem 5.3. By the uniform expansivity condition on the functions bn and the fact that max{|f (z) − fn (z)| : z ∈ An } → 0 as n → ∞ (see (5.22)), we deduce using Cauchy’s estimate that there exists η > 1 such that, for sufficiently large n ∈ N, |f 0 | ≥ η > 1 on a neighborhood of An ; (7.5) in particular, f has no critical points in a neighborhood of An for such n. Relabeling the Un if necessary, we can assume that the above conditions hold for any n ≥ 0. Step 1 For each n ≥ 0, we let A bn denote the pre-image component of An+1 under f in An , given by Lemma 7.3, with inner and outer boundary components γ bn and Γ b n , say, respectively. b Then let Ăn denote the annulus lying between An and Γn (see Figure 7). Our first claim is that there exists M = M (η, d) > 0 such that, for n ≥ 0, each geodesic in the vertical foliation Fn of the annulus Ăn has Euclidean length at most M . We start by showing that each Jordan curve Γ b n has length which is uniformly bounded by 3πd/η. Indeed, we can parametrize Γ b n : [t0 , t1 ] ∪ [t1 , t2 ] ∪ · · · ∪ [td−1 , td ] → C, where t0 < t1 < b b · · · < td , with Γn (t0 ) = Γn (td ), in such a way that f is univalent on Γ b n (ti , ti+1 ), 0 ≤ i < d − 1. This can be done because the degree of bn (and hence of f on Ăn , for n sufficiently large) is bounded above by d, and Γ b n is a Jordan curve. Notice that f (Γ b n [ti , ti+1 ]) ⊆ Γn+1 . For n large, we have, by (7.5), Z t1 Z td ! 1 0 b b (t)| dt + · · · + 0 0 b n (t))||Γ b (t)| dt 0 3π ≥ `Eucl (Γn+1 ) ≥ |f (Γn (t))||Γ n |f (Γ n d t0 td−1 Z η td b 0 η b n ). ≥ |Γn (t)| dt = `Eucl (Γ d t0 d 43 b n [ti , ti+1 ] → Γn+1 is surjective for every i.) (The second inequality becomes an equality if f : Γ Since, by construction, the bounded component of C \ A bn contains the circle γn = {z : |z| = b rn } and rn ≥ 1/2 for n ≥ 0, the annulus An satisfies the hypotheses of Theorem 7.5, giving the claim. f (Γn ) f γn Γn f (γn ) γn+1 Γn+1 bn γ bn A bn An An+1 Γ Ăn Figure 7: Sketch of the setup in Step 1. Step 2 For n ∈ N, let Γ en , γ en be those pre-images under f n of Γn , γn , respectively, that are contained in A0 and are such that Γ e n surrounds Γ e n+1 for every n, while γ en is surrounded by γen+1 for every n. The existence of γ en and Γe n , and the fact that they are Jordan curves, follows by applying inductively Lemma 7.3, since there are no critical points in An for any n by the uniform expansivity condition. We now concentrate on the family of Jordan curves Γ e n and construct a continuous map Σ e from Γ0 to the limit of the Γn , defined in an appropriate way. Fix z0 ∈ Γ0 . Let Σ0 (z0 ) be the (unique) geodesic in F0 which connects z0 to some point z1 ∈ Γ e 1 , and let us parametrize it as a curve Σ0 (z0 , t), t ∈ [0, 1], with Σ0 (z0 , 0) = z0 , Σ0 (z0 , 1) = z1 . Consider f (z1 ) ∈ Γ1 , and the geodesic ω1 ∈ F1 which connects f (z1 ) to some point z20 say in f −1 (Γ2 ). Notice that the definition of F1 automatically specifies the connected component of f −1 (Γ2 ) to which z20 belongs. The preimage of ω1 under f which contains z1 is an arc that can be parametrized as Σ1 (z0 , t), t ∈ [1, 2], connecting z1 to some e 2 . Proceeding in this way, for each n we can construct a point zn and point z2 ∈ f −1 (z20 ) ∩ Γ a curve Σn (z0 , t), t ∈ [n, n + 1], connecting zn to zn+1 . This can be repeated for any starting point z ∈ Γ0 to construct a continuous injective curve Σ(z, t) : Γ0 × [0, ∞] → A0 , such that Σ(z, n) ∈ Γe n and, for each z ∈ Γ0 and each j ≤ n, n ∈ N, we have f j (Σ(z, [n, n + 1])) ⊂ Aj and f n (Σ(z, [n, n + 1])) is a geodesic in Fn . Recalling that the Euclidean length of elements in Fn is bounded uniformly in n by a 44 constant M = M (η, d) > 0, and using the expansivity estimate (7.5) on f , we obtain 1 `Eucl (Σ(z, [n, n + 1])) ≤ M, for z ∈ Γ0 , n ∈ N. ηn It follows that for each z ∈ Γ0 the curve Σ(z, t) converges to Σ(z), say, as t → ∞, and moreover that the map Σ : Γ0 → Σ(Γ0 ) is continuous in z, so Σ(Γ0 ) is a closed curve. Note, however, that we have not shown that Σ is a Jordan curve, since the map z 7→ Σ(z) has not been shown to be injective. We can construct an analogous map σ : γ0 → σ(γ0 ) and obtain a closed curve σ(γ0 ) as a en in a similar manner. uniform limit using the Jordan curves γ Step 3 We now do the final step of showing that ∂U0 is indeed a Jordan curve. It is sufficient (see [New61, Chapter VI, Theorem 16.1]) to show that each point of ∂U0 is accessible from both complementary components. By the construction in Step 2, it is enough to show that ∂U0 ⊂ σ(γ0 ) ∩ Σ(Γ0 ). Let Cn , cn denote the bounded complementary components of the Jordan curves Γ fn , γ fn , respectively. Then (Cn ) and (cn ) each form a sequence of nested topological disks which are respectively decreasing and increasing, because for each n we have that f (Γn ) surrounds Γn+1 and f (γn ) is surrounded by γn+1 . Notice that for each n, the annulus A en := Cn \ cn contains ∂U0 . This is because, for n ≥ 0, we have Γn surrounds Un and γn ⊂ Un . Now suppose that ∂U0 6⊂ Σ(Γ0 ), and let ζ ∈ ∂U0 \ Σ(Γ0 ). Then for some r > 0 the disk D(ζ, r) does not meet the closed curve Σ(Γ0 ), so there is some open disk D(ζ 0 , r0 ) ⊂ D(ζ, r) that lies in both the exterior of U0 and in the bounded complementary component of Σ(Γ0 ) which contains U0 . Hence D(ζ 0 , r0 ) ⊂ A en , for all n ∈ N, and by construction f n (D(ζ 0 , r0 )) ⊂ An for every n. Now, the maximal radii of the Euclidean disks contained in the annuli An are bounded, and indeed converge to 0 as n → ∞. 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