This is a preprint of: “Achievable connectivities of Fatou components for a family of singular perturbations”, Jordi Canela, Xavier Jarque, Dan Alexandru Paraschiv, Discrete Contin. Dyn. Syst., vol. 42(9), 4237–4261, 2022. DOI: [10.3934/dcds.2022051] ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF SINGULAR PERTURBATIONS JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV Abstract. In this paper we study the connectivity of Fatou components for maps in a large family of singular perturbations. We prove that, for some parameters inside the family, the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and we determine precisely these connectivities. In particular, these results extend the ones obtained in [Can17, Can18]. Keywords: holomorphic dynamics, Fatou and Julia sets, singular perturbation, connectiv- ity of Fatou components. 1. Introduction In the recent decades there has been an increasing interest in studying families of rational maps usually called singular perturbations. Roughly speaking, a family is called a singular perturbation if it is defined by a base family (called the unperturbed family and for which we have a deep understanding of the dynamical plane) plus a local perturbation, that is, a perturbation which has a significant effect on the orbits of points in some part(s) of the dynamical plane, but a very small dynamical relevancy on other regions. Singular perturbations, no matter the concrete formulas, have some common properties which make their study interesting. On the one hand, the degree of the unperturbed family is smaller than the degree of the perturbed one. Consequently, one should expect richer dynamics for singular perturbations than for the unperturbed maps. On the other hand, most of this new freedom arising from the perturbation may be captive of the dynamical properties of the unperturbed family. The balance between these two scenarios has become very successful in finding new dynamical phenomena. The relation between the topology of the dynamically invariant sets (Fatou and Julia set) and the behaviour of the critical orbit(s) is an important issue when studying the dynamical plane of a particular rational map. A paradigmatic example of this is the Dichotomy Theorem for the quadratic family. In this way, singular perturbations are somehow a perfect scenario to observe new phenomena for the invariant sets with respect to the unperturbed maps, for which we usually observe a tame topology. Indeed, the main goal of this paper is to investigate in this direction and to prove that for a certain family of singular perturbations we can construct examples for which, in the same dynamical plane, there are Fatou components with given arbitrarily high connectivities. Date: February 1, 2021. The first author was supported by Spanish Ministry of Economy and Competitiveness, through the Marı́a de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445), by BGSMath Banco de Santander Postdoctoral 2017, and by the project UJI-B2019-18 from Universitat Jaume I. The second and third authors were supported by MINECO-AEI grant MTM-2017-86795-C3-2-P. The second author was also supported by AGAUR grant 2017 SGR 1374. The third author was also supported by the Spanish government grant FPI PRE2018-086831. 1 2 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV Let f : Ĉ → Ĉ be a rational map acting on the Riemann sphere. Then f partitions the dynamical plane into J (f ), the Julia set, and F(f ), the Fatou set. The Julia set is closed and coincides with the set of points z ∈ Ĉ where the family of iterates {f n |U }n≥0 is not a normal family for any neighbourhood U of z. Its complement, the Fatou set, is an open set and its connected components are called Fatou components. By the No Wandering Domains Theorem, Fatou components of rational maps are either preperiodic or periodic (see [Sul85]). By the Classification Theorem of periodic Fatou components (see [Mil06], Theorem 16.1), every periodic Fatou component either belongs to the immediate basin of attraction of an attracting or parabolic cycle, or is a simply connected rotation domain (Siegel disk) or is a doubly connected rotation domain (Herman ring). The existence of periodic Fatou components can by studied by analysing the orbits of critical points, i.e. points where f 0 vanishes. Indeed, every immediate basin of attraction contains, at least, a critical point while Siegel disks and Herman rings have critical orbits accumulating on their boundaries. The connectivity of a domain D ⊂ Ĉ is defined as the number of connected components of its boundary. It is known that periodic Fatou components have connectivity 1, 2, or ∞. Indeed, Siegel disks have connectivity 1, Herman rings have connectivity 2, and immediate basins of attraction have connectivity 1 or ∞. Preperiodic Fatou components can have finite connectivity greater than 2. The first such example, with connectivities 3 and 5, was presented in [Bea91]. Moreover, for any given n ∈ N, there are examples of rational maps with Fatou components of connectivity n. These examples can either be obtained by quasiconformal surgery (see [BKL91]) or by giving explicit families of rational maps (see [QG04] and [Ste93]). However, the degree of the rational maps obtained in all previous examples grows rapidly with n. To our knowledge, the first example of rational map whose dynamical plane contains Fatou components of arbitrarily large finite connectivities was presented in [Can17] (see also [Can18]) by using singular perturbations. However, in these papers it is not shown which precise connectivities can actually be attained. The goal of this paper is to study the attainable connectivities for a wider family of singular perturbations which includes the ones studied in [Can17, Can18]. We also want to remark that while this paper was being prepared we knew that, independently, professor Hiroyuki has obtained another family of rational maps with Fatou components of arbitrarily large connectivity [Hir]. Singular perturbations of rational maps were introduced by McMullen in [McM88]. He proposed the study of the family λ (1) Qn,d,λ (z) = z n + , zd where n, d ≥ 2 and λ ∈ C, |λ| small. Observe that in (1) the unperturbed map is the simplest possible: z n . He considered the case n = 2 and d = 3 and he proved that if |λ| is small enough then the Julia set is a Cantor sets of quasicircles (the result actually holds for n and d satisfying 1/n + 1/d < 1 (compare [DLU05])). Later, Devaney, Look, and Uminsky [DLU05] considered (1) as a λ-family of rational maps and they extended McMullen’s result by proving the Escape Trichotomy. More specifically, they showed that if all critical points belong to the basin of attraction of infinity then the Julia set is a Cantor set, a Sierpinski carpet, or a Cantor set of quasicircles (McMullen’s case). The proof relays on the fact that there is a symmetry in the dynamical plane which implies that there is a unique free critical orbit (the symmetry forces all critical points to follow symmetric orbits). Other models similar to (1) have also been considered. For instance, in [BDGR08, GMR13] the authors consider singular perturbations of polynomials of the form z n + c, c ∈ C, choosing c appropriately. Those ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 3 examples have shown Julia and Fatou sets with new and rich topology, but the connectivity of the Fatou components is kept as 1, 2 or ∞. The examples mentioned in the previous paragraph are done perturbing maps with no free critical points: one or more poles are added to superattracting cycles which contain no critical points, other than the critical points of the cycle, in their basins of attraction. A next natural step is to consider singular perturbations of maps with free critical points. A good candidate for such a perturbation is the family of Blaschke products z−a Bn,a (z) = z n , where a ∈ C and n ≥ 2. 