Absorbing sets and Baker domains for holomorphic maps

We consider holomorphic maps $f: U \to U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $\zeta$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has parabolic I type in the sense of the Baker--Pommerenke--Cowen classification of its lift by a universal covering (and $\zeta$ is not an isolated boundary point of $U$). Moreover, we provide counterexamples for other types of the map $f$ and give an exact characterization of parabolic I type in terms of the dynamical behaviour of $f$.


Introduction
In this paper, we study iterates f n = f • · · · • f n times of a holomorphic map where U is a hyperbolic domain in the complex plane C (that is, a domain whose complement in C contains at least two points) and f has no fixed points, that is, f (z) = z for z ∈ U . In the special case when U is the unit disc D (or, equivalently, the right half-plane H), the dynamical behaviour of f has been extensively studied, starting from the works of Denjoy, Valiron and Wolff in the 1920s and 1930s (see [14,[24][25][26] and a more detailed explanation in Section 2). In particular, the celebrated Denjoy-Wolff Theorem asserts that under this assumption, the iterates of f converge almost uniformly (that is, uniformly on compact subsets of U ) as n → ∞ to a point ζ in the boundary of U . Changing the coordinates by a Möbius map, we can conveniently assume in this case U = H, ζ = ∞. Baker and Pommerenke [2,19] and Cowen [13] proved that f on H is semi-conjugate to a Möbius map T : Ω → Ω by a holomorphic map ϕ : H → Ω, where the following three cases can occur: (i) Ω = H, T (ω) = aω for some a > 1 (hyperbolic type); (ii) Ω = H, T (ω) = ω ± i (simply parabolic type); (iii) Ω = C, T (ω) = ω + 1 (doubly parabolic type) (see Section 2 for a precise formulation). The terms 'simply' and 'doubly' are used due to the following fact: if f has, respectively, simply or doubly parabolic type and extends holomorphically to a neighbourhood of infinity in the Riemann sphere, then ∞ becomes a parabolic fixed point with one or two petals, respectively (see, for example, [9,15]). An alternative terminology for simply and doubly parabolic types, used in [11], is 'parabolic type II' and 'parabolic type I', respectively.
For an arbitrary hyperbolic domain U ⊂ C, the problem of describing the dynamics of a holomorphic map f : U → U without fixed points is more complicated. To this aim, one can consider a lift g : H → H of f by a universal covering π : H → U . Some results on the dynamics of f were obtained by Marden and Pommerenke [17] and Bonfert [10], who proved that if f has no isolated boundary fixed points (that is, points ζ in the boundary of U in C such that f extends holomorphically to U ∪ {ζ} with f (ζ) = ζ, see Definition 2.11), then it is semiconjugate to a Möbius map on C or on a hyperbolic domain in C. In 1999, König [16] extended the Baker-Pommerenke-Cowen result on the semi-conjugacy of f to a Möbius map for the case when f n → ∞ as n → ∞, and every closed loop in U is eventually contractible in U under iteration of f (see Theorem 2.9).
One can extend the classification of f into the three types (hyperbolic, simply parabolic and doubly parabolic), defining its type by the type of its lift g (see Section 3). In [16], König characterized the three types of f (under the restriction on eventual contractibility of loops in U ) in terms of the behaviour of the sequence |f n+1 (z) − f n (z)|/ dist(f n (z), ∂U) for z ∈ U , where ∂U denotes the boundary of U in C and dist(f n (z), ∂U) = inf u∈∂U |f n (z) − u| (see Theorem 2.9).
In this paper, we present a characterization of maps f of doubly parabolic type in terms of their dynamical properties in the general case, where f is an arbitrary holomorphic map without fixed points on a hyperbolic domain U ⊂ C. More precisely, we prove the following.
Theorem A. Let U be a hyperbolic domain in C and let f : U → U be a holomorphic map without fixed points and without isolated boundary fixed points. Then the following statements are equivalent: (a) f has doubly parabolic type; where U denotes the hyperbolic distance in U .
For the other two types of f, we prove that if inf z∈U lim n→∞ U (f n+1 (z), f n (z)) > 0, then f has hyperbolic type (see Proposition 3.3 and Remark 3.5).
Another question we consider in this paper is the existence and properties of absorbing domains in U for f .
