Achievable connectivities of Fatou components for a family of singular perturbations
AACHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR AFAMILY 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 largefamily of singular perturbations. We prove that, for some parameters inside the family, thedynamical planes for the corresponding maps present Fatou components of arbitrarily largeconnectivity and we determine precisely these connectivities. In particular, these resultsextend the ones obtained in [Can17, Can18].
Keywords: holomorphic dynamics, Fatou and Julia sets, singular perturbation, connectiv-ity of Fatou components. Introduction
In the recent decades there has been an increasing interest in studying families of rationalmaps usually called singular perturbations . Roughly speaking, a family is called a singularperturbation if it is defined by a base family (called the unperturbed family and for whichwe 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 thedynamical plane, but a very small dynamical relevancy on other regions.Singular perturbations, no matter the concrete formulas, have some common propertieswhich make their study interesting. On the one hand, the degree of the unperturbed familyis smaller than the degree of the perturbed one. Consequently, one should expect richerdynamics 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 dynamicalproperties of the unperturbed family. The balance between these two scenarios has becomevery 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 dynamicalplane of a particular rational map. A paradigmatic example of this is the Dichotomy Theoremfor the quadratic family. In this way, singular perturbations are somehow a perfect scenario toobserve new phenomena for the invariant sets with respect to the unperturbed maps, for whichwe usually observe a tame topology. Indeed, the main goal of this paper is to investigate in thisdirection and to prove that for a certain family of singular perturbations we can constructexamples for which, in the same dynamical plane, there are Fatou components with givenarbitrarily high connectivities.
Date : February 2, 2021.The first author was supported by Spanish Ministry of Economy and Competitiveness, through the Mar´ıade Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445), by BGSMath Banco de SantanderPostdoctoral 2017, and by the project UJI-B2019-18 from Universitat Jaume I. The second and third authorswere supported by MINECO-AEI grant MTM-2017-86795-C3-2-P. The second author was also supported byAGAUR grant 2017 SGR 1374. The third author was also supported by the Spanish government grant FPIPRE2018-086831. a r X i v : . [ m a t h . D S ] F e b JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV
Let f : ˆ C → ˆ C be a rational map acting on the Riemann sphere. Then f partitionsthe dynamical plane into J ( f ), the Julia set, and F ( f ), the Fatou set. The Julia set isclosed and coincides with the set of points z ∈ ˆ C where the family of iterates { f n | U } n ≥ isnot a normal family for any neighbourhood U of z . Its complement, the Fatou set, is anopen set and its connected components are called Fatou components. By the No WanderingDomains Theorem, Fatou components of rational maps are either preperiodic or periodic (see[Sul85]). By the Classification Theorem of periodic Fatou components (see [Mil06], Theorem16.1), every periodic Fatou component either belongs to the immediate basin of attractionof 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 Fatoucomponents can by studied by analysing the orbits of critical points, i.e. points where f (cid:48) vanishes. Indeed, every immediate basin of attraction contains, at least, a critical point whileSiegel disks and Herman rings have critical orbits accumulating on their boundaries.The connectivity of a domain D ⊂ ˆ C is defined as the number of connected componentsof its boundary. It is known that periodic Fatou components have connectivity 1 , , or ∞ .Indeed, Siegel disks have connectivity 1, Herman rings have connectivity 2, and immediatebasins of attraction have connectivity 1 or ∞ . Preperiodic Fatou components can have finiteconnectivity greater than 2. The first such example, with connectivities 3 and 5, was presentedin [Bea91]. Moreover, for any given n ∈ N , there are examples of rational maps with Fatoucomponents of connectivity n . These examples can either be obtained by quasiconformalsurgery (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 Fatoucomponents 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 preciseconnectivities can actually be attained. The goal of this paper is to study the attainableconnectivities 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 knewthat, independently, professor Hiroyuki has obtained another family of rational maps withFatou components of