1 − az See [CFG15] for an introduction to the dynamics of these maps for n = 3. If a belongs to the punctured unit disk D∗ := D \ {0}, the maps Bn,a restrict to automorphisms of the unit disk whose dynamical plane is trivial. Indeed, its Fatou set consists of two invariant components given by the immediate basin of attraction A∗ (0) of z = 0 (the unit disk) and the immediate basin of attraction A∗ (∞) of z = ∞ (the complement of the closed unit disk). Their common boundary component, the unit circle, is the Julia set of these maps. Moreover, if a ∈ D∗ the map Bn,a has only two simple critical points c− ∈ A∗ (0) and c+ ∈ A∗ (∞), other than the superatracting fixed points z = 0 and z = ∞. In [Can17, Can18] the author studied the family of singular perturbations of the maps Bn,a given by z−a λ Bn,d,a,λ (z) = z n + , 1 − az z d where a ∈ D∗ and λ ∈ C∗ := C \ {0}, for n = 3 and d = 2. Compared to McMullen’s singular perturbations, these maps can present a much richer dynamics since their free critical points (which come from the the singular perturbation and the continuous extension of c± ) are not tied by any kind of symmetry. Despite that, in [Can17] it was proven that if |λ| is small enough the family B3,2,a,λ (z) is essentially unicritical: all critical points but the continuous extension c− (λ) of c− belong to the basin of attraction of infinity A(∞). In that case, if c− (λ) belongs to a Fatou component in A(∞) which surrounds z = 0, the dynamical plane has Fatou components of arbitrarily large finite connectivity. The actual existence of parameters for which this actually happens was proven in [Can18]. We want to point out that the same results can be proven for n, d ≥ 2 such that 1/n + 1/d < 1. In Figure 1 we illustrate the dynamical plane of Bn,d,a,λ (z) for a = 0.5, d = 3, and different values of n and λ. The goal of this paper is to extend the results in [Can17, Can18] to a wider family of singular perturbations and to study which connectivities are attainable for such family. With this aim we consider the family of degree n + 1 rational maps given by z n (z − a) (2) Sn,a,Q (z) = , Q(z) where n ≥ 2, a ∈ C∗ , and Q is a polynomial of degree at most n. On the one hand it is clear that the family Sn,a,Q contains the family Bn,a . On the other hand it is worth to be noticed that Sn,a,Q also includes the family Mn,a (z) = z n (z − a), where n ≥ 2 and a ∈ C. This family was first introduced by Milnor in 1991 (see [Mil09]) when studying cubic polynomials (n = 2) and was later studied by Roesch [Roe07] for n ≥ 2. If a 6= 0, these maps have z = 0 and z = ∞ as superattracting fixed points of local degree n and n + 1, respectively. Moreover, they have a unique free critical point ca 6= 0 and the global 4 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV phase portrait settles down on its dynamical behaviour. It is easy to see that, if |a| is small enough, ca belongs to the immediate basin of attraction of z = 0 and the Julia set consists of a quasicircle which separates the immediate basins of attraction of z = 0 and z = ∞ (see Corollary 2.2). In this sense, for |a| small the family Mn,a can be understood as a simplified version of Bn,a , |a| < 1, where there is no free critical point in A∗ (∞) but the Julia set is a quasicicle instead of a circle. We now turn to the unperturbed family Sn,a,Q . Inspired by the work in [Can17, Can18] we will impose the following conditions to be satisfied for the maps in Sn,a,Q . (a) The point z = 0 is a superattracting fixed point of degree n of Sn,a,Q . In particular Q(0) 6= 0. (b) The fixed point z = ∞ is (super)attracting. In particular the coefficient of z n of Q, say bn , satisfies 0 ≤ |bn | < 1. (c) There are exactly two Fatou components: the immediate basins of attraction A∗ (0) and A∗ (∞) of z = 0 and z = ∞, respectively. Remark 1. We can deduce the following observations from the above conditions. Since the maps Sn,a,Q have degree n + 1, the immediate basins of attraction are mapped onto themselves with degree n + 1 and, hence, each of them contains exactly n critical points counting multi- plicity. In particular, the basin of attraction of z = 0 (which is a critical point of multiplicity n − 1) contains a simple critical point ν0 6= 0. Once the unperturbed family has been described, we now consider the singular perturbation λ (3) Sn,d,λ (z) = Sn,a,Q (z) + , λ ∈ C∗ , d ≥ 2. zd Notice that to simplify notation we do not specify the dependence on a and Q of the family Sn,d,λ . Notice also that the family Sn,d,λ includes the family Bn,d,a,λ . It follows immediately that all maps Sn,d,λ have degree n + d + 1 and that for λ 6= 0 the point z = 0 is a pole of degree d. We will say that Sn,d,λ satisfies (a), (b), and (c) if Sn,a,Q satisfies the conditions (a), (b), and (c) explained above. Analogously to the condition needed to obtain a Cantor set of quasicircles for McMullen’s family (see [DLU05]), we have to add a fourth condition to the family: (d) The numbers n, d ≥ 2 are such that n1 + d1 < 1. In other words, we exclude n = d = 2. Since the critical points are not tied by any relation, for |λ| big the dynamics can be very rich. Despite that, if |λ| is small the family is essentially unicritical. Indeed, there exists a constant C > 0 such that if |λ| < C, λ 6= 0, the following hold (see Proposition 3.5): • The continuous extensions of the n critical points which belong to the immediate basin of attraction A∗0 (∞) of z = ∞ before perturbation belong to the immediate basin of attraction A∗λ (∞) of z = ∞ after perturbation. Moreover, A∗λ (∞) is a quasidisk. • The pole z = 0 belongs to a quasidisk Tλ (usually called trap door) which is mapped onto A∗λ (∞) under Sn,d,λ . • The n + d critical points which appear around z = 0 after perturbation belong to a doubly connected Fatou component Aλ which is mapped onto Tλ under Sn,d,λ . The previous points actually coincide with the skeleton of the dynamics in the Cantor set of quasicircles case of McMullen’s family (see [McM88]). This is why the dynamical planes for this perturbed family resemble the dynamical planes for the Cantor set of quasicircles with extra decorations (see Figure 1 and Figure 2). These decorations come from the presence of ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 Figure 1. Dynamical planes of the family Bn,d,a,λ (z) for d = 3. The top-left figure corresponds to n = 2 and λ = 2 · 10−8 ; the top-right corresponds to n = 3 and λ = −5 · 10−8 ; the bottom-left figure corresponds to n = 4 and λ = −6.3 · 10−9 ; and bottom-right corresponds to n = 5 and λ = −1.2 · 10−10 . In all cases we can see the triply connected regions (where the critical point νλ lies) and their eventual preimages, which are Fatou components with increasing connectivity. the extra critical point νλ , which comes from the continuous extension of the critical point ν0 that belongs to the basin of attraction of z = 0 before perturbation. This is the only critical point which may not belong to the basin of attraction Aλ (∞) of z = ∞ after perturbation if |λ| is small. We want to remark that the main difference between Sn,d,λ and the particular family Bn,d,a,λ is that we allow certain degrees of freedom in the n critical points that lie in 6 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1 1.5 Figure 2. Left figure illustrates the dynamical planes of Mn,a for n = 2 and a = (0.9 + 0.6i). Right picture illustrates the dynamical plane of the (perturbed) family Sn,d,λ when the unperturbed map is precisely M2,a , and the pertubation corresponds to d = 3 and λ = −10−7 . We can see in the right figure the triply connected Fatou component which contains νλ and its eventual preimages with higher connectivity. A∗λ (∞). For instance, if the degree of Q is 0, then z = ∞ is a superatracting fixed point of local degree n. On the other hand, if the degree of Q is n, then z = ∞ is attracting (but not superattracting) and there are n critical points which move in A∗λ (∞). Also, the shape of the Julia set before perturbation affects the shape of the Julia set of the perturbed map (see Figure 1 and Figure 2). Recall that in the Blaschke case the unperturbed Julia set is the unit circle. The goal of the paper is to analyse the connectivities which can be achieved with these singular perturbations. The critical point νλ is crucial in order to increase the connectivities beyond 2. Indeed, if νλ belongs to a preimage Uν of Aλ then the Fatou component Uν is triply connected. Moreover, if Uν surrounds z = 0 then we can find sequences of iterated preimages of Uν which increase the connectivity with every iteration. The next theorems describe the connectivities which can be achieved with this process. We denote by Bdd(Aλ ) the union of the connected component of the complement of Aλ not containing z = ∞ and the annulus itself. Theorem A. Let Sn,d,λ satisfying (a), (b), (c), and (d) and let λ 6= 0, |λ| < C. Assume also that νλ ∈ Uν , where Uν is an iterated preimage of Aλ which surrounds z = 0. Let k be the minimal number of iterations needed by the free critical point νλ to be mapped into Bdd(Aλ ). Let U be a Fatou component of connectivity κ > 