This is a preprint of: “Fatou components and singularities of meromorphic functions”, Krzysztof Baranski, Nuria Fagella, Xavier Jarque, Boguslawa Karpinska, Proc. Roy. Soc. Edinburgh Sect. A, 1–22, 2017. DOI: [10.1017/prm.2018.142] FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA Abstract. We prove several results concerning the relative position of points in the postsin- gular set P (f ) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević- Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that dist(pn , ∂Un ) → 0 as n → ∞. We also prove that if Un ∩ P (f ) = ∅ arXiv:1706.01732v1 [math.DS] 6 Jun 2017 and the postsingular set of f lies at a positive distance from the Julia set (in C) then any sequence of iterates of wandering domains must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values. 1. Introduction and statement of the results We consider dynamical systems defined by the iteration of a meromorphic function b f : C → C, on the complex plane, especially those with an essentially singularity at infinity (transcenden- tal). Motivating examples of such maps are given by root-finding algorithms like, for instance, Newton’s method applied to any entire transcendental map. In this setting the Riemann sphere C b splits into two invariant sets: the Fatou set F (f ), which consists of points for which the family of iterates (f n ) is well defined for all n > 0 and normal in some neighborhood; and its complement J(f ) known as the Julia set. The connected components of the Fatou set or Fatou components which are periodic can be completely classified based on the possible limit functions of the family of iterates. Indeed, if U is a periodic Fatou component, then either U is a component of a basin of attraction of an attracting or parabolic cycle; or U is a topological disk (Siegel disk) or annulus (Herman ring) on which the map is conjugate to an irrational rigid rotation; or the iterates of f (or multiples of them) tend uniformly on compact subsets to the essential singularity at infinity, in which case U is a component of a cycle of Baker domains. If a Fatou component is neither periodic nor preperiodic, then it is called a wandering domain. Neither Baker domains nor wandering components are present in the dynamical plane when f is a rational map [Sul85]. For classical background on the dynamics of meromorphic functions we refer to the survey [Ber93a] or the articles [BKY91a, BKY90, BKY91b, BKY92]. Date: June 7, 2017. 2010 Mathematics Subject Classification. Primary 30D05, 37F10, 30D30. The first and fourth authors were partially supported by the Polish NCN grant decision DEC- 2014/13/B/ST1/04551. The second and third authors were partially supported by the Spanish grant MTM2014-52209-C2-2-P. The fourth author was partially supported by PW grant 504/02465/1120. 1 http://www.gsd.uab.cat 2 KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA The set S(f ) of finite singular values of f plays an important role in determining the dynamics of the map. Recall that singular values are either critical or asymptotic values. Among well known classes of transcendental maps are those with a finite set of singular values (the Speisser class S) or with a bounded set of singular values (the Eremenko–Lyubich class B). The orbits of the singular values and their accumulation points form the postsingular set [ [ ∞ P (f ) = C ∩ f n (s), s∈S(f ) n=0 where we neglect the terms which are not defined or are infinite. The importance of singular values lies on the fact that periodic Fatou components are in some way associated to the postsingular set. Indeed, any basin of attraction of an attracting or parabolic cycle must contain a singular value, while the boundary components of Siegel disks and Herman rings are contained in the closure P (f ) of the postsingular set. The relation between singular values and Fatou components that are specific for transcen- dental maps, namely Baker and wandering domains, is less clear. This problem is nowadays one of the challenges in transcendental dynamics, and the object of this paper. It is worth noticing that infinitely many singular values are necessary for these types of components to exist at all [Bak84, GK86, EL92, BKY92], and that entire maps in class B do not have Baker domains, nor wandering domains tending to infinity under iteration [BKY91b, EL92, RS99]. Hence, results in this direction must go beyond these classes of functions. It is known [BHK+ 93, Zhe03] that for a wandering domain U , all finite limit functions of {f n |U } lie in the derived set of P (f ). Moreover, if for a Baker or wandering domain U we have