The problem of the existence of suitable absorbing domains for holomorphic maps has a long history, and is related to the study of the local behaviour of a holomorphic map near a fixed point and properties of the Fatou components in the theory of the dynamics of rational, entire and meromorphic maps. (For basic information about the dynamics of holomorphic maps, we refer the reader to [6,12].) For instance, if U is a neighbourhood of an attracting fixed point ζ of f (for example, if U is the immediate basin of an attracting periodic point ζ of period p of a meromorphic mapf , where f =f p | U ), then f is conformally conjugate (by a map φ) to the map w → f (ζ)w (if 0 < |f (ζ)| < 1) or w → w k for some integer k > 1 (if f (ζ) = 0) near w = 0, and W = φ −1 (D(0, ε)) for a small ε > 0 is a simply connected absorbing domain in U for f , such that f (W ) ⊂ W and n 0 f n (W ) = {ζ} (see, for example, [12]).
From now on, assume that is a holomorphic map on a hyperbolic domain U ⊂ C, and the iterates of f converge to a boundary point ζ of U . Changing the coordinates by a Möbius map, we can assume ζ = ∞, so f n −→ ∞ as n −→ ∞ almost uniformly on U . Since the above definition of an absorbing domain is quite wide (observe for instance that the whole domain U is always absorbing for f ), we introduce a notion of a nice absorbing domain.
An example of a nice absorbing domain is an attracting petal W in a basin U of a parabolic p-periodic point ζ = ∞ for a rational mapf , where f =f p | U (see, for example, [12]).
The question of the existence of absorbing regions in hyperbolic domains U is particularly interesting in studying the dynamics of entire and meromorphic maps with Baker domains. Recall that a p-periodic Baker domain for a transcendental meromorphic mapf : Note that periodic Baker domains for entire maps are always simply connected (see [1]), while in the transcendental meromorphic case they can be multiply connected. The dynamical properties of Baker domains have been studied in many papers, see, for example, [3,5,8,9,15,16,20,22,23] and a survey [21].
The Baker-Pommerenke-Cowen results [2,13,19] imply that for a holomorphic map f : H → H with f n → ∞ as n → ∞, there exists a nice simply connected absorbing domain W in U for f , such that the map ϕ, which semi-conjugates f to a Möbius map T : Ω → Ω, is univalent on W . Hence, by the use of a Riemann map, one can construct nice simply connected absorbing domains for f : The existence of such absorbing regions in non-simply connected hyperbolic domains U , in particular Baker domains for transcendental meromorphic maps, was an open question addressed, for example, in [7,11,18], related to the question of the existence of so-called virtual immediate basins for Newton's root-finding algorithm for entire functions.
In [16], König showed that if U is an arbitrary hyperbolic domain in C, and every closed loop in U is eventually contractible in U under iteration of f , then there exists a nice simply connected absorbing domain in U for f . In particular, this holds if U is a p-periodic Baker domain for a transcendental meromorphic mapf with finitely many poles, where f =f p | U (see Theorem 2.9).
In a recent paper [4], the authors constructed nice absorbing domains for f : U → U with f n → ∞ for an arbitrary hyperbolic domain U ⊂ C (see Theorem 2.13). In particular, the construction was used to prove that the Baker domains of Newton's method for entire functions are always simply connected.
In this paper, we consider the question of the existence of simply connected absorbing domains W in U for f . In fact, this is equivalent to the condition that every closed loop in U is eventually contractible in U under iteration of f (see Proposition 4.1). We prove the following.
Theorem B. Let U be a hyperbolic domain in C and let f : U → U be a holomorphic map, such that f n → ∞ as n → ∞ and ∞ is not an isolated point of the boundary of U in the Riemann sphere C. If f has doubly parabolic type, then there exists a nice simply connected absorbing domain W in U for f .
Note that the assumption on the point at infinity is necessary. In fact, if ∞ is an isolated point of the boundary of U in C, then a simply connected absorbing domain cannot exist for any type of the map f (see Proposition 4.3).
We also provide counterexamples for maps which are not of doubly parabolic type.
Theorem C. There exist transcendental meromorphic maps f : We also provide examples of simply connected absorbing domains W in U for f of doubly parabolic type. In all three types of examples, the map f has the form The plan of the paper is the following. In Section 2, we present notation, definitions and a more detailed description of the classical results mentioned in the introduction, together with some other facts used in the proofs of Theorems A-C. In Section 3, we characterize doubly parabolic type (Theorem A), and in Section 4 we prove Theorem B. The examples described in Theorem C are constructed in Section 5.