2. Then, there exist i, j, ` ∈ N such that κ = (n + 1)i dj n` + 2 and ` ≤ jk. In other words, Theorem A is telling us all potential connectivities κ > 2 for a Fatou component of a map in Sn,d,λ for |λ| sufficiently small; but it is not claiming the existence of a Fatou component of each (i, j, `)-connectivity. The next result complements Theorem A ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 7 and it gives the connectivities that are certainly achieved for any parameter λ as long as |λ| is sufficiently small and νλ satisfies certain dynamical conditions. Theorem B. Let Sn,d,λ satisfying (a), (b), (c), and (d) and let λ 6= 0, |λ| < C. Assume also that νλ ∈ Uν , where Uν is an iterated preimage of Aλ which surrounds z = 0. Let k ≥ 1 be the minimal number of iterations needed by the free critical point νλ to be mapped into Bdd(Aλ ). For any given i, j, ` ∈ N such that ` ≤ j(k − 1), there exists a Fatou component U of connectivity κ = (n + 1)i dj n` + 2. In Theorem A and Theorem B the achievable connectivities depend on the minimal number of iterations k > 0 needed by the free critical point νλ to be mapped into Bdd(Aλ ). However, choosing λ appropriately we can make this k as big as desired. Therefore, for any ` and j we can find λ so that the inequality ` ≤ j(k − 1) is satisfied. From this, we obtain Theorem C. Theorem C. Let Sn,d,λ satisfying (a), (b), (c), and (d) and let λ 6= 0, |λ| < C. For any given i, l ≥ 0 and j > 0, there exists a parameter λ such that Sn,d,λ (z) has a Fatou component of connectivity κ = (n + 1)i dj n` + 2, and a Fatou component of connectivity κ = (n + 1)i + 2. The paper is organised as follows. In Section 2 we briefly introduce the tools later used in the paper. In Section 3 we describe in detail the skeleton of the dynamical plane of Sn,d,λ satisfying the conditions (a), (b), (c) and (d) for |λ| small enough. In Section 4 we prove Theorems A and B. Finally, in Section 5 we prove Theorem C. 2. Preliminaries In this section we present the main tools that we use along the paper. Before, we introduce some notation. In the introduction we used the notation Bdd(A) to denote the set bounded by an annulus A, including itself. It will be useful to generalise this concept for other multiply connected sets. Let U ⊂ C be a multiply connected open set. We denote by Bdd(U ) the minimal simply connected open set which contains U but not z = ∞. Let γ ∈ C be a Jordan curve. We denote by Ext (γ) and Int (γ) the connected components of Ĉ \ γ that contain z = ∞ and do not contain z = ∞, respectively. We denote by A(γ1 , γ2 ) the open annulus bounded by Jordan curves γ1 and γ2 with γ1 ⊂ Int (γ2 ). We denote the circle centered at the origin and of radius c > 0 by Sc . Finally, if U ⊂ C we denote by U its closure. We now proceed to introduce the needed tools. The next result provides a sufficient criterion for the Julia set of a map to be a quasicircle. Theorem 2.1. [CG93, Theorem 2.1, page 102] If the Fatou set of a rational map R contains exactly two Fatou components and the map R is hyperbolic on its corresponding Julia set J(R), then J(R) is a quasicircle. We can immediately conclude that the Julia sets of the maps Sn,a,Q are quasicircles. Corollary 2.2. Let Sn,a,Q satisfying (a), (b), and (c). Then, its Julia set is a quasicircle. Finally we recall the Riemann-Hurwitz formula (see for instance [Ste93]), which we use in order to study connectivities of Fatou components. Theorem 2.3. (Riemann-Hurwitz formula) Let U, V ⊂ Ĉ be two connected domains of con- nectivity mU , mV ∈ N∗ and let f : U → V be a degree d proper map branched over r critical points, counted with multiplicity. Then, mU − 2 = d(mV − 2) + r. 8 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV Along the text we also use the following corollary of the Riemann-Hurwitz formula (compare [CFG15, Corollary 2.2]). Corollary 2.4. Let U ⊂ Û be an open set and let f : U → f (U ) be a proper holomorphic map. Then, the following statements hold: (i) If f (U ) is doubly connected and f has no critical points in U , then U is doubly con- nected. (ii) If f (U ) is simply connected and f has at most one critical point in U (not counting multiplicities), then U is simply connected. 3. The perturbed family Let Sn,d,λ satisfying conditions (a), (b), and (c) described above. This section describes the main properties of the dynamical plane for parameters belonging to a neighbourhood of λ = 0. We first describe the immediate basin of attraction of z = ∞, which we further denote by A∗λ (∞), and its boundary. The proof of Proposition 3.1 uses the theory of polynomial-like mappings introduced by Douady and Hubbard in [DH85] (see also [BF14]). Remark 2. Along the paper, when we say that a compact set moves continuously with respect to parameters, we use the topology induced by the Hausdorff metric for compact sets. Proposition 3.1. Let Sn,d,λ satisfying conditions (a), (b), and (c). Then, for |λ| small enough, the following hold: (i) The Fatou component A∗λ (∞) is mapped onto itself with degree n + 1. (ii) The boundary of A∗λ (∞) is a quasicircle that moves continuously with respect to λ. (iii) The set A∗λ (∞) contains exactly n critical points counting multiplicity. Each critical point of Sn,d,λ in A∗λ (∞) is a continuous extension of a critical point of Sn,a,Q in A∗ (∞). Proof. Observe that the three statements are trivially satisfied (by definition and Corollary 2.2) for the unperturbed family. So this proposition says that this conditions are still true if the perturbation is small enough. To prove the proposition we show the existence of an analytic family of polynomial-like maps which ensures the continuous deformation of the key dynamical objects. Fix Sn,a,Q and let U be the maximal domain of Bottcher coordinates around z = 0. The critical point ν0 lies on ∂U ⊂ A∗ (0). Let γ be an analytic Jordan curve surrounding the origin such that γ ∈ U \ Sn,a,Q (U ). We now show that the preimage of γ has a unique connected component. Let A := A(γ, ∂A∗ (0)). Notice that A ⊂ A∗ (0). Observe that the annulus A does not include any critical value. By Corollary 2.4(i), its preimage A−1 (under Sn,a,Q ) is also an annulus. Since ∂A∗ (0) is mapped by Sn,a,Q onto itself with degree n + 1, A−1 is also mapped onto A with degree n + 1. Let γ0−1 be the connected component of ∂A−1 other than ∂A∗ (0). Since A−1 is mapped onto A with degree n + 1 under Sn,a,Q , then γ0−1 is mapped onto γ with degree n + 1 under Sn,a,Q . Since Sn,a,Q has (global) degree n + 1, we conclude that there is no other preimage of γ than γ0−1 under the map Sn,a,Q . Let V = Ext(γ) and U0 = Ext(γ0−1 ). It follows from the construction that U0 is compactly contained in V and Sn,a,Q |U0 : U0 7→ V is a proper map of degree n + 1. Therefore, the triple (Sn,a,Q |U0 , U0 , V) is a degree n + 1 polynomial-like map (see [DH85], see also [BF14]). We want to extend (for |λ| small enough) this map to a J -stable analytic family of polynomial like mappings. Observe that the map Sn,d,λ depends analytically on λ for all z ∈ C b \ Dε , ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 9 where Dε denotes the disk of radius ε centered at z = 0, for all λ ∈ C and all ε > 0. Recall that Sn,a,Q = Sn,d,0 . Therefore, if |λ| is small enough, the continuous extensions of the n critical points (counting multiplicity) which lie in A∗ (∞) = A∗0 (∞) for Sn,a,Q lie in A∗λ (∞) for Sn,d,λ . Moreover, if |λ| is small enough then there exists a unique connected component γλ−1 of γ under the map Sn,d,λ−1 which is an analytic Jordan curve. In fact, γλ−1 is a continuous deformation of γ0 and it is mapped with degree n+1 onto γ under Sn,d,λ . Let Uλ = Ext(γλ−1 ). −1 Decreasing |λ| if necessary, we can ensure the following. The set Uλ is compactly contained in V and the only critical points of Sn,d,λ in Uλ are the ones which come from the continuous extension of the critical points in A∗ (∞). Moreover, Sn,d,λ |Uλ : Uλ 7→ V is a proper map of degree n + 1 and the triple (Sn,d,λ |Uλ , Uλ , V) is a degree n + 1 polynomial-like mapping. Let Λ be an open disk centered at λ = 0 compactly contained in the open set of parameters for which the previous conditions hold. Then, {(Sn,d,λ |Uλ , Uλ , V)}λ∈Λ defines an analytic fam- ily of polynomial like mappings (see [DH85], see also [BF14]). Let Kλ := {z ∈ Uλ | Sn,d,λ n (z) ∈ Uλ for all n ≥ 0} and Jλ = ∂Kλ denote the filled Julia set and the Julia set of the polynomial like map (Sn,d,λ , Uλ , V), respectively. Notice that K0 = A∗ (∞). Notice also that all connected components of the interior of Kλ are Fatou components of Sn,d,λ . Therefore, since the point z = ∞ belongs to Kλ for all λ ∈ Λ, we conclude that A∗λ (∞) ⊂ Kλ . To finish the proof, we observe that since all critical points of Sn,d,λ |Uλ belong to A∗λ (∞) it follows that the analytic family of {(Sn,d,λ , Uλ , V)}λ∈Λ is J -stable. In particular, the Julia sets Jλ are quasicircles which are continuous deformations of J0 = ∂A∗ (∞) (see [DH85, Proposition 10]). Notice that, by Corollary 2.2, ∂A∗ (∞) is a quasicircle. Since A∗λ (∞) ⊂ Kλ , we can conclude that ∂A∗λ (∞) = Jλ for all λ ∈ Λ. This proves (ii). Statements (i) and (iii) follow from the choice of the set of parameters Λ.  