f np |U → ∞ as S n → ∞ for some p ≥ 1, then ∞ is in the derived set of p−1 k k=0 f (S(f )) [Bak02, Zhe03]. These results can be used to rule out the existence of wandering domains in certain situations. The existence of such domains was also excluded for some entire maps studied in [Sta91] and some meromorphic maps with finitely many poles, which are Newton methods of entire functions [Ber93b]. There are known examples of Baker domains on which f is univalent (and hence contains no critical values), has finite degree larger than 1 or infinite degree. We refer the reader to [BF01, FH06, Rip06, Rip08, BFJK15] for classification and examples of such components. Bergweiler in [Ber95] gave an example of a Baker domain of an entire map lying at a positive (Euclidean) distance from the postcritical set and therefore showing that a very mild relation between these objects is possible. Examples of wandering domains, with the first one given by Baker in [Bak76], are not so numerous. Most of them are constructed either using approximating theory [EL87], by the lifting method [FH09] (that is by lifting a function with no zeros by the logarithm and therefore converting periodic components into wandering domains), or by quasiconformal surgery [KS08]. In the first case the method does not allow much control on the postsingular set of the map, while in the second case the relation of the postsingular set to the wandering domain is completely determined by the type of periodic Fatou component of the original map. As an example, take the function considered in [FH09], defined as f (z) = 2 − λ − log(2 − λ) + 2z − ez √ 2−λ with λ = e2πi(1− 5)/2 , which is the lift of g(w) = e2−λ we−w (see Figure 1). The map f has infinitely many orbits of simply connected wandering domains on which f is univalent, while the postsingular set is dense in the boundary of each of these wandering domains. To FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS 3 our knowledge, this is the only explicit example of a simply connected wandering domain on which f is univalent. All other (lifting) examples contain critical points inside wandering components lifted from basins of attraction of attracting or parabolic cycles. We mention also the inspiring examples by Kisaka and Shishikura in [KS08], where they construct wandering domains of eventual connectivity two for maps whose all singular values lie in preperiodic components (see also [BRS13]). 2−λ √ Figure 1. Left: Dynamical plane of g(w) = e2−λ we−w with λ = e2πi(1− 5)/2) . There is a superattracting basin centered at w = 0 and a Siegel disk centered at w0 = 2 − λ. The only singular values are at the origin and at c = 2 (on the boundary of the Siegel disk). Right: Dynamical plane of f (z) satisfying exp(f (z)) = g(exp(z)). The superattracting basin lifts to a Baker domain (white), while the Siegel disk lifts to infinitely many orbits of wandering domains on which f is univalent (grey). Figures are extracted from [FH06]. The above examples show different possibilities for the relationship between Baker or wan- dering domains and the set P (f ). Our goal in this paper is to present some new results in this direction. Our first result concerns Baker domains. In [Ber95], Bergweiler showed the following result for invariant Baker domains of entire functions. Theorem 1.1 ([Ber95, Theorem 3]). Let f be a transcendental entire function with an in- variant Baker domain U . If U ∩ S(f ) = ∅, then there exists a sequence pn ∈ P (f ) such that, as n → ∞, (1) |pn | → ∞, (2) |pn+1 /pn | → 1, (3) dist(pn , U ) = o(|pn |). Here and in the sequel dist denotes the Euclidean distance. Additionally, as we mentioned above, Bergweiler provided an example (which inspired the example in Figure 1), where the postcritical set is located at a positive distance from the Baker domain, showing in some sense the sharpness of the theorem. We remark that Theorem 1.1 holds for entire 4 KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA functions and therefore all considered Baker domains are simply connected. A version of Theorem 1.1 for meromorphic transcendental maps (and hence for Baker domains which are not necessarily simply connected) was proven by Mihaljević-Brandt and Rempe-Gillen in [MBRG13, Theorem 1.5], with conclusion (2) replaced by pn+1 sup < ∞. n≥1 pn They also asked a question, whether the stronger conclusion (3) holds in this setting. In this paper we answer this question positively under a mild additional hypothesis, proving the following. Theorem A. Let U be a Baker domain of period p ≥ 1 for a meromorphic map f , such that f pn |U → ∞ as n → ∞ and C \ U has an unbounded connected component. Then there exists a sequence of points pn ∈ P (f ) such that pn+1 dist(pn , U ) |pn | → ∞, → 1, →0 pn |pn | as n → ∞. We remark that the assumption on an unbounded component of C \ U (which is equivalent to the condition that {∞} is not a singleton component of C b \ U ) is trivially satisfied if the Baker domain is simply connected. This is always the case if f is Newton’s method of an entire map, as it was shown by the authors in [BFJK14b]. Theorem A is a consequence of a technical lemma (Lemma 2.1) which generalizes [Ber95, Lemma 3] and, additionally, has several applications concerning the relation between points in P (f ) and wandering domains. The following consequence of Lemma 2.1, although stated for a general Fatou component, is most meaningful when U is a wandering or Baker domain. In particular, it contributes to an answer to Question 11 from [Ber93a]. Theorem B. Let f be a transcendental meromorphic map and U be a Fatou component of f . Denote by Un the Fatou component such that f n (U ) ⊂ Un . Then for every z ∈ U there exists a sequence pn ∈ P (f ) such that dist(pn , Un ) → 0, as n → ∞. dist(f n (z), ∂Un ) In particular, if for some d > 0 we have dist(f n (z), ∂Un ) < d for all n (for instance if the diameter of Un is uniformly bounded), then dist(pn , Un ) → 0 as n tends to ∞. The second application of Theorem B concerns functions for which the Julia set and the postsingular set are apart from each other. More precisely, following Mayer and Urbański [MU07, MU10] we consider the class of topologically hyperbolic meromorphic functions. This class was also considered by Stallard in [Sta90] in the context of entire maps, where she proved that in this case |(f n )0 (z)| → ∞ for all z ∈ J(f ). Definition 1.2. A meromorphic transcendental function f is called topologically hyperbolic if dist(P (f ), J(f ) ∩ C) > 0. This condition can be regarded as a kind of weak hyperbolicity in the context of tran- scendental meromorphic functions, while hyperbolicity is (usually) defined by the condition that P (f ) is bounded and disjoint from the Julia set. In particular, hyperbolic maps are FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS 5 always in class B. On the contrary, topologically hyperbolic maps could have sequences of (post)singular points in the Fatou set accumulating at infinity. See e.g. the discussion on various kinds of hyperbolicity for transcendental maps in [RS99]. Note also that topologically hyperbolic maps do not possess parabolic cycles, rotation domains or wandering domains which do not tend to infinity (otherwise they would have a constant finite limit function in P (f ) ∩ J(f ), see [Bak02]). Examples of topologically hyperbolic maps include many Newton’s methods of entire functions (see Propositions 4.1 and 4.2). The following corollary shows that Baker or wandering domains of topologically hyperbolic maps either contain postsingular points or arbitrarily large disks. Corollary C. Let f be a topologically hyperbolic meromorphic map and U be a Fatou com- ponent of f . Denote by Un the Fatou component such that f n (U ) ⊂ Un and suppose that Un ∩ P (f ) = ∅ for n > 0. Then for every compact set K ⊂ U , every z ∈ K and every r > 0 there exists n0 such that for all n ≥ n0 D(f n (z), r) ⊂ Un . In particular, diam Un → ∞ and dist(f n (z), ∂Un ) → ∞ for every z ∈ U , as n → ∞. Corollary C may be used in many instances to exclude the presence of wandering domains for a given map. Examples can be found in Section 4, where we show that certain Newton maps outside class B, with infinitely many poles, do not possess wandering Fatou components. In particular, we prove (see Theorem D and Corollary E) that Newton’s method of the function h(z) = aez + bz + c has no wandering domains for many parameters a, b, c, including the case when a, b, c ∈ R, a, b 6= 0 and the map h has a real zero. Acknowledgement. The authors would like to thank the Centro Internazionale per la Ricerca Matematica (CIRM) in Trento for the hospitality during their visit within Research in Pairs program. 