arbitrarily large connectivity [Hir].Singular perturbations of rational maps were introduced by McMullen in [McM88]. Heproposed the study of the family(1) Q n,d,λ ( z ) = z n + λz d , where n, d ≥ 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 smallenough then the Julia set is a Cantor sets of quasicircles (the result actually holds for n and d satisfying 1 /n + 1 /d < λ -family of rational maps and they extended McMullen’s result by provingthe Escape Trichotomy. More specifically, they showed that if all critical points belong tothe basin of attraction of infinity then the Julia set is a Cantor set, a Sierpinski carpet, ora Cantor set of quasicircles (McMullen’s case). The proof relays on the fact that there is asymmetry in the dynamical plane which implies that there is a unique free critical orbit (thesymmetry 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 singularperturbations of polynomials of the form z n + c, c ∈ C , choosing c appropriately. Those CHIEVABLE 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 connectivityof the Fatou components is kept as 1, 2 or ∞ .The examples mentioned in the previous paragraph are done perturbing maps with no freecritical points: one or more poles are added to superattracting cycles which contain no criticalpoints, other than the critical points of the cycle, in their basins of attraction. A next naturalstep is to consider singular perturbations of maps with free critical points. A good candidatefor such a perturbation is the family of Blaschke products B n,a ( z ) = z n z − a − az , where a ∈ C and n ≥ . See [CFG15] for an introduction to the dynamics of these maps for n = 3. If a belongs to thepunctured unit disk D ∗ := D \ { } , the maps B n,a restrict to automorphisms of the unit diskwhose dynamical plane is trivial. Indeed, its Fatou set consists of two invariant componentsgiven by the immediate basin of attraction A ∗ (0) of z = 0 (the unit disk) and the immediatebasin of attraction A ∗ ( ∞ ) of z = ∞ (the complement of the closed unit disk). Their commonboundary component, the unit circle, is the Julia set of these maps. Moreover, if a ∈ D ∗ the map B n,a has only two simple critical points c − ∈ A ∗ (0) and c + ∈ A ∗ ( ∞ ), other thanthe superatracting fixed points z = 0 and z = ∞ . In [Can17, Can18] the author studied thefamily of singular perturbations of the maps B n,a given by B n,d,a,λ ( z ) = z n z − a − az + λz d , where a ∈ D ∗ and λ ∈ C ∗ := C \ { } , for n = 3 and d = 2. Compared to McMullen’s singularperturbations, 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 nottied by any kind of symmetry. Despite that, in [Can17] it was proven that if | λ | is small enoughthe family B , ,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 ) belongsto a Fatou component in A ( ∞ ) which surrounds z = 0, the dynamical plane has Fatoucomponents of arbitrarily large finite connectivity. The actual existence of parameters forwhich this actually happens was proven in [Can18]. We want to point out that the sameresults can be proven for n, d ≥ /n + 1 /d <
1. In Figure 1 we illustrate thedynamical plane of B n,d,a,λ ( z ) for a = 0 . 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 ofsingular perturbations and to study which connectivities are attainable for such family. Withthis aim we consider the family of degree n + 1 rational maps given by(2) S n,a,Q ( z ) = z n ( z − a ) Q ( z ) , where n ≥ a ∈ C ∗ , and Q is a polynomial of degree at most n . On the one hand it is clearthat the family S n,a,Q contains the family B n,a . On the other hand it is worth to be noticedthat S n,a,Q also includes the family M n,a ( z ) = z n ( z − a ) , where n ≥ 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 ≥ a (cid:54) = 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 c a (cid:54) = 0 and the global JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV phase portrait settles down on its dynamical behaviour. It is easy to see that, if | a | is smallenough, c a belongs to the immediate basin of attraction of z = 0 and the Julia set consistsof a quasicircle which separates the immediate basins of attraction of z = 0 and z = ∞ (seeCorollary 2 . | a | small the family M n,a can be understood as a simplifiedversion of B n,a , | a | <
1, where there is no free critical point in A ∗ ( ∞ ) but the Julia set is aquasicicle instead of a circle.We now turn to the unperturbed family S n,a,Q . Inspired by the work in [Can17, Can18]we will impose the following conditions to be satisfied for the maps in S n,a,Q .(a) The point z = 0 is a superattracting fixed point of degree n of S n,a,Q . In particular Q (0) (cid:54) = 0.(b) The fixed point z = ∞ is (super)attracting. In particular the coefficient of z n of Q ,say b n , satisfies 0 ≤ | b n | < A ∗ (0)and A ∗ ( ∞ ) of z = 0 and z = ∞ , respectively. Remark 1.