The first part of the following lemma describes a neighbourhood of z = ∞ which, for |λ| small enough, always lies in the interior of A∗λ (∞). The second part shows that z = 0 lies in a preimage of A∗λ (∞), different from it. Lemma 3.2. Let Sn,d,λ satisfying conditions (a), (b), and (c). Then, for |λ| small enough, the following happen: (i) There exists a constant K, which only depends on n, a, and Q, such that z ∈ A∗λ (∞) if |z| > K. (ii) Assume that Sn,d,λ also satisfies n o (d). For any constant K1 > 0, if |λ| condition n+d is small enough, the disk |z| < K1 |λ| nd belongs to a Fatou component Tλ . The Fatou component Tλ is mapped onto Aλ (∞) and it is different from A∗λ (∞). ∗ Proof. We begin with statement (i). From condition (b) we know that, for fixed Sn,a,Q , there exists a constant K such that the set {z ∈ C| |z| > K} is compactly contained in the immediate basin of attraction of ∞. By continuity with respect to λ, for |λ| small enough, this set is also contained in A∗λ (∞). For statement (ii), let K1 > 0. Assume that λ is such that (i) is satisfied for the constant n K above. Let z ∈ C such that |z| < K1 |λ| n+d . It follows that nd 1 λ z n (z − a) |λ|1− n+d |λ| n+d K1n (|a| + 1) |Sn,d,λ (z)| > d − > − =: C1 (λ) + C2 (λ). z Q(z) K1d M 10 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV Notice that C2 (λ) tends to 0 as λ tends to 0. Because of assumption (d), C1 (λ) tends to n ∞ as λ tends to 0. Shrinking n |λ| if necessary, o if |z| < |λ| n+d K1 , then |Sn,d,λ (z)| > K. We n+d conclude that the set |z| < K1 |λ| nd belongs to a Fatou component. This Fatou component contains z = 0, which is mapped to ∞ with degree d. By continuity with respect to λ and Proposition 3.1, ∂A∗λ (∞) is a quasicircle which surrounds z = 0. It follows that A∗λ (∞) does not contain the origin and z = 0 belongs to a preimage of A∗λ (∞), different from A∗λ (∞), which we denote by Tλ .  Recall that each map of the perturbed family has global degree n + d + 1. Hence, it has 2(n + d) critical points (counting multiplicity). By Proposition 3.1, n + d − 1 of them lie in A∗λ (∞) ∪ {0}. By continuity with respect to λ, there is a (simple) critical point νλ which is the continuous extension of the critical point ν0 of Sn,a,Q in A∗ (0). Each map has n + d + 1 zeros, one of which, say wλ , corresponds to the continuous extension of w0 = a. We now give a description of the position of the remaining n + d critical points and the n + d preimages of z = 0 for Sn,d,λ . Lemma 3.3. Let Sn,d,λ satisfying conditions (a), (b), and (c). Assume ξ ∈ C is a (n + d)th- root of unity, ξ n+d = 1. Then, for |λ| small enough, there exist n + d free critical points, cλ,ξ , and n + d zeros, zλ,ξ , given by   1  1  dQ(0) n+d n+d 1 cλ,ξ = ξ λ + o λ n+d , −na   1  1  Q(0) n+d n+d1 zλ,ξ = ξ λ + o λ n+d . a Proof. Let us start with the zeros. Notice that all the zeros of Sn,d,λ (except for wλ ) must converge to z = 0 when λ tends to 0. The zeros of Sn,d,λ (z) are the solutions of z n+d (z − a) = −λQ(z). Since a is away from z = 0, there are n + d zeros bifurcating from z = 0, for |λ| small enough. They are the fixed points of n + d operators   1 Q(z) n+d n+d1 1 Tλ,ξ (z) = ξ λ = R(z)λ n+d , a−z where ξ n+d = 1 are roots of the unity. Observe that in a sufficiently small neighbourhood of z = 0, R(z) is holomorphic and bounded (notice that R(0) 6= 0), so Tλ,ξ (z) → 0 as λ → 0.   1 1 We can approximate zλ,ξ by Tλ,ξ (0) = ξ Q(0) n+d a λ n+d . Indeed, 1 0 |zλ,ξ − Tλ,ξ (0)| = |Tλ,ξ (zλ,ξ ) − Tλ,ξ (0)| ≤ sup |Tλ,ξ (ω)||zλ,ξ −0| = |λ| n+d sup |R0 (ω)||zλ,ξ |. ω∈[0,zλ,ξ ] ω∈[0,zλ,ξ ] For |λ| small enough, there is no pole of R0 in a neighbourhood of z = 0 containing the line segment [0, zλ,ξ ], so it is bounded by a constant, say K2 . It follows that   1 1 Q(0) n+d n+d zλ,ξ − ξ a λ 1 ≤ K2 |zλ,ξ |. λ n+d ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 11 Finally, since lim K2 |zλ,ξ | = 0, it follows that λ→0   1  1  Q(0) n+d 1 zλ,ξ = ξ λ n+d + o λ n+d . a It can be shown analogously that the n + d free critical points are solutions of the equation 1   λd 2 (n + 1)z n Q(z) − z n+1 Q0 (z) − anz n−1 Q(z) + az n Q0 (z) − d+1 = 0. Q (z) z As before, we write the operators Sλ,ξ as   n+d 1 dQ2 (z) 1 Sλ,ξ (z) = ξ 2 0 0 λ n+d , (n + 1)zQ(z) − anQ(z) − z Q (z) + azQ (z) which have the critical points of Sn,d,λ as fixed points. The argument made is identical since Q(0) 6= 0, so Sλ,ξ are holomorphic in the neighbourhood of z = 0. Finally, the critical points of the perturbation map are of the form   1  1  dQ(0) n+d n+d 1 cλ,ξ = ξ λ + o λ n+d . −na  Next we show that there exists a straight annulus (we will show later that it belongs to a doubly connected Fatou component) which is mapped into Tλ under Sn,d,λ . Let ( 1 1 ) ( 1 1 ) 1 dQ(0) n+d Q(0) n+d dQ(0) n+d Q(0) n+d c1 = min , and c2 = 2 max , . 2 na a na a Lemma 3.4. Let Sn,d,λ satisfying conditions (a), (b), (c), and (d). Then, for |λ| small enough, the straight annulus   (4) Ωλ := A S 1 ,S 1 c1 |λ| n+d c2 |λ| n+d contains the points cλ,ξ and zλ,ξ introduced in Lemma 3.3 and it is mapped into Tλ under Sn,d,λ . Proof. The first part of the statement follows directly from the algebraic expression of the points cλ,ξ and zλ,ξ in Lemma 3.3. The rest of the proof is devoted to show that Sn,d,λ (Ωλ ) ⊂ Tλ . Let m = min{|z|, Q(z) = 0} and let M = min{|Q(z)|, |z| < m/2} (notice that M > 0 since z = 0 is not a root of Q). Let z ∈ Ωλ . For |λ| small enough we have n n cn |λ| n+d (|a| + 1) |λ| n+d |Sn,d,λ (z)| < 2 + . M cd1 n We can rewrite this as |Sn,d,λ (z)| < K1 |λ| n+d , where K1 depends on Q, c1 and c2 , but it does not depend on z and λ. By Lemma 3.2, for |λ| small enough, the disk centered at z = 0 and n of radius K1 |λ| n+d lies in Tλ , as desired.  12 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV A∗λ (∞) Dλ νλ Aλ Tλ Figure 3. Partition of the dynamical plane with respect to A∗λ (∞), Aλ , Tλ , and Dλ , described in Proposition 3.5. Blue and purple points denote zeros and critical points, respectively. In the next proposition we describe the skeleton of the dynamical plane for |λ| small (compare Figure 3). Recall that wλ is the zero of Sn,d,λ which is the continuous extension of w0 = a and νλ is the continuous extension of the critical point ν0 in A∗ (0) of Sn,a,Q . Proposition 3.5. Let Sn,d,λ satisfying conditions (a), (b), (c), and (d). Then, there exists a constant C = C(a, Q, n, d) such that if λ 6= 0 and |λ| < C the following statements are satisfied: (i) The Fatou component Tλ is simply connected and it is mapped with degree d onto A∗λ (∞) under Sn,d,λ . There are no other preimages of A∗λ (∞). (ii) There exists a Fatou component Aλ which is doubly connected and contains exactly n + d simple critical points, given by cλ,ξ , and n + d zeros, given by zλ,ξ . Moreover, Aλ is mapped with degree n + d onto Tλ and surrounds the origin. (iii) Let Aout be the annulus bounded by Aλ and ∂A∗λ (∞). There exists a Fatou component Dλ ⊂ Aout which is simply connected, is mapped with degree 1 onto Tλ , and contains wλ . (iv) The critical point νλ lies in Aout \ Dλ . (v) There are no preimages of Tλ other than Dλ and Aλ . (vi) Let Ain be the annulus bounded by ∂Tλ and Aλ . Then, Ain is mapped onto the annulus A(∂Tλ , ∂A∗λ (∞)) with degree d. Proof. Before proving the statements of the proposition we study the location and distribution of the critical points of Sn,d,λ . By Proposition 3.1, A∗λ (∞) is simply connected (in the Riemann sphere) and it is mapped onto itself with degree n + 1. Since the global degree of the map Sn,d,λ is n + d + 1, there exist exactly d preimages of ∞ outside A∗λ (∞), counting multiplicity. Since z = 0 is mapped to ∞ with degree d, there exist no other preimages of ∞ (different from the ones in A∗λ (∞) and z = 0). Moreover Tλ is mapped with degree d onto A∗λ (∞) (observe that up to this point we still do not know if Tλ is simply connected). ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 13 Let Ωλ be the annulus defined in (4). By Lemma 3.4, we know that Sn,d,λ (Ωλ ) ⊂ Tλ and Sn,d,λ (Tλ ) ⊂ A∗λ (∞). Thus, Ωλ ∩ Tλ = ∅ and Ωλ is part of a multiply connected Fatou component which is a preimage of Tλ . We denote this Fatou component by Aλ (observe that up to this point we still do not know if Aλ is doubly connected). We claim that wλ and νλ do not belong to Tλ ∪ Aλ . To see the claim we will prove that, for sufficiently small values of |λ|, wλ and νλ belong to the annulus bounded by ∂A∗λ (∞) and Aλ , denoted in what follows by Aout . Let γ be a smooth Jordan curve which separates z = 0 from ν0 and w0 , and such that Sn,a,Q2 (γ) is a Jordan curve that surrounds z = 0 and lies in Int(γ). Its existence follows from the Bötcher coordinates of the fixed point z = 0 for the unperturbed map. Notice that, by construction, γ does not depend on λ and it has a definite 2 distance to z = 0. By continuity with respect to λ, for |λ| small enough, Sn,d,λ (γ) ⊂ Int(γ). 2 ∗ Since Sn,d,λ (Aλ ) ⊂ Aλ (∞) we conclude that γ ∩ Aλ = ∅. Shrinking |λ|, if necessary, we claim that γ lies outside Bdd (Ωλ ). Indeed, according to (4) the annulus Ωλ collapses to the origin as λ → 0 while γ keeps in a definite distance to z = 0. Finally, notice that for |λ| small νλ and wλ remain as close as we want to ν0 and w0 , respectively. Therefore, γ separates νλ and wλ from Tλ and Aλ . Let C be a constant such that if |λ| < C all the above is true. Now we are ready to prove the statements. Since Tλ contains only one critical point at z = 0 with multiplicity d − 1 and it is mapped with degree d onto the topological disk A∗λ (∞), it follows from the Riemann-Hurwitz formula that Tλ is simply connected. This proves (i). Similarly, Aλ contains exactly n + d simple critical points and it is mapped with degree n + d onto the topological disk Tλ . Thus, it is doubly connected by Riemann-Hurwitz formula. This proves (ii). The point wλ is a preimage of z = 0 which lies in Aout , so it must belong to a Fatou component, denoted by Dλ , different from Tλ and Aλ . Moreover, Sn,d,λ (Dλ ) = Tλ . Since wλ is the only (simple) preimage of z = 0 in Dλ we conclude that Dλ is mapped with degree 1 onto Tλ and is a conformal copy of Tλ . In particular, νλ ∈ / Dλ and all preimages of z = 0 belong to either Aλ and Dλ . This proves (iii), (iv), and (v). Finally, to prove statement (vi) we just notice that Sn,d,λ |Ain : Ain → A(∂Tλ , ∂A∗λ (∞)) is a proper map. Since its degree is accomplished on the boundaries and ∂Tλ is mapped onto ∂A∗λ (∞) with degree d, it follows that Ain is mapped onto its image with degree d.  We now prove that Sn,d,λ is conjugate to a finite Blaschke product on the annulus Aout introduced in the previous proposition. Proposition 3.6. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Then, there exist an analytic Jordan curve Γ ⊂ Aλ which surrounds z = 0, b ∈ D∗ , θ ∈ [0, 1), and a quasiconformal map ϕ : Ĉ → Ĉ such that ϕ ◦ Rb,θ = ϕ ◦ Sn,d,λ on A(Γ, ∂A∗λ (∞)), where z−b Rb,θ = e2πiθ z n 1 − bz is a Blaschke product. Proof. We claim that there exists an analytic Jordan curve Γ ⊂ Aλ which surrounds z = 0 and the n + d critical points cλ,ξ and which is mapped with degree n to its image under Sn,d,λ . To see the claim let γ be an analytic Jordan curve in the interior of Tλ surrounding z = 0 and the n + d critical values, images of the critical points cλ,ξ . Clearly, the annulus A(γ, ∂Tλ ) contains no critical values and, from the Riemann-Hurwitz formula (compare Corollary 2.4), 14 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV any connected component of its preimage is an annulus bounded by preimages of γ and ∂Tλ . It follows from Proposition 3.5 that two (among a total of three) of those preimages are disjoint annuli in Aλ , one associated to the internal boundary of Aλ and another associated to the external one. Denote them by G in and G out . By construction, Sn,d,λ restricted to those two preimages is a proper map. We know that Sn,d,λ restricted to Aλ is proper of degree n + d (see Proposition 3.5 (ii)) while Sn,d,λ restricted to G in is proper of degree d (notice that the degree is achieved in the boundary, compare with Proposition 3.5(vi)). All together implies that Sn,d,λ restricted to G out is proper of degree n. Let Γ be the inner boundary of G out . Then, Γ is an analytic Jordan curve, it maps to γ with degree n, and it surrounds the origin as well as all critical points cλ,ξ , as desired. The remaining part of the proof is analogous to the one of [Can18, Proposition 3.1], so we provide the main idea and leave the details to the reader. The strategy is to use a similar construction to the one of the Straightening Theorem for polynomial-like mappings (compare [BF14, Theorem 7.4]) to glue a dynamics conjugated to the one of the map z → z n inside the curve γ, keep Sn,d,λ outside Γ, and interpolate using a quasi-conformal map in the annulus A (γ, Γ). In this way we obtain a quasiregular map F of the Riemann sphere which has z = 0 as superattracting fixed point of local degree n (F is actually holomorphic around z = 0). The map F coincides with Sn,d,λ on Ext(Γ), all points in Int(∂A∗λ (∞)) converge to z = 0 under iteration of F , and it maps Int(∂A∗λ (∞)) onto itself with degree n + 1 (since we have that z = 0 maps to itself with degree n and wλ is the only further preimage of z = 0). The map F is conjugate to a holomorphic function f via a quasiconformal ϕ map fixing z = 0. The basin of attraction of z = 0 under f is given by ϕ(Int(∂A∗λ (∞))). Since the basin of attraction is simply connected, f is conjugate to a Blaschke product in ϕ(Int(∂A∗λ (∞))). Since z = 0 is superattracting of local degree n and f maps ϕ(Int(∂A∗λ (∞))) onto itself with z−b degree n + 1, the Blaschke product has the form Rb,θ = e2πiθ z n 1−bz , where b ∈ D∗ and θ satisfies |e2πiθ | = 1. Since F coincides with Sn,d,λ in A(Γ, ∂A∗λ (∞)), it follows that Sn,d,λ is conjugate to Rb,θ in A(Γ, ∂A∗λ (∞)).  It will be crucial in what follows to have a deep understanding of the preimages of curves which surround the origin z = 0 (as well as Fatou components). The following proposition describes this in a precise way. Proposition 3.7. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Let −1 γ ⊂ A(∂Tλ , ∂A∗λ (∞)) be a Jordan curve which surrounds z = 0. Then, Sn,d,λ (γ) contains a in single connected component in A , which surrounds z = 0 and is mapped with degree d onto −1 γ. The other components of Sn,d,λ (γ) lie in Aout and, depending on the location of the free critical value, i.e. Sn,d,λ (νλ ), one of the following holds: −1 (i) If Sn,d,λ (νλ ) ∈ Int(γ), then Sn,d,λ (γ) has a single connected component in Aout . Indeed, it is a Jordan curve which surrounds z = 0 and it is mapped with degree n + 1 onto γ under Sn,d,λ . −1 (ii) If Sn,d,λ (νλ ) ∈ γ, then Sn,d,λ (γ) has a single connected component in Aout consisting of 2 Jordan curves intersecting precisely at νλ . One is a Jordan curve γ0−1 which surrounds z = 0, but not wλ , and it is mapped with degree n onto γ. The other is a Jordan curve γw −1 which surrounds w , but not z = 0, and it is mapped with degree 1 λ onto γ. The curve γ0−1 does not surround γw −1 . ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 15 −1 (iii) If Sn,d,λ (νλ ) ∈ Ext(γ), then Sn,d,λ (γ) has 2 disjoint components in Aout . One is a Jordan curve γ0−1 which surrounds z = 0, but not wλ , and it is mapped with degree n onto γ. The other is a Jordan curve γw −1 which surrounds w , but not z = 0, and it λ is mapped with degree 1 onto γ. The curve γ0−1 does not surround γw −1 . Proof. We first notice that given any Jordan curve in A(∂Tλ , ∂A∗λ (∞)) all preimages should be located either in Ain or Aout since Tλ , Aλ and A∗λ (∞) are mapped outside A(∂Tλ , ∂A∗λ (∞)). Moreover, by Proposition 3.5(vi) any Jordan curve in A(∂Tλ , ∂A∗λ (∞)) should have preim- age(s) in Ain as well as in Aout . Let γ ∈ A(∂Tλ , ∂A∗λ (∞)) be a Jordan curve surrounding the origin. First we study the topology of the preimage(s) of γ in Ain . By Proposition 3.5, γ has exactly d preimages in Ain . Let γ0 be one of the preimages of γ in Ain . The goal is to show that in fact γ0 is mapped by Tλ with degree d, so there are no other preimages whatsoever. Observe that Int (γ0 ) should contain either a pole, a zero, or z = 0, otherwise it cannot be mapped to γ which surrounds z = 0. Therefore, γ0 surrounds z = 0. Take the annulus A(γ, ∂A∗λ (∞)) and consider its preimage in Ain . Since the only preimage of ∂A∗λ (∞) in Ain is ∂Tλ and Ain contains no critical point, the preimage should be the annulus A(∂Tλ , γ0 ). The map Sn,d,λ |A(∂Tλ ,γ0 ) is proper of degree d since Sn,d,λ maps ∂Tλ onto ∂A∗λ (∞) with degree d. We conclude that γ0 is mapped with degree d onto γ, as desired. The proof of statements (i)-(iii) about the topology of the preimage(s) of γ in Aout is analogous to the one of [Can18, Proposition 3.3] by using that Sn,d,λ is conjugate to the Blaschke product Rθ,b in Aout (see Proposition 3.6).  