Background
For z ∈ C and A, B ⊂ C, we write The symbols A, ∂A denote, respectively, the closure and boundary of A in C. The Euclidean disc of radius r centred at z ∈ C is denoted by D(z, r), and the unit disc D(0, 1) is simply written as D.
Let U ⊂ C be a hyperbolic domain, that is, a domain whose complement in C contains at least two points. By the Uniformization Theorem, there exists a universal holomorphic covering π from the right half-plane H onto U . Every holomorphic map f : U → U can be lifted by π to a holomorphic map g : H → H, such that the diagram commutes. By U (·) and U (·, ·), we denote, respectively, the density of the hyperbolic metric and the hyperbolic distance in U , defined by the use of the hyperbolic metric in H. The disc of radius r centred at z with respect to the hyperbolic metric in U is denoted by D U (z, r).
Recall the classical Schwarz-Pick Lemma and Denjoy-Wolff Theorem.
with strict inequality unless z = z or f lifts to a Möbius automorphism of H. The following estimate relates the hyperbolic density U to the quasi-hyperbolic density 1/ dist(z, ∂U ).

Lemma 2.3 ([12, Theorem 4.3]). Let U ⊂ C be a hyperbolic domain. Then
and Moreover, if U is simply connected, then The above lemma implies the following standard estimate of the hyperbolic distance. We include the proof for completeness.
Proof. Suppose that there exist z, z ∈ U such that and let γ be the straight line segment connecting z and z . In particular, (2.3) implies that |z − z | < dist(z, ∂U ), so γ ⊂ U and |u − z| < dist(z, ∂U ) for u ∈ γ. Thus, by (2.1), The lower bounds from Lemma 2.3 can be improved in the presence of dynamics. The following result was proved by Rippon in [20] (actually, it was formulated under an additional assumption f n → ∞ as n → ∞, but the proof does not use this).
for every z, z ∈ K and every n 0.
The following result proved by Bonfert in [10] describes a relationship between the dynamical behaviour of f and its lift g.
An obvious consequence of this theorem is that the left-hand side of the equivalence is either satisfied for every z ∈ U or for none.
The next theorem summarizes the results of Baker-Pommerenke-Cowen [2,13,19] on the dynamics of holomorphic maps in H. We use the notation from [13]. ( Moreover, ϕ and T depend only on g. In fact (up to a conjugation of T by a Möbius transformation preserving Ω), one of the following cases holds: Remark 2.8. An equivalent description of the three cases can be given by taking (i) Ω = {z ∈ C : 0 < Im(z) < b} for some b > 0 (hyperbolic type); (ii) Ω = {z ∈ C : Im(z) > 0} (simply parabolic type); (iii) Ω = C (doubly parabolic type); and T (ω) = ω + 1 in all three cases.
The following theorem gathers König's results from [16]. Moreover, Furthermore, iff : C → C is a meromorphic map with finitely many poles, and U is a periodic Baker domain of period p, then the above assumptions are satisfied for f =f p | U , and consequently, there exists a simply connected domain W in U with the properties (a)-(d) for f =f p .
In fact, if under the assumptions of Theorem 2.9, we take V and ϕ from Cowen's Theorem 2.7 for a lift g of f by a universal covering π : H → U , then π is univalent in V and we can set W = π(V ) and ψ = ϕ • π −1 , which is well defined in U . Remark 2.10. It follows from results proved in [4] that under the conditions of Cowen's Theorem 2.7 or Theorem 2.9, one can choose the absorbing domain W to be nice.
The existence of nice absorbing regions in arbitrary hyperbolic domains was proved by the authors in [4].

Characterization of doubly parabolic type: proof of Theorem A
Let U be a hyperbolic domain in C, and let f : U → U be a holomorphic map without fixed points. Consider a universal covering π : H → U and a lift g : H → H of the map f by π. Then g has no fixed points, so by the Denjoy-Wolff Theorem 2.2, g n → ζ for a point ζ in the boundary of H in C. Conjugating g by a Möbius map, we can assume ζ = ∞. Consider the map T : Ω → Ω from Cowen's Theorem 2.7 for the map g. By properties of a universal covering, for different choices of π and g, the suitable maps T are conformally conjugate, so in fact the type of T does not depend on the choice of π and g. Hence, we can state the following definition.