2. Preliminaries and tools We use the notation D(z, r) = {w ∈ C : |w − z| < r} and [ D(A, r) = D(z, r) z∈A for z ∈ C, A ⊂ C, r > 0. By [z1 , z2 ] we denote the straight line segment connecting z1 to z2 . The length of a rectifiable curve γ is denoted by `(γ). The symbol diam denotes the Euclidean diameter. We set dist(z, B) = inf{|z − w| : w ∈ B}, dist(A, B) = inf{|z − w| : z ∈ A, w ∈ B} for z ∈ C, A, B ⊂ C. The closure and boundary of a set A ⊂ C are denoted, respectively, by A and ∂A. We use the following classical results on the distortion of conformal maps (see e.g. [CG93]). Koebe’s Distortion Theorem. Let g : D(a, r) → C be a univalent holomorphic map, ρ ∈ (0, 1) and z ∈ D(a, ρr). Then 1−ρ 1+ρ 3 |g 0 (a)| ≤ |g 0 (z)| ≤ |g 0 (a)|. (1 + ρ) (1 − ρ)3 6 KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA Koebe’s One-quarter Theorem. Let g : D(a, r) → C be a univalent holomorphic map. Then  g(D(a, r)) ⊃ D g(a), 41 |g 0 (a)|r . In this section we prove the following lemma, which is the main tool in the proofs of our results. It is a generalization of [Ber95, Lemma 3]. The estimates are similar to the ones in [MBRG13, Proposition 7.4], which were formulated in terms of the hyperbolic metric. Lemma 2.1. Let f be a transcendental meromorphic map and U be a Fatou component of f . Denote by Un the Fatou component such that f n (U ) ⊂ Un . Then for every compact set K ⊂ U and every ε > 0, M ≥ 1 there exists n0 > 0 such that for every n ≥ n0 , every z ∈ K and every curve γ in C connecting f n (z) to a point w ∈ ∂Un with `(γ) ≤ M dist(f n (z), ∂Un ) there exists a point p ∈ D(γ, ε`(γ)) ∩ P (f ). Proof. We argue by contradiction. If the assertion of the lemma does not hold, then there exist a compact set K ⊂ U , numbers ε > 0, M ≥ 1, sequences nj → ∞, zj ∈ K and curves γj connecting the point ξj := f nj (zj ) to wj ∈ ∂Unj , such that (1) `(γj ) ≤ M dist(ξj , ∂Unj ) and D(γj , ε`(γj )) ∩ P (f ) = ∅. Replacing ε by 2ε, we can actually assume (2) D(γj , ε`(γj )) ∩ P (f ) = ∅. Take the branch gj of f −nj defined on D(ξj , ε`(γj )), such that gj (ξj ) = zj and extend it along γj . Then gj is well-defined (single-valued) on Dj := D(ξj , ε`(γj )) ∪ γj ∪ D(wj , ε`(γj )). We claim that the distortion of gj on γj (i.e. supu1 ,u2 ∈γj |gj0 (u1 )|/|gj0 (u2 )|) is bounded by a constant independent of j. To see the claim, take a parameterization of γj by t ∈ [0, 1] and define the sequence t0 , . . . , tk ∈ [0, 1] inductively in the following way. Let t0 = 0 and let   ε`(γj ) ts+1 = inf t ∈ (ts , 1] : |γj (t) − γj (ts )| = 2 for s ≥ 0, as long as the infimum is defined (i.e. the above set is non-empty). We notice that if ts is defined, then ε`(γj ) s = |γj (t1 ) − γj (t0 )| + · · · + |γj (ts ) − γj (ts−1 )| ≤ `(γj ), 2 which implies s ≤ 2/ε. This means that there exists k > 0 such that k ≤ 2/ε and tk+1 is not defined, i.e. |γj (t) − γj (tk )| < ε`(γj )/2 for every t ∈ (tk , 1]. Setting tk+1 = 1, we have a sequence 0 = t0 < · · · < tk+1 = 1, such that   ε`(γj ) γj ([ts , ts+1 ]) ⊂ D γj (ts ), 2 for every s = 0, . . . , k. Since gj extends to D(γj (ts ), ε`(γj )), the Koebe Distortion Theorem implies that the distortion of gj on γj ([ts , ts+1 ]) is bounded by a constant c0 > 1, which is 1+2/ε independent of j and s. Then the distortion of gj on γj is bounded by c1 := ck+1 0 ≤ c0 . Let vj = gj (wj ), rj = sup{r > 0 : D(vj , r) ⊂ gj (Dj )}. FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS 7 Since D(wj , ε`(γj )) ⊂ Dj , by the Koebe One-quarter Theorem, we have ε`(γj ) 0 (3) rj ≥ |gj (wj )|. 4 Moreover, (4) |zj − vj | ≤ sup |gj0 (u)| `(γj ) ≤ c1 |gj0 (wj )| `(γj ). u∈γj As vj ∈ / U , zj ∈ K and K ⊂ U is compact, we have |zj − vj | > c2 for some c2 > 0 independent of j, which together with (3) and (4) gives εc2 (5) rj > c3 := 4c1 independent of j. By (1) and (2), we have D(ξj , min(ε, 1/M )`(γj )) ⊂ Dj ∩ Un , and so, by the Koebe One-quarter Theorem, we conclude   min(ε, 1/M )`(γj ) 0 (6) D zj , |gj (ξj )| ⊂ U. 4 On the other hand, as K ⊂ U is compact, we have maxz∈K dist(z, ∂U ) < ∞, so (6) implies (7) `(γj )|gj0 (ξj )| < c4 for some c4 > 0 independent of j. This, together with (4) and the compactness of K, gives |vj | ≤ |zj − vj | + |zj | ≤ c5 for some c5 > 0 independent of j. Hence, taking a subsequence of nj we can assume that vj → v ∈ ∂U ∩ C, and by (5) we have  c  3 D := D v, ⊂ gj (Dj ) 2 for every sufficiently large j. In particular, v ∈ D ∩ J(f ) and f nj are defined and univalent on D for all large j, with the distortion bounded by a constant independent of j. By the density of periodic sources in J(f ), we have |(f nj )0 | → ∞ on D as j → ∞, so by the Koebe 1/4 Theorem, every bounded set in C is contained in f nj (D) for sufficiently large j, which clearly contradicts the univalency of f nj on D.  3. Proofs of main results Proof of Theorem A. Fix a point z ∈ U and let K be a curve connecting z to f p (z) in U . Then [∞ Γ= f pm (K) ⊂ U m=0 is a curve starting at z and tending to ∞. Since C \ U has an unbounded component, there exists r0 > 0 such that for every r > r0 there is a point w ∈ ∂U with |w| = r. Replacing z by f pn (z) for some n > 0 we can assume (8) |ζ| > r0 for every ζ ∈ Γ. 8 KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA Take a positive integer k and use Lemma 2.1 for the set K and 1 ε= , M = 2πk, k to find N (k) such that for every m ≥ N (k) and every curve γ connecting f pm (z) to a point in ∂U with `(γ) ≤ M dist(f pm (z), ∂U ) there exists a point in D(γ, ε`(γ)) ∩ P (f ). Increasing N (k) inductively, we can assume |f pN (k) (z)| < |f pN (k+1) (z)| for all k. uj (k) wj (k) γj (k) f pN (k) (z) Γ U ζj (k) f pN (k+1) (z) r1 (k) rlk (k) = r1 (k + 1) Figure 2. Construction of points uj (k). Choose positive numbers h i r1 (k) < . . . < rlk (k) ∈ |f pN (k) (z)|, |f pN (k+1) (z)| such that r1 (k) = |f pN (k) (z)|, rlk (k) = |f pN (k+1) (z)| and rj+1 (k) 1 (9) <1+ , j = 1, . . . , lk − 1 rj (k) k Note that, by definition, (10) rlk (k) = r1 (k + 1) FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS 9 and (11) lim min rj (k) = lim r1 (k) = ∞. k→∞ j k→∞ SN (k+1) pm (K) Take j ∈ {1, . . . , lk − 1}. Then there is a point ζj (k) ∈ m=N (k) f with |ζj (k)| = rj (k). Now define a curve γj (k) joining ζj (k) to ∂U in the following way. If dist(ζj (k), ∂U ) ≤ rj (k)/k, then we set γj (k) to be the straight line segment from ζj (k) to the nearest point of ∂U . In this case we have rj (k) `(γj (k)) = dist(ζj (k), ∂U ) ≤ . k In the complementary case dist(ζj (k), ∂U ) > rj (k)/k, note that rj (k) > r0 by (8), so there is a point wj (k) ∈ ∂U with |wj (k)| = rj (k). Let γj (k) be a circle arc in ∂D(0, rj (k)) connecting ζj (k) to wj (k). Changing wj (k) if necessary, we can assume γj (k) \ {wj (k)} ⊂ U . In this case we have `(γj (k)) ≤ 2πrj (k) ≤ M dist(ζj (k), ∂U ). Since in both cases ζj (k) ∈ f pm (K) for m ≥ N (k) and `(γj (k)) ≤ M dist(ζj (k), ∂U ), by Lemma 2.1 and the definition of N (k) we conclude that there is a point   `(γj (k)) uj (k) ∈ P (f ) ∩ D γj (k), . k By the definition of γj (k),     rj (k) rj (k) γj (k) ⊂ D 0, rj (k) + \ D 0, rj (k) − k k and `(γj (k)) ≤ 2πrj (k), so 2π + 1 |uj (k)| 2π + 1 (12) 1− < <1+ , k rj (k) k and by (9) and (10), uj+1 (k) c (13) <1+ uj (k) k for some c > 0 independent of k, j, where we define ulk (k) = u1 (k + 1). Moreover, by the definition of γj (k), 2πrj (k) (14) dist(uj (k), U ) ≤ . k To define a suitable sequence of points pn ∈ P (f ) which satisfies the assertions of the theorem, it is sufficient to renumber the sequence u1 (1), . . . , ul1 −1 (1), u1 (2), . . . , ul2 −1 (2), . . . . 10 KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA This can be done formally by setting pn = uj(n) (k(n)), . . . , n = 1, 2, . . . , where j(n), k(n) are the unique numbers satisfying k(n)−1 k(n) k(n)−1 X X X 1+ (ls − 1) ≤ n < 1 + (ls − 1) and j(n) = n − (ls − 1). s=1 s=1 s=1 Clearly, k(n) → ∞ as n → ∞. By (11) and (12), |pn | → ∞ as n → ∞, while (13) implies |pn+1 /pn | → ∞. Finally, (12) and (14) give dist(pn , U )/|pn | → 0. The proof of Theorem A is complete.  Remark 3.1. It would be interesting to determine whether the assumption that C \ U has an unbounded connected component is necessary. Proof of Theorem B. Take a positive integer k and z ∈ U . We use Lemma 2.1 for ε = 1/k, M = 1 and γn = [f n (z), wn ], where |f n (z) − wn | = dist(f n (z), ∂Un ) =: rn . By Lemma 2.1 we know that there exists n0 (k), such that for every n ≥ n0 (k) one can find a point  rn  pn ∈ D [f n (z), wn ], ∩ P (f ). k Clearly, we can assume n0 (k) → ∞ as k → ∞. Since [f n (z), wn ] \ {wn } ⊂ Un by definition, we have rn (15) dist(pn , Un ) < . k Now for any n > 0 define k(n) = max{k : n0 (k) ≤ n}. Note that the maximum is attained since n0 (k) → ∞ as k → ∞. Obviously, k(n) → ∞ as n → ∞ and n ≥ n0 (k(n)), so by (15) we have dist(pn , Un ) 1 < →0 rn k(n) as n → ∞, which ends the proof.  Proof of Corollary C. By Theorem B, there exists a sequence of points pn ∈ P (f ) such that dist(pn , Un ) < εn , dist(f n (z), ∂Un ) where εn → 0 as n → ∞. Since f is topologically hyperbolic and Un ∩ P (f ) = ∅, we have dist(pn , Un ) > c for some fixed c > 0, which implies dist(f n (z), ∂Un ) > c/εn → ∞.  