Remark 3. It follows from Proposition 3.7 that each Fatou component different from Tλ and A∗λ (∞) which surrounds z = 0 (and so it contains a Jordan curve which surrounds z = 0) has exactly two preimages which also surround z = 0. One of them lies in Ain and the other lies in Aout . The following lemma shows that Fatou components which do not surround the origin do not have preimages which surround it. Lemma 3.8. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Let U be a multiply connected Fatou component. If U does not surround z = 0, then no component of its −1 preimage Sn,d,λ (U ) surrounds z = 0. Proof. Suppose that U does not surround z = 0 and let V be a preimage of U which surrounds z = 0. Let U 0 = Bdd(U ) and V 0 the preimage of U 0 which contains V . Observe that U 0 ⊂ A(∂Tλ , ∂A∗λ (∞)). Since V 0 ⊂ Ain ∪ Aout , it can contain at most one critical point. Since Sn,d,λ |V 0 : V 0 → U 0 is proper and V’ contains at most one critical point, it follows from the Riemann Hurwitz formula that V 0 is simply connected (compare Corollary 2.4). Since V surrounds the origin, then z = 0 lies in V 0 . However, this is impossible since z = 0 is mapped to ∞ and U 0 is bounded.  Proposition 3.5 tells us that for |λ| small enough the map Sn,d,λ is essentially uni-critical since all critical points except νλ belong to Aλ (∞). Up to now, however, we have not imposed any particular dynamical behaviour for νλ . With the aim of proving the main results of this paper from now on we restrict ourselves to parameters for which the free critical point νλ belongs to Aλ (∞) (sometimes the term captured parameters is used). 16 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV A∗λ (∞) A∗λ (∞) U1 Uν Sn,d,λ (Uν ) Aλ Tλ Ud Un Un+1 Vn+1 Vn Tλ (a) The partition with respect to Uν and Aλ . (b) The partition with respect to Sn,d,λ (Uν ). Figure 4. Partitions of the dynamical plane introduced in Theorem 3.9. Under this assumption, Proposition 3.5 implies that νλ ∈ Aλ (∞)\(A∗λ (∞) ∪ Tλ ∪ Dλ ∪ Aλ ). We further assume that νλ belongs to a Fatou component which is an eventual preimage of Aλ that surrounds z = 0. The following result gives relevant notation and determines a partition of the dynamical plane (compare Figure 4) that will be extremely useful to study achievable connectivities of Fatou components. Theorem 3.9. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Assume that νλ ∈ Uν , where Uν is a Fatou component which is eventually mapped onto Aλ and surrounds z = 0. Then, Uν is triply connected and Uν ⊂ Aout . Moreover, the following statements hold. (i) The set Uν bounds an open disk U1 which is mapped with degree 1 onto the open disk Vn ∪ Tλ , where Vn is the annulus bounded by ∂Tλ and Sn,d,λ (Uν ). In particular, wλ ∈ U1 . (ii) The annulus Un+1 bounded by Uν and ∂A∗λ (∞) is mapped with degree n + 1 onto the annulus Vn+1 bounded by Sn,d,λ (Uν ) and ∂A∗λ (∞). (iii) The annulus Un bounded by Aλ and Uν is mapped with degree n onto the annulus Vn bounded by ∂Tλ and Sn,d,λ (Uν ). (iv) The annulus Ud bounded by ∂Tλ and Aλ (i.e., the annulus Ain ) is mapped with degree d onto the annulus A(∂Tλ , ∂A∗λ (∞)). (v) The Fatou component Uν lies in Vn+1 and it is mapped under Sn,d,λ with degree n + 1 onto its image. In particular, Uν surrounds Sn,d,λ (Uν ). Remark 4. Statement (iv) in Theorem 3.9 coincides with statement (vi) of Proposition 3.5. We use this double naming (Ud and Ain ) in order to uniformize notation in what follows. Notice that every set Ui , i = d, n, n + 1, is mapped onto its image with degree i. This notation is particularly useful in Section 4. Also, notice that in order to simplify notation we ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 17 avoid indicating the dependence of Uν and Ui , i = 1, d, n, n + 1, with respect to the parameter λ. Proof. By the Riemann-Hurwitz formula, the iterated preimages of Aλ are doubly connected unless they contain a critical point. Under our hypothesis, this occurs precisely at Uν (since the only critical point eventually mapped in Aλ is νλ ). A direct application of the Riemann- Hurwitz formula implies that, since νλ is a simple critical point, Uν is triply connected. Moreover, Uν ⊂ Aout since νλ ∈ Aout . This proves the first part of the statement. From above ∂Uν has three components. Since Uν separates z = 0 from z = ∞, there should be exactly two components of ∂Uν surrounding z = 0. We denote them by γcin and γcout , where γcin ⊂ Int(γcout ). The other component of ∂Uν , denoted by γc1 , does not surround z = 0. Set U1 = Int(γc1 ). Since γc1 is mapped onto a component of ∂Sn,d,λ (Uν ), U1 is mapped either to the bounded or the unbounded component of the complement of Sn,d,λ (Uν ) (which is an annulus by hypothesis). However, since all poles are in A∗λ (∞) ∪ Tλ , then U1 should be mapped onto the bounded component of Sn,d,λ (Uν ). Therefore, U1 contains the zero wλ (and no other preimages of z = 0). We conclude that Sn,d,λ |U1 has degree 1. In particular, γ1 is mapped onto its image with degree 1. This proves (i). Let Un+1 be the annulus bounded by Uν and ∂A∗λ (∞), and let Vn+1 = Sn,d,λ (Un+1 ). By construction, Vn+1 is the annulus bounded by Sn,d,λ (Uν ) and ∂A∗λ (∞). It is immediate that the map Sn,d,λ |Un+1 : Un+1 → Vn+1 is proper. Since the degree is accomplished on the boundaries and ∂A∗λ (∞) is mapped onto itself with degree n + 1, Un+1 is mapped onto Vn+1 with degree n + 1. This proves (ii). The proof of statement (iii) is similar and statement (iv) was already proven in Proposition 3.5. Finally, we prove statement (v) by contradiction. Assume that Uν does not lie in Vn+1 . Then, either Uν maps onto itself (which is impossible) or Vn+1 ⊂ Un+1 . This would imply that Un+1 is mapped under Sn,d,λ on itself and, hence, there exists a periodic Fatou component different from Aλ (∞). This is impossible since, by assumption, the orbits of all critical points converge to z = ∞.  Remark 5. Under the assumptions of Theorem 3.9, if U is a Fatou component which sur- rounds z = 0 and lies in Un+1 or Un , then it follows from Proposition 3.6 that its image lies in Bdd(U ). Indeed, Un and Un+1 are contained in Aout (see Figure 4), and Aout belongs to the region where the dynamics are conjugate to the ones of a Blaschke product. As it will become clear in the next sections devoted to prove the main results of this paper, the presence of Fatou components with high connectivity in the dynamical plane is based on taking special iterated preimages of Uν . With this in mind we end the section by stating the following corollary of Proposition 3.7 and Theorem 3.9. Corollary 3.10. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Assume that νλ ∈ Uν , where Uν is an iterated preimage of Aλ which surrounds z = 0. Let W be a Fatou component which surrounds z = 0, different from Tλ and Aλ . Then, the following statements hold. (i) If W ⊂ Vn+1 , then it has a unique preimage in Ud , which surrounds z = 0 and is mapped under Sn,d,λ onto W with degree d, and a unique preimage in Un+1 , which surrounds z = 0 and is mapped under Sn,d,λ onto W with degree n + 1. (ii) If W ⊂ Vn , then it has a unique preimage in Ud , which surrounds z = 0 and is mapped under Sn,d,λ onto W with degree d, and two further preimages. One lies in Un , surrounds z = 0 and is mapped under Sn,d,λ onto W with degree n. The other one 18 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV lies in U1 , does not surround z = 0, and is mapped under Sn,d,λ onto W with degree 1. 4. Proofs of theorems A and B In this section we prove Theorem A and Theorem B. We first show that Fatou components which do not surround z = 0 cannot be used to achieve higher connectivities. Lemma 4.1. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Assume also that νλ ∈ Uν . Let V be a Fatou component which does not surround z = 0. Then V and all of its eventual preimages have the same connectivity. Proof. Let V be a Fatou component which does not surround z = 0 and let U be a preimage of it. By Lemma 3.8, U does neither surround z = 0. It follows that Bdd(U ) does not contain any critical point. Therefore, the map Sn,d,λ |Bdd(U ) : Bdd(U ) → Bdd(V ) is a proper map of degree 1. We can conclude that Sn,d,λ |U : U → V is conformal, so U and V have the same connectivity.  