Hence, Ω = C, T (ω) = ω + 1 and g has doubly parabolic type. The implication (c) ⇒ (b) is trivial. To show (b) ⇒ (c), note first that by Theorem 2.6, the pointwise convergence holds for every z ∈ U . Take a compact set K ⊂ U and suppose that the convergence is not uniform on K. This means that there exist sequences z j ∈ K, n j → ∞ as j → ∞, and a constant c > 0, such that Passing to a subsequence, we can assume z j → z for some z ∈ K. Then U (z j , z) → 0, so by the Schwarz-Pick Lemma 2.1, for every n 0 as j → ∞. Hence, since U (f nj +1 (z), f nj (z)) → 0 by the pointwise convergence, we have which is a contradiction. This ends the proof of (b) ⇔ (c).
By Theorems A and 2.12, we immediately obtain the following. The following proposition gives a sufficient condition for a map f to be of hyperbolic type.
Remark 3.5. If the images under f n of any closed curve in U are eventually contractible in U (for example, when U is a Baker domain of a meromorphic map with finitely many poles), then by Theorem 2.9, one obtains a characterization of all three types of f in terms of its dynamical behaviour. In the general case, apart from the characterization of doubly parabolic type in Theorem A, Proposition 3.3 gives a sufficient condition for f to be of hyperbolic type. It would be interesting to determine whether the condition is necessary, and to obtain a characterization of all three types of f in terms of its dynamical behaviour in the general case.

Simply connected absorbing domains: proof of Theorem B
With the goal of proving Theorem B, we present a condition equivalent to the existence of a simply connected absorbing domain W in U for f .   Proof. By assumption, U is a punctured neighbourhood of ∞ in C. Since U is hyperbolic and f (U ) ⊂ U , the set f (U ) omits at least three points in C, so by the Picard Theorem, the map f extends holomorphically to U ∪ {∞}. Let V = {z ∈ C : |z| > R} ∪ {∞} for a large R > 0. Since f n → ∞ uniformly on the boundary of V in C, by the openness of f n , the closure of f n (V ) in C is contained in V for every sufficiently large n. This easily implies that ∞ is an attracting fixed point of the extended map f . Hence, f in a neighbourhood of ∞ is conformally conjugate (by a map ψ) to the map z → λz for some λ ∈ C, 0 < |λ| < 1 or z → z k for some integer k 2 in a neighbourhood of z = 0. Let γ = ψ −1 (∂D(0, r)) for a small r > 0. Then for every n > 0, we have f n (γ) ⊂ ext(γ) and K(f n (γ)) ⊃ K(γ), so f n (γ) is not contractible in U and we can use Proposition 4.1 to end the proof. Now we prove the main result of this section.
Theorem (Theorem B). Let U be a hyperbolic domain in C and let f : U → U be a holomorphic map, such that f n → ∞ as n → ∞, and ∞ is not an isolated point of the boundary of U in C. If f has doubly parabolic type, then there exists a nice simply connected absorbing domain W in U for f .
Proof. Note first that f has no fixed points in U . Moreover, we will show that f has no isolated boundary fixed points. Indeed, suppose that ζ 0 is an isolated point of the boundary of U in C, and f extends holomorphically to U ∪ {ζ 0 } with f (ζ 0 ) = ζ 0 . By assumption, ζ 0 = ∞. Take a Jordan curve γ 0 ⊂ U surrounding ζ 0 in a small neighbourhood of ζ 0 , such that D \ {ζ 0 } ⊂ U , where D is the component of C \ γ 0 containing ζ 0 . Since f n → ∞ uniformly on γ 0 and ζ 0 = f n (ζ 0 ) ∈ f n (D), by the Maximum Principle we obtain Hence, f has no isolated boundary fixed points.