4. Examples As an application of the above results we will consider some examples of topologically hyperbolic maps for which we can rule out the existence of wandering domains. All the examples are Newton’s methods for some transcendental entire functions. Some of the maps considered in this section were studied in detail in [BFJK, Examples 7.2–7.3]. Here we show additionally that none of them has wandering domains. FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS 11 The first proposition was proved previously by Bergweiler and Terglane in [BT96] by the use of Sullivan’s quasiconformal deformations technique. Here we present an elementary proof based on Corollary C. Proposition 4.1. The map Nf (z) = z − tan(z), which is Newton’s method of the map f (z) = sin(z), has no wandering domains. Proof. The map Nf has all its fixed points located at ck := kπ, k ∈ Z (the zeros of sin(z)). All of them are critical points (in fact Nf0 (ck ) = Nf00 (ck ) = 0), and they are the only ones. Moreover, Nf has no finite asymptotic values. One can also prove that the vertical lines π rk (t) := + kπ + it, t ∈ R, k ∈ Z, 2 are invariant, contain all poles of Nf (located at pk := π2 + kπ) and are contained in the Julia set of Nf (see [BFJK]). In each of the vertical strips bounded by the lines rk there is an (unbounded) attracting basin of the super-attracting fixed point ck . Since 1 Nf0 (z) = 1 − , cos2 (z) it is clear from the periodicity of the function that there exists r > 0 such that for every k, we have |Nf0 (z)| ≤ 1/2 for z in D(ck , r) and we conclude that this disk is contained in the Fatou set of Nf . It follows that Nf is topologically hyperbolic and therefore if W is a wandering domain for Nf and z ∈ W , then Nfn (z) tends to infinity as n → ∞. Moreover, Corollary C implies that for any R > 0 we have D(Nfn (z), R) ⊂ Wn for every sufficiently large n, where Wn is the Fatou component such that Nfn (W ) ⊂ Wn . This contradicts the fact that the lines rk , which are at distance π from each other, are contained in the Julia set.  The second proposition is proved in a similar way. Proposition 4.2. The map Ng (z) = z + i + tan(z),  R  z du which is the Newton’s method of the map g(z) = exp − 0 i+tan(u) , has no wandering do- mains. Proof. In [BFJK] it is shown that Ng has an invariant Baker domain U such that H+ ∪ L ⊂ U, where H+ is the upper halfplane and [ L= lk for lk = {Re(z) = kπ, k ∈ Z}. k∈Z Note also that the vertical lines n π o sk = z ∈ C : Re(z) = + kπ 2 are invariant under Ng . Each of the lines contains the pole pk := π/2 + kπ and the two critical points ck = π/2 + kπ ± iyk , where ±yk ≈ ±0.8814 are, respectively, the solutions of 12 KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA the equation exp(−y) − exp(y) = ±2. Moreover, Ng has no other critical points and no finite asymptotic values. From the equality   sin 2x sinh 2y tan(x + iy) = +i cos 2x + cosh 2y cos 2x + cosh 2y it follows that all points z in the halflines s− k := {z ∈ sk : Im(z) < 0} satisfy Im(Ng (z)) < Im(z). Since Ng has no fixed points, this implies that Im(Ngn (z)) tends to −∞ as n → ∞ for z ∈ s−k . Moreover, one can check that the sets   π π ln 2 Sk := z ∈ C : Re(z) − − kπ < , Im(z) < − 2 8 2 satisfy Ng (Sk ) ⊂ Sk (see [BFJK]), which implies that s− k ∪ Sk is contained in an invariant Baker domain of Ng . We conclude that Ng is topologically hyperbolic. Hence, Ng has no wandering domains since for every such domain W , its forward images Wn := f n (W ) would contain arbitrary large discs which cross the lines lk , a contradiction.  The main result of this section is to exclude the presence of wandering domains for the following class of maps. Theorem D. Let h(z) = aez + bz + c for a, b, c ∈ C, a, b 6= 0 and let h(z) z − 1 − αe−z (16) Nh (z) = z − = , h0 (z) 1 + βe−z where α = c/a, β = b/a, be its Newton’s method. If a, b, c ∈ C, a, b 6= 0, and the asymptotic value u = −c/b = −α/β of Nh satisfies inf n≥0 dist(Nhn (u), J (Nh )) > 0, then Nh has no wandering domains. Remark 4.3. Checking the relation α = c/a, β = b/a is straightforward. Observe that for a = 0 the map Nh degenerates to a constant. If a 6= 0 and b = c = 0, then Nh (z) = z−1, and if a, c 6= 0 and b = 0, then Nh is conjugated to a Fatou function of the form F (z) = z + e−z + λ, for some λ ∈ C. In [KU05] it is proved that such maps have no wandering domains for Re(λ) ≥ 1. Here we assume a, b 6= 0. Since α = c/a, β = b/a, it is enough to consider the case a = 1 (and set α = c, β = b 6= 0). Moreover, it is easy to check that Nh has exactly one asymptotic value u = −α/β, and it satisfies u = limt→−∞ Nh (t + 2kπi) for k ∈ Z. The following corollary shows explicit values of parameters for which the assumptions of Theorem D are satisfied. Corollary E. If a, b, c ∈ R, a, b 6= 0 and the map h(z) = aez + bz + c has a real zero, then its Newton’s method Nh has no wandering domains. The condition hold if and only if ab > 0 or ab < 0, c/b ≥ 1 − ln(−b/a) (equivalently, if β > 0 or β < 0, α ≤ β(1 − ln(−β))). To prove Theorem D we will apply Corollary C. Hence, the first step is to show that Nh is a topologically hyperbolic map. Precisely, we have the following. Proposition 4.4. If α, β ∈ C, β 6= 0 and inf n≥0 dist(Nhn (u), J(Nh )) > 0, then Nh is topolog- ically