We can deduce the following observations from the above conditions. Since themaps S n,a,Q have degree n + 1 , the immediate basins of attraction are mapped onto themselveswith 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 − ) contains a simple critical point ν (cid:54) = 0 . Once the unperturbed family has been described, we now consider the singular perturbation(3) S n,d,λ ( z ) = S n,a,Q ( z ) + λz d , λ ∈ C ∗ , d ≥ . Notice that to simplify notation we do not specify the dependence on a and Q of the family S n,d,λ . Notice also that the family S n,d,λ includes the family B n,d,a,λ . It follows immediatelythat all maps S n,d,λ have degree n + d + 1 and that for λ (cid:54) = 0 the point z = 0 is a pole ofdegree d . We will say that S n,d,λ satisfies (a), (b), and (c) if S n,a,Q satisfies the conditions(a), (b), and (c) explained above. Analogously to the condition needed to obtain a Cantorset of quasicircles for McMullen’s family (see [DLU05]), we have to add a fourth condition tothe family:(d) The numbers n, d ≥ n + d <
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 veryrich. Despite that, if | λ | is small the family is essentially unicritical. Indeed, there exists aconstant C > | λ | < C , λ (cid:54) = 0, the following hold (see Proposition 3 . • The continuous extensions of the n critical points which belong to the immediate basinof attraction A ∗ ( ∞ ) of z = ∞ before perturbation belong to the immediate basin ofattraction A ∗ λ ( ∞ ) of z = ∞ after perturbation. Moreover, A ∗ λ ( ∞ ) is a quasidisk. • The pole z = 0 belongs to a quasidisk T λ (usually called trap door ) which is mappedonto A ∗ λ ( ∞ ) under S n,d,λ . • The n + d critical points which appear around z = 0 after perturbation belong to adoubly connected Fatou component A λ which is mapped onto T λ under S n,d,λ .The previous points actually coincide with the skeleton of the dynamics in the Cantor setof quasicircles case of McMullen’s family (see [McM88]). This is why the dynamical planes forthis perturbed family resemble the dynamical planes for the Cantor set of quasicircles withextra decorations (see Figure 1 and Figure 2). These decorations come from the presence of CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 5 − − . . − − . . − − . . − − . . − − . . − − . . − − . . − − . . Figure 1.
Dynamical planes of the family B n,d,a,λ ( z ) for d = 3. The top-leftfigure corresponds to n = 2 and λ = 2 · − ; the top-right corresponds to n = 3 and λ = − · − ; the bottom-left figure corresponds to n = 4 and λ = − . · − ; and bottom-right corresponds to n = 5 and λ = − . · − .In all cases we can see the triply connected regions (where the critical point ν λ lies) and their eventual preimages, which are Fatou components with increasingconnectivity.the extra critical point ν λ , which comes from the continuous extension of the critical point ν that belongs to the basin of attraction of z = 0 before perturbation. This is the only criticalpoint which may not belong to the basin of attraction A λ ( ∞ ) of z = ∞ after perturbation if | λ | is small. We want to remark that the main difference between S n,d,λ and the particularfamily B n,d,a,λ is that we allow certain degrees of freedom in the n critical points that lie in JORDI CANELA, XAVIER JARQUE, AND DAN PARASCHIV − − . . . − − . . . − − . . . − − . . . Figure 2.
Left figure illustrates the dynamical planes of M n,a for n = 2and a = (0 . . i ). Right picture illustrates the dynamical plane of the(perturbed) family S n,d,λ when the unperturbed map is precisely M ,a , andthe pertubation corresponds to d = 3 and λ = − − . We can see in theright figure the triply connected Fatou component which contains ν λ and itseventual preimages with higher connectivity. A ∗ λ ( ∞ ). For instance, if the degree of Q is 0, then z = ∞ is a superatracting fixed point oflocal degree n . On the other hand, if the degree of Q is n , then z = ∞ is attracting (butnot superattracting) and there are n critical points which move in A ∗ λ ( ∞ ). Also, the shapeof 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 theunit circle.The goal of the paper is to analyse the connectivities which can be achieved with thesesingular perturbations. The critical point ν λ is crucial in order to increase the connectivitiesbeyond 2. Indeed, if ν λ belongs to a preimage U ν of A λ then the Fatou component U ν is triplyconnected. Moreover, if U ν surrounds z = 0 then we can find sequences of iterated preimagesof U ν which increase the connectivity with every iteration. The next theorems describe theconnectivities which can be achieved with this process. We denote by Bdd( A λ ) the union ofthe connected component of the complement of A λ not containing z = ∞ and the annulusitself. Theorem A.