Take a closed curve γ ⊂ U . We will show that there exists n > 0 such that f n (γ) is contractible in U . (By Proposition 4.1, this will prove the existence of a nice simply connected absorbing domain W in U for f .) Suppose that this is not true. By Theorem 2.5 for K = γ ∪ f (γ), there exists a constant C > 0 such that for every z ∈ γ and every n 0, so by the assertion (c) of Theorem A, there exists n 0 0 such that for every z ∈ γ and every n n 0 , |f n+1 (z) − f n (z)| < 1 2 dist(f n (z), ∂U). Take an arbitrary point v ∈ C \ U . By (4.1), for every z ∈ γ and every n n 0 we have |f n+1 (z) − f n (z)| < 1 2 |f n (z) − v|. This implies that for n n 0 , the point f n+1 (z) − v lies in a disc D of centre f n (z) − v and radius 1 2 |f n (z) − v|. Clearly, 0 / ∈ D and a simple calculation shows that D is included in a sector of vertex 0 an angle of measure π/3, symmetric with respect to the straight line containing 0 and f n (z) − v. Hence, there exists a branch Arg of the argument function in D such that in a neighbourhood of z we have Taking the analytic continuation of this branch while z goes around γ, we see that the winding number of f n (γ) around v is the same as the winding number of f n+1 (γ) around v. In particular, v is in a bounded component of C \ f n (γ) if and only if v is in a bounded component of C \ f n+1 (γ). Using this inductively, we show that for every v ∈ C \ U and every m n n 0 , v ∈ K(f n (γ)) if and only if v ∈ K(f m (γ)). Take n n 0 . By assumption, f n (γ) is not contractible in U , so there exists a point v 0 ∈ K(f n (γ)) \ U . Since f k → ∞ as k → ∞ uniformly on γ, there exists m > n such that f m (γ) ∩ K(f n (γ)) = ∅. We cannot have K(f m (γ)) ⊂ ext(f n (γ)), because then we would have v 0 / ∈ K(f m (γ)), which contradicts (4.2) for v = v 0 . Hence, K(f n (γ)) ⊂ K(f m (γ)), so On the other hand, because otherwise there would exists a point v 1 ∈ C \ U , such that v 1 ∈ K(f m (γ)) \ K(f n (γ)), which contradicts (4.2) for v = v 1 . We conclude that for every n n 0 there exists m > n, such that K(f n (γ)) ⊂ K(f m (γ)) and K(f n (γ)) \ U = K(f m (γ)) \ U.
Using this inductively, we construct a strictly increasing sequence n j , j 0, such that for every j. Since f nj → ∞ as j → ∞ uniformly on γ, (4.3) implies that ∞ j=0 K(f nj (γ)) = C and the set is a compact subset of C. Hence, U contains a punctured neighbourhood of ∞ in C, so ∞ is an isolated point of the boundary of U in C, which is a contradiction.

Examples
Throughout this section, let where P ⊂ C has one of the following three forms: It is obvious that for sufficiently small |a p |, the map (5.1) is transcendental meromorphic, with the set of poles equal to P. LetP (In the case P = Z or P = Z + iZ, we haveP = P.) The assertions of Theorem C and other results mentioned in Section 1 are gathered in the following theorem. (1) In case (i), we have U (f n+1 (z), f n (z)) −→ 0 as n −→ ∞ for z ∈ U and f | U has doubly parabolic type, so there exists a simply connected absorbing domain W in U for f .
(2) In case (ii), we have and note for further purposes that for all n 1, To prove Theorem 5.1, we will use the following two lemmas.
First, consider case (iii). Then, writing P p = j + im for j, m ∈ Z, we have Note that which proves the claim (5.6) for n + 1, completing the induction. Finally, note that the second part of (5.6) implies f n (z) ∈ V , since z + n ∈Ṽ and dist(Ṽ , C \ V ) = ε, by the definitions of V andṼ . This ends the proof of the lemma.
Proof of Theorem 5.1. Take a small 0 < δ < 1/4. Set ε = δ/2 and consider a map of the form (5.1) for the numbers a p , p ∈ P satisfying the conditions of Lemma 5.3 for this value of ε. In particular, Lemma 5.3 implies that f n → ∞ as n → ∞ almost uniformly onṼ . SinceṼ is connected, it follows that C \ p∈P D(p, δ) =Ṽ is contained in an invariant Baker domain U of f . Note that U ∩ P = ∅ since the poles are contained in the Julia set of f . Now we characterize cases (i)-(iii). In case (i), we have U ⊃Ṽ ⊃ {z ∈ C : Re(z) 2ε}, in particular 1 ∈Ṽ ⊂ U . By Lemma 5. as n → ∞. Therefore, Theorem A implies that U (f n+1 (z), f n (z)) → 0 as n → ∞ for z ∈ U and f | U has doubly parabolic type. It is easy to check that W = {z ∈ C : Re(z) > 1} is a nice simply connected absorbing domain in U for f . Now consider case (ii). Then U ⊃Ṽ ⊃ {z ∈ C : |Im(z)| 2ε}, in particular ik ∈Ṽ ⊂ U for every positive integer k. Hence, by Lemma 5.3, for n 0. Hence, by Theorem 2.5 for K = {ik, f (ik)}, we have