hyperbolic. Proof. As Nh is Newton’s method of h, the fixed points of Nh coincide with the zeroes of h. We have h(z)h00 (z) h(z) = ez + βz + α, h0 (z) = ez + β, h00 (z) = ez , Nh0 (z) = . (h0 (z))2 FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS 13 In particular, h00 6= 0, so all critical points of Nh , denoted by ck , k ∈ Z, are superattracting fixed points of Nh and satisfy (17) h(ck ) = eck + βck + α = 0. If e1−α/β + β = 0, then the point (18) c̃ = 1 − α/β is the unique double zero of h and the unique (attracting and non-superattracting) fixed point of Nh outside {ck }k∈Z . Otherwise, all zeroes of h are simple and the fixed points of Nh coincide with the points ck , k ∈ Z. Therefore, the set of the singular values of Nh consists of the superattracting fixed points ck , k ∈ Z, and the asymptotic value u. By hypothesis, it is enough to show inf k∈Z dist(ck , J(f )) > 0. To this end, we will prove that there exists r > 0 such that (19) |Nh0 (z)| < 1 for every z ∈ D(ck , r), k ∈ Z. Indeed, in this case D(ck , r) is forward invariant, so it is contained in the basin of ck , which ends the proof. To show (19), note that by (17), if z ∈ D(ck , 1), then |ez + βz + α| = |ez − eck + β(z − ck )| ≤ |ez |(1 + eRe(ck −z) ) + |β||z − ck | ≤ (e + 1)|ez | + |β| and |ez | = |ez−ck ||eck | = eRe(z−ck ) ||βck + α| ≥ e−1 |βck + α| −−−−→ ∞. |k|→∞ This implies that if z ∈ D(ck , 1) for large |k|, then h00 (z) h(z)h000 (z) 2h(z)(h00 (z))2 |Nh00 (z)| = + − h0 (z) (h0 (z))2 (h0 (z))3 ez ez (ez + βz + α) 2e2z (ez + βz + α) ≤ z + + ≤C e +β (ez + β)2 (ez + β)3 for some C > 1. Since Nh0 (ck ) = 0, for z ∈ D(ck , 1/C) we have |Nh0 (z)| ≤ sup |Nh00 (ζ)||z − ck | < 1, ζ∈D(ck ,1/C) which gives (19) with r = 1/C and ends the proof.  Proof of Theorem D. By Proposition 4.4, we know that under our assumptions Nh is topo- logically hyperbolic. We want to show that Nh has no wandering domains. We assume, to get a contradiction, that W0 is a wandering domain. Let w0 ∈ W0 and let wn := Nhn (w0 ) ⊂ Wn where Wn denotes the Fatou component such that Nhn (W ) ⊂ Wn , n ≥ 0. Recall that Wn , n ≥ 0 must be simply connected (see [BT96], [BFJK14a]). The idea of the proof is to show that for n large enough there exists a vertical segment η contained in Wn such that its image by Nh contains a closed curve surrounding η, which implies that Wn+1 is not simply connected, a contradiction. Now we proceed to the technical details. By Corollary C, (20) D(wn , Rn ) ⊂ Wn for some sequence Rn → ∞. Note that the poles of Nh are the solutions of the equation ez + β = 0 (with the exception of the point c̃ defined in (18) in the case e1−α/β + β = 0), so 14 KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA they are located at the points ln |β| + i arg(−β) + 2kπi, k ∈ Z. In particular, since the poles of Nh are outside Wn , (20) implies (21) |Re(wn )| → ∞. Note that if Re(wn ) < −c for a large c > 0, then |wn+1 | |wn | + 1 |α| 1 + 1/c |α| 1 ≤ −Re(w ) + Re(w ) ≤ c + −c < , |wn | |wn |(|β|e n − 1) |wn |(|β| − e n ) |β|e − 1 c(|β| − e ) 2 which together with (21) implies (22) lim sup Re(wn ) = ∞. n→∞ Moreover, if Re(wn ) > c for a large c > 0 and |Im(wn )| ≤ eRe(wn ) /(3|β|), then, since α + β(wn − 1) wn+1 = wn − 1 − , ewn + β we have   α + β(wn + 1) |Re(wn+1 ) − Re(wn ) + 1| = Re ewn + β |α| + |β|(Re(wn ) + 1 + |Im(wn )|) ≤ eRe(wn ) − |β| |α| + |β|(c + 1) + ec /3 1 ≤ < , ec − |β| 2 so −3/2 < Re(wn+1 ) − Re(wn ) < −1/2. This together with (21) and (22) implies that for a sequence nk → ∞ there holds eRe(wnk ) (23) Re(wnk ) → ∞, |Im(wnk )| > . 3|β| Fix a large k > 0 and let M = Rnk , p = ln(3|β|M ), R = Re(wnk ) − p. Take also q ∈ [−2π, 2π] such that Im(wnk ) + q = 2lπ for some l ∈ Z and 2|l|π ≥ |Im(wnk )|. Define z0 = wnk − p + iq = R + 2lπi. Then by (23), |z0 | |Im(z0 )| 2|l|πi |Im(wnk )| eRe(wnk ) ep (24) ≥ = ≥ > = = M. eR eR eR eR 3|β|eR 3|β| Note that by (20) and (23), we have p > 0 and we can assume that the constants M, R > 0 are arbitrarily large. Let η(t) = z0 + it, t ∈ [−3π, 3π] and note that η ⊂ D(wnk , p + |q| + 3π), where p + |q| + 3π ≤ ln(3|β|Rnk ) + 5π < Rnk (if k is chosen large enough), so by (20), (25) η ⊂ Wn k . FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS 15 Let γ(t) = Nh (η(t)) − z0 + 1 − ζ, where z0 ζ= . (e2R /|β|2 ) − 1 By the definition of Nh , we can write γ in the form γ(t) = γ1 (t) + γ2 (t) + it for −z0 β − α − iβt γ1 (t) = R+it − ζ, γ2 (t) = . 