Next we give the form of the connectivities of Fatou components of Sn,d,λ . We want to remark that not all these connectivities are achievable (see Theorem A). Proposition 4.2. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Assume also that νλ ∈ Uν , where Uν is an iterated preimage of Aλ which surrounds z = 0. All Fatou components have connectivity 1, 2, or κ = (n + 1)i nj d` + 2, where i, j, ` ∈ N. Proof. By Proposition 3.5, Dλ and all its eventual preimages have connectivity 1 since none of them can contain critical points. Analogously, all eventual preimages of Aλ other than Uν and its preimages have connectivity 2 since none of them contain critical points (see Corollary 2.4). Finally, we study the connectivity of the preimages of Uν . By Lemma 4.1, it suffices to study preimages of Uν which surround z = 0. It follows from the Riemann-Hurwitz formula that if f : U → V is proper of degree q without critical points and V has connectivity p + 2, then U has connectivity qp + 2. By Corollary 3.10, all preimages of Uν which surround z = 0 map to their images with degree d, n, or n + 1. Since Uν has connectivity 3 = 1 + 2, the possible connectivities of the preimages of Uν surrounding z = 0 are of the form κ = (n + 1)i nj d` + 2, where i, j, ` ∈ N.  According to Corollary 3.10, if U is a iterative preimage of Uν which surrounds z = 0, its preimages which surround z = 0 may be located in Ud , Un or Un+1 . Next lemma shows that there are achievable upper bounds for the itineraries of iterated preimages of Uν . Recall from Remark 5 that if U ⊂ Un is an iterated preimage of Uν , then either Sn,d,λ (U ) ⊂ Un k or Sn,d,λ (U ) ⊂ Ud . Let k ≥ 1 be minimal such that Sn,d,λ (Uν ) ⊂ Bdd(Aλ ). The first half of Lemma 4.3 shows that if U ⊂ Un is a preimage of Uν which surrounds z = 0, then the itinerary of U intersects Ud in p ≤ k iterations. The second half of Lemma 4.3 shows that if U ⊂ Ud is a preimage of Uν which surrounds z = 0, then there exist at least k − 1 consecutive backwards iterates of U which lie in Un . Lemma 4.3. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Assume also k j that νλ ∈ Uν . Let k ≥ 1 such that Sn,d,λ (Uν ) ⊂ Bdd(Aλ ) and Sn,d,λ (Uc ) ⊂ Un for 1 ≤ j < k. ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 19 B1 U1 Uν B2out W1 Aλ B2in W2 Tλ Figure 5. Description of the situation in the proof of Lemma 4.3, where k = 2 and W2 ⊂ Ud . In this case, B2 = B2out ∪ Aλ ∪ B2in . (i) Let U ⊂ Un be an iterated preimage of Uν which surrounds z = 0. Then, there exists 0 p ≥ 1 such that S p (U ) ⊂ Ud and S p (U ) ∈ Un for 0 ≤ p0 < p. Moreover, p satisfies p ≤ k. (ii) Let U ⊂ Ud be a preimage of Uν which surrounds z = 0. Then, there exists U 0 ⊂ Un j k−1 such that Sn,d,λ (U 0 ) ⊂ Un for 0 ≤ j < k − 1 and Sn,d,λ (U 0 ) = U . Proof. Set Wi = Sn,d,λ i (Uν ), i = 0, . . . , k. By Remark 5, Wi surrounds Wi+1 , i = 0, . . . , k − 1. Let Bi+1 be the annulus bounded by Wi and Wi+1 , i = 0, . . . , k−1. It follows that Sn,d,λ (Bi ) = Bi+1 , i = 1, . . . , k − 1. Observe that if Wk ⊂ Ud , then Bk = Bkout ∪ Aλ ∪ Bkin , where Bkout = Bk ∩ Un and Bkin = Bk ∩ Ud (see Figure 5). Along the proof we distinguish the cases Wk = Aλ and Wk ⊂ Ud . We now prove statement (i). Assume first that Wk = Aλ . Let U ⊂ Bi , i = 1, . . . , k, be k+1−i an eventual preimage of Uν which surrounds z = 0. Then Sn,d,λ (U ) ⊂ Ud . We can conclude that if U ⊂ Un is a preimage of Uν which surrounds z = 0, then there exists p ≤ k such p that Sn,d,λ (U ) ⊂ Ud . In fact, p = k − i if U ⊂ Bi . Now assume Wk ⊂ Ud . For k = 1, Sn,d,λ (Un ) ⊂ Ud and the conclusion follows. For k ≥ 2 we have W1 ⊂ Un (so Bk−1 and Bk exist). Let U ⊂ Bi , i = 1, . . . , k − 1, be a preimage of Uν which surrounds z = 0 and let k−1−i V = Sn,d,λ (U ). Observe that V ⊂ Bk−1 is an eventual preimage of Uν which surrounds z = 0 and Ud is the disjoint union of Bkin , Wk , and Sn,d,λ (Bkout ). Then, Sn,d,λ (V ) ⊂ Bkout 2 k−1−i (and Sn,d,λ (V ) ⊂ Ud ) or Sn,d,λ (V ) ⊂ Bkin (and Sn,d,λ (V ) ⊂ Ud ). Since Sn,d,λ (Bi ) = Bk−1 , i = 0, . . . , k − 1, this concludes the proof of statement (i). Now we prove (ii). Let U ⊂ Ud be a preimage of Uν which surrounds z = 0. Assume first that Wk = Aλ . Then Ud = Sn,d,λ (Bk ) = Sn,d,λ k (B1 ). So there exists U 0 ⊂ U1 such that j Sn,d,λ (U 0 ) ⊂ Bj+1 ⊂ Un for 0 ≤ j < k and Sn,d,λ k (U 0 ) = U . Now let Wk ⊂ Ud . For k = 1 there is nothing to prove. For k ≥ 2 we have W1 ⊂ Un (so Bk−1 and Bk exist). Moreover, Ud = Bkin ∪ Wk ∪ Sn,d,λ (Bkout ). Since U is a preimage of Uν we have that U 6= Wk . We 20 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV distinguish 2 cases. If U ⊂ Bkin , then U ⊂ Sn,d,λ (Bk−1 ) and we can take preimages through j the sets Bi so that there exists U 0 ⊂ B1 such that Sn,d,λ (U 0 ) ⊂ Bj+1 ⊂ Un for 0 ≤ j < k − 1 k−1 and Sn,d,λ (U 0 ) = U . Finally, if U ⊂ Sn,d,λ (Bkout ), then U ⊂ Sn,d,λ (Bk ) and there exists j U 0 ⊂ B1 such that Sn,d,λ (U 0 ) ⊂ Bj+1 ⊂ Un for 0 ≤ j < k and Sn,d,λ k (U 0 ) = U . This concludes the proof of statement (ii).  We can now proceed with proof of Theorem A. Proof of Theorem A. By Corollary 3.10, Lemma 4.1, and Proposition 4.2, if the connectivity of a Fatou component is different from 1 or 2, then it has to be of the form κ = (n+1)i dj n` +2 where i, j, ` ∈ N. It follows from the Riemann-Hurwitz formula that if f : U → V is proper of degree q without critical points and V has connectivity p + 2, then U has connectivity qp + 2. Moreover, these connectivities are achieved through preimages of Uν . In order to increase the connectivity we have to take preimages of Uν , which has connectivity 3 = 1 + 2. It follows from Remark 5 that if U ⊂ Us , s ∈ {n + 1, d, n}, is a Fatou component that surrounds z = 0, then it is mapped onto its image with degree s. Therefore, in order to increase the coeficient n in the expression of the connectivity, we have to take preimages in Un . By Remark 5, every Fatou component U ⊂ Un is eventually mapped to Ud , without passing through Un+1 . By Lemma 4.3 (i), for every backwards iteration through Ud there are at most k backwards iterations in Un . Since Uν 6⊂ Ud ∪ Un , it follows that ` ≤ jk.  The final part of this section is devoted to the proof of Theorem B. The following lemma shows that there are no restrictions to the exponents of n + 1 and d in respect to achievable connectivities. Lemma 4.4. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Assume also that νλ ∈ Uν , where Uν is an iterated preimage of Aλ which surrounds z = 0. Then, the following hold: (i) There exists an eventual preimage of Uν which lies in Un+1 , surrounds z = 0, and has connectivity κ = (n + 1)i + 2, ∀i ∈ N. (ii) Let V be an eventual preimage of Uν which surrounds z = 0 and let κ be the con- nectivity of V . Then, there exists a Fatou component, which surrounds z = 0, of connectivity d(κ − 2) + 2. In particular, there exists a Fatou component which surrounds z = 0 and has connectivity κ = (n + 1)i dj + 2, ∀i, j ∈ N. Proof. First we prove (i). Recall that, by Theorem 3.9, Uν ⊂ Vn+1 . By Corollary 3.10 and Remark 5, for any i ≥ 1 there exists a Fatou component U which surrounds z = 0 such that j i Sn,d,λ (U ) ⊂ Un+1 , for j = 0, . . . i − 1, and Sn,d,λ (U ) = Uν . Since Uν has connectivity 3 and no eventual preimage of Uν contains a critical point, it follows by succesively applying the Riemann-Hurwitz formula that the connectivity of U is (n + 1)i + 2. This proves (i). To prove (ii), let V be an eventual preimage of Uν which surrounds z = 0, of connectivity κ. By Corollary 3.10, V has a preimage in Ud which surrounds z = 0 and which is mapped onto it with degree d. Since V cannot contain any critical value, by the Riemann-Hurwitz formula the connectivity of this preimage of V is d(κ − 2) + 2. This concludes the proof of (ii).  We can now proceed with proof of Theorem B. ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 21 Proof of Theorem B. Fix i ≥ 0, j ≥ 0, ` ≥ 0 such that ` ≤ j(k − 1). We want to show that there exists a Fatou component U (which will be an iterated preimage of Uν ) of connectivity κ = (n + 1)i dj n` + 2. If k = 1, then by Lemma 4.4 the conclusion holds. Otherwise, Sn,d,λ (Uν ) ⊂ Un and Ud ⊂ Vn . By Lemma 4.4 (i), there exists a Fatou component V which is an iterated preimage of Uν , surrounds z = 0, lies in Un+1 , and has connectivity (n + 1)i + 2. This concludes the proof for j = 0. Assume that j 6= 0 (remember that k > 1). By Corollary 3.10 (i), there exists a preimage of V in Ud which surrounds z = 0. By Lemma 4.3 (ii), there exists V (1) ⊂ Un which surrounds z = 0 such that r k−1 Sn,d,λ (V (1) ) ⊂ Un for 0 ≤ r < k − 1, Sn,d,λ (V (1) ) ⊂ Ud and k Sn,d,λ (V (1) ) = V. Recall that no iterated preimage of Uν contains a critical point and so, by Corollary 3.10, if they lie in Ui , i ∈ {n + 1, n, d}, they map forward with degree i. Applying this criteria to the iterated preimages of V up to V (1) , we get from the Riemann-Hurwitz formula that   r κ Sn,d,λ (V (1) ) = (n + 1)i dnk−1−r + 2, r = 0, . . . k − 1. Starting the process with V (1) instead of V we can take a preimage of V (1) in Ud and then we can take up to k − 1 iterated preimages in Un to land on, say, V (2) . As above, we get that   r κ Sn,d,λ (V (2) ) = (n + 1)i d2 n2(k−1)−r + 2, r = 0, . . . k − 1. Repeating the same process j-times we conclude that there exist Fatou components with connectivity   r κ Sn,d,λ (V (s) ) = (n + 1)i ds ns(k−1)−r + 2, s = 1, . . . j, r = 0, . . . k − 1. If (j −1)(k −1) < ` ≤ j(k −1) we are done (take s = j in the previous formula). If ` ≤ t(k −1) with t ≤ (j − 1), then we stop the process at level t and take j − t preimages in Ud to get the desired connectivity.  5. Proof of theorem C In this section we prove Theorem C. We first show that there is a sequence of preimages of Aλ which surround z = 0 and accumulate on ∂A∗λ (∞). We want to remark that the set Aout depends on λ even if we do not indicate it in its notation. Lemma 5.1. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Let A0,λ := Aλ . Then there exist {Am,λ }m≥1 iterated preimages of A0,λ , Sn,d,λ (Am+1,λ ) = Am,λ , such that the following properties are satisfied: (i) Each Fatou component Am,λ is surrounded by Am+1,λ , that is, Am,λ ⊂ Bdd(Am+1,λ ). In particular, all {Am,λ }m≥1 lie in Aout . (ii) The sequence of Fatou components {Am,λ }m≥1 accumulate on ∂A∗λ (∞) as m → ∞. Proof. Every Fatou component which surrounds z = 0 has exactly 2 boundary components which surround z = 0. It follows from Proposition 3.7 that every Fatou component in Aout which surrounds z = 0 has exactly a preimage in Aout which surrounds z = 0. Let {Am,λ }m≥1 be the sequence of Fatou components obtained by taking consecutive preimages of A0,λ in Aout which surround z = 0. Since Sn,d,λ is conjugated to a Blaschke product on Aout , by 22 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV Proposition 3.6, the Fatou components Am,λ accumulate on ∂A∗λ (∞) as m goes to ∞. It also follows from the conjugation with the Blaschke product that Am,λ ⊂ Bdd(Am+1,λ ) for all m ≥ 0.  The multiply connected sets Am,λ surround z = 0. Therefore, there are exactly 2 boundary components of Am,λ which surround z = 0. We denote them by ∂ Int Am,λ and ∂ Ext Am,λ , where ∂ Int Am,λ ⊂ Int(∂ Ext Am,λ ). Next lemma tells as that if m is large enough then there are parameters λ such that νλ ∈ Am,λ . Lemma 5.2. Let Sn,d,λ satisfying (a), (b), (c), and (d). Let λ 6= 0, |λ| < C. Then, if m ∈ N∗ is big enough, there exists a parameter λ such that νλ ∈ Am,λ . Proof. The idea of the proof is to show that, if m is big enough, we can find a parameter λ0 such that νλ0 ∈ Bdd(Am,λ0 ) and a parameter λm such that νλm belongs to the unbounded component of C∗ \ Am,λm . We will then conclude that there exists a parameter λ0m such that νλ0m ∈ Am,λ0m . Fix λ0 such that all hypothesis hold. Then A0,λ0 is well defined, and so are Am,λ0 , m > 0. Let m0 be such that νλ0 ∈ Bdd(Am0 ,λ0 ). Then, for all m ≥ m0 we have νλ0 ∈ Bdd(Am,λ0 ). For fixed m ≥ m0 , we want to find the parameter λm . If λ = 0, the critical point ν0 belongs to the boundary of the maximum domain of definition of the Böttcher coordinate of A∗ (0) under Sn,a,Q . Therefore, the orbit of ν0 under Sn,a,Q accumulates on z = 0 but never maps onto it. Observe that Sn,d,λ converges uniformly to Sn,a,Q on compact subsets of C∗ \ D as λ → 0, where  > 0 is arbitrarily small and D denotes the disk of radius  centered at z = 0. Consequently, for fixed m ≥ 0 and  > 0, if |λ| is small enough then Am,λ ⊂ D . Since νλ → ν0 as λ → 0, it follows that if |λ| is small enough then νλ belongs to the unbounded component of C∗ \ Am,λ . It is enough to take λm to be any such λ. To finish the proof we need to show that when we move continuously the parameter from λ0 to λm we need to find intermediate parameters λ0m such that νλ0m ∈ Am,λ0m . By Proposition 3.1 we know that ∂A∗∞ moves continuously with respect to λ. Since ∂A∗∞ and ∂Tλ cannot contain critical values, it follows that both boundary components of A0,λ move continuously with respect to λ. For fixed λ0 , the set ∂Am,λ , m ≥ 1, moves continuously with respect to λ in a neighbourhood of λ0 unless ∂Am,λ0 (or an iterated image of ∂Am,λ0 ) contains a critical point. Here by moving continuously we mean that every connected component of ∂Am,λ is a Jordan curve that moves continuously with respect to the Hausdorff metric (in particular, it does not pinch itself or split in several connected components). Notice that since there is only a free critical point, at most one of the 2 components of ∂Am,λ which surround z = 0 may not move continuously for λ in a neighbourhood of λ0 . Using Proposition 3.6 it can be proven that Sn,d,λ (∂ Ext Am,λ0 ) = ∂ Ext Am−1,λ0 and Sn,d,λ (∂ Int Am,λ0 ) = ∂ Int Am−1,λ0 . Assume that for λ0 we have νλ0 ∈ ∂Am,λ0 . Then Am−1,λ is an annulus that moves continuously with respect to λ for all λ in a neighbourhood of λ0 . If νλ0 ∈ ∂ Int Am,λ0 then Sn,d,λ (νλ0 ) ∈ ∂ Int Am−1,λ0 . By Proposition 3.7, for λ in a neighbourhood of λ0 exactly one of the following holds (see the three upper figures in Figure 6): • If Sn,d,λ (νλ ) ∈ Int(∂ Int Am−1,λ ) then νλ ∈ Int(∂ Int Am,λ ) and Am,λ is doubly connected. • If Sn,d,λ (νλ ) ∈ ∂ Int Am−1,λ then νλ ∈ ∂ Int Am,λ . Then, Am,λ is doubly connected and ∂ Int Am,λ consists of the union of 2 Jordan curves. • If Sn,d,λ (νλ ) ∈ Am−1,λ then νλ ∈ Am,λ and Am,λ is triply connected. ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 23 νλ νλ νλ Sn,d,λ (νλ ) Sn,d,λ (νλ ) Sn,d,λ (νλ ) Ak−1,λ Ak−1,λ Ak−1,λ Ak,λ Ak,λ Ak,λ A0 νλ νλ νλ Sn,d,λ (νλ ) Sn,d,λ (νλ ) Sn,d,λ (νλ ) Ak−1,λ Ak−1,λ Ak−1,λ Ak,λ Ak,λ Ak,λ Figure 6. The top figures correspond to the possible cases of νλ lying in a neighbourhood of ∂ Int Am,λ . The top figures correspond to the possible cases of νλ lying in a neighbourhood of ∂ Ext Am,λ . On the other hand, if νλ0 ∈ ∂ Ext Am,λ0 then Sn,d,λ (νλ0 ) ∈ ∂ Ext Am−1,λ0 . By Proposition 3.7, for λ in a neighbourhood of λ0 exactly one of the following holds (see the three lower figures in Figure 6): • If Sn,d,λ (νλ ) ∈ Am−1,λ then νλ ∈ Am,λ and Am,λ is triply connected. • If Sn,d,λ (νλ ) ∈ ∂ Ext Am−1,λ then νλ ∈ ∂ Ext Am,λ . Then, Am,λ is doubly connected, ∂ Ext Am,λ is a Jordan curve, and there is an extra preimage A0 of Am−1,λ such that ∂Am,λ ∩ ∂A0 = νλ . • If Sn,d,λ (νλ ) ∈ Ext(∂ Int Am−1,λ ) then νλ ∈ Ext(∂ Int Am,λ ) and Am,λ is doubly con- nected. It follows from the previous configurations that if we move continuously the parameter λ from λ0 until λm we can find parameters λ0m such that νλ0m ∈ Am,λ0m . This finishes the proof of the result.  We can now proceed with proof of Theorem C. Proof of Theorem C. Fix i, j, `. We have to prove that there exists λ for which there is a Fatou component of connectivity κ = (n + 1)i dj n` + 2 and a Fatou component of connectivity κ = (n + 1)i + 2. Recall that the results in Section 4 required the free critical point νλ to 24 JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV lie in a preimage of Aλ which surrounds z = 0. By Lemma 5.2, there exists m > ` such that νλ ∈ Am,λ . 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Steinmetz, The formula of Riemann-Hurwitz and iteration of rational functions, Complex Vari- ables Theory Appl. 22 (1993), no. 3-4, 203–206. [Sul85] D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. ACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 25 Departament de Matemàtiques, Universitat Jaume I, 12071 Castelló de la Plana, Spain Email address: canela@uji.es Departament de Matemàtiques i Informàtica at Universitat de Barcelona and Barcelona Graduate School of Mathematics, 08007 Barcelona, Catalonia. Email address: xavier.jarque@ub.edu Departament de Matemàtiques i Informàtica at Universitat de Barcelona and Barcelona Graduate School of Mathematics, 08007 Barcelona, Catalonia. Email address: dan.paraschiv@ub.edu