1+e /β β + eR+it Note that when t varies along an interval of length 2π, the curve γ1 describes the image of the circle ∂D(0, eR ) under the Möbius map z 7→ −z0 /(1 + z/β) − ζ. Easy computations show that this image is the circle ∂D(0, r) for |z0 | r= . eR /|β| − |β|e−R Hence, |β||z0 | (26) |γ1 (t)| = r ≥ eR and |α| + (3π + 1)|β| (27) |γ2 (t)| ≤ < ε1 eR − |β| for a small ε1 > 0, if we can choose R large enough. By (24), (26) and (27), |β||z0 | |β||z0 | (28) |γ(t)| ≥ |γ1 (t)| − |γ2 (t)| − 3π ≥ − ε1 − 3π > , eR 2eR if M is chosen large enough. Let −βz0 γ̃(t) = eR+it and note that, by (27) and (24), γ(t) γ1 (t) + ζ ζ γ2 (t) 3π −1 ≤ −1 + + + γ̃(t) γ̃(t) γ̃(t) γ̃(t) |γ̃(t)| 1 |β|eR (|α| + (3π + 1)|β|)eR 3πeR ≤ + + + |1 + eR+it /β| e2R − |β|2 |β||z0 |(eR − |β|) |β||z0 | 1 |β| |α/β| + 3π + 1 3π ≤ R + R 2 −R + R + < ε2 e /|β| − 1 e − |β| e M (e − |β|) |β|M for a small ε2 > 0, if M, R are chosen sufficiently large. Hence, there exists a branch Arg of the argument defined along the curve γ(t), such that (29) Arg(γ(t)) = −t + A + δ(t), where A = Arg(−βz0 ) ∈ [0, 2π] and |δ(t)| < ε2 . 16 KRZYSZTOF BARAŃSKI, NÚRIA FAGELLA, XAVIER JARQUE, AND BOGUSLAWA KARPIŃSKA Figure 3. The curve γ. By (26) and (27), (30) γ1 + γ2 ⊂ D(0, r + ε1 ) \ D(0, r − ε1 ) + − − for a small ε1 > 0 (provided R is chosen large enough) and by (29), there exist t+ 1 , t2 , t1 , t2 ∈ [−3π, 3π] such that t+ 1 = min{t ∈ [−π/2 + A − ε2 , −π/2 + A + ε2 ] : Arg(γ(t)) = π/2}, t+ 2 = min{t ∈ [−5π/2 + A − ε2 , −5π/2 + A + ε2 ] : Arg(γ(t)) = 5π/2}, t− 1 = max{t ∈ [π/2 + A − ε2 , π/2 + A + ε2 ] : Arg(γ(t)) = −π/2}, t− 2 = max{t ∈ [−3π/2 + A − ε2 , −3π/2 + A + ε2 ] : Arg(γ(t)) = 3π/2}. By definition, γ(t+ + 1 ), γ(t2 ) ∈ {z : Re(z) = 0, Im(z) > 0}, γ(t− − 1 ), γ(t2 ) ∈ {z : Re(z) = 0, Im(z) < 0}, γ|(t− ,t+ ) ⊂ {z : Re(z) < 0} for j = 1, 2. j j Since γ(t) = γ1 (t) + γ2 (t) + it, by (29) and (30) we have |γ(t+ + + + + + 1 )| − |γ(t2 )| = Im(γ(t1 )) − Im(γ(t2 )) < t1 − t2 + 2ε1 < −2π + 2ε2 + 2ε1 < 0 and |γ(t− − − − − − 1 )| − |γ(t2 )| = −Im(γ(t2 )) + Im(γ(t2 )) > t2 − t1 − 2ε1 > 2π − 2ε2 − 2ε1 > 0 (see Figure 4). Hence, by standard topological arguments, the curves γ|(t− ,t+ ) and γ|(t− ,t+ )  1 1 2 2 intersect. Together with (28), this shows thatD 0, |β||z0 |/(2eR ) lies in a bounded component of C \ γ, so D := D z0 − 1 + ζ, |β||z0 |/(2eR ) lies in a bounded component of C \ Nh (η). By (24), we have |z0 | |β||z0 | |β|M |β||z0 | |ζ − 1| ≤ |ζ| + 1 = +1≤ + ≤ (e /|β|2 ) 2R −1 4e R 4 2eR FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC FUNCTIONS 17 (provided M, R are chosen sufficiently large), which implies z0 ∈ D, so z0 in a bounded component of C \ Nh (η). By (25), we have z0 ∈ Wnk ⊂ C \ Wnk +1 and f (η) ⊂ Wnk +1 . Hence, Wnk +1 is not simply connected, which makes a contradiction.  This ends the proof of Theorem D. We finish this section proving Corollary E. Proof of Corollary E. We will check that under the hypothesis, the unique asymptotic value u of Nh is in the basin of attraction of a real fixed point of Nh and so inf dist (Nhn (u), J(Nh )) > 0. n≥0 Consider Nh restricted to the real axis. Note first that since α + β(t − 1) Nh (t) = t − 1 − , et + β we have Nh (t) < t for sufficiently large t ∈ R. Assume first β > 0. Then h has one (simple) real zero and h0 has no real zeroes, so Nh |R has no poles and exactly one fixed point, say c0 , which is superattracting, since e1−α/β +β > 0 (see the proof of Proposition 4.4) and is also the unique critical point of Nh |R , such that Nh00 (c0 ) 6= 0. Then Nh (t) < t for t > c0 and Nh |R attains the minimum at c0 . As u = limt→−∞ Nh (t), we have u > c0 , so Nhn (u) converges monotonically to c0 as n → ∞. Assume from now on β < 0. Consider first the case α = β(1−ln(−β)). Then e1−α/β +β = 0, so h has one real zero equal to c̃ defined in (18), which is also a zero of h0 . Hence, Nh |R has no poles, no critical points and exactly one fixed point c̃, which is attracting. Consequently, Nh |R is increasing and Nhn (u) converges monotonically to c̃ as n → ∞. Consider now the case α < β(1 − ln(−β)). Then h has two (simple) real zeroes, so Nh |R has one (simple) pole, say p and exactly two superattracting fixed points, say c0 < c1 , which are the unique critical points of Nh |R , such that Nh00 (c0 ), Nh00 (c1 ) 6= 0. Since Nh (t) < t for large t and u = limt→−∞ Nh (t), we have c0 < p < c1 , Nh (c0 ) = maxt

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Gran Via 585, 08007 Barcelona, Catalonia, Spain E-mail address: nfagella@ub.edu Departament de Matemàtiques i Informàtica, Institut de Matemàtiques de la Universitat de Barcelona (IMUB), and Barcelona Graduate School of Mathematics (BGSMath). Gran Via 585, 08007 Barcelona, Catalonia, Spain E-mail address: xavier.jarque@ub.edu Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Ko- szykowa 75, 00-662 Warszawa, Poland E-mail address: bkarpin@mini.pw.edu.pl