Let S n,d,λ satisfying (a), (b), (c), and (d) and let λ (cid:54) = 0 , | λ | < C . Assume alsothat ν λ ∈ U ν , where U ν is an iterated preimage of A λ which surrounds z = 0 . Let k be theminimal number of iterations needed by the free critical point ν λ to be mapped into Bdd ( A λ ) .Let U be a Fatou component of connectivity κ > . Then, there exist i, j, (cid:96) ∈ N such that κ = ( n + 1) i d j n (cid:96) + 2 and (cid:96) ≤ jk . In other words, Theorem A is telling us all potential connectivities κ > S n,d,λ for | λ | sufficiently small; but it is not claiming the existenceof a Fatou component of each ( i, j, (cid:96) )-connectivity. The next result complements Theorem A CHIEVABLE 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 ν λ satisfies certain dynamical conditions. Theorem B.
Let S n,d,λ satisfying (a), (b), (c), and (d) and let λ (cid:54) = 0 , | λ | < C . Assumealso that ν λ ∈ U ν , where U ν is an iterated preimage of A λ which surrounds z = 0 . Let k ≥ be the minimal number of iterations needed by the free critical point ν λ to be mapped intoBdd ( A λ ) . For any given i, j, (cid:96) ∈ N such that (cid:96) ≤ j ( k − , there exists a Fatou component U of connectivity κ = ( n + 1) i d j n (cid:96) + 2 . In Theorem A and Theorem B the achievable connectivities depend on the minimal numberof iterations k > ν λ to be mapped into Bdd( A λ ). However,choosing λ appropriately we can make this k as big as desired. Therefore, for any (cid:96) and j wecan find λ so that the inequality (cid:96) ≤ j ( k −
1) is satisfied. From this, we obtain Theorem C.
Theorem C.
Let S n,d,λ satisfying (a), (b), (c), and (d) and let λ (cid:54) = 0 , | λ | < C . For anygiven i, l ≥ and j > , there exists a parameter λ such that S n,d,λ ( z ) has a Fatou componentof connectivity κ = ( n + 1) i d j n (cid:96) + 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 inthe paper. In Section 3 we describe in detail the skeleton of the dynamical plane of S n,d,λ satisfying the conditions (a), (b), (c) and (d) for | λ | small enough. In Section 4 we proveTheorems 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 introducesome notation. In the introduction we used the notation Bdd( A ) to denote the set boundedby an annulus A , including itself. It will be useful to generalise this concept for other multiplyconnected sets. Let U ⊂ C be a multiply connected open set. We denote by Bdd( U ) theminimal simply connected open set which contains U but not z = ∞ . Let γ ∈ C be a Jordancurve. We denote by Ext ( γ ) and Int ( γ ) the connected components of ˆ C \ γ that contain z = ∞ and do not contain z = ∞ , respectively. We denote by A ( γ , γ ) the open annulusbounded by Jordan curves γ and γ with γ ⊂ Int ( γ ). We denote the circle centered at theorigin and of radius c > S c . Finally, if U ⊂ C we denote by U its closure.We now proceed to introduce the needed tools. The next result provides a sufficient criterionfor 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 containsexactly 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 S n,a,Q are quasicircles. Corollary 2.2.
Let S n,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 inorder to study connectivities of Fatou components.
Theorem 2.3. (Riemann-Hurwitz formula) Let
U, V ⊂ ˆ C be two connected domains of con-nectivity m U , m V ∈ N ∗ and let f : U → V be a degree d proper map branched over r criticalpoints, counted with multiplicity. Then, m U − d ( m V −
2) + r. 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 ⊂ ˆ U be an open set and let f : U → f ( U ) be a proper holomorphicmap. 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 countingmultiplicities), then U is simply connected. The perturbed family
Let S n,d,λ satisfying conditions (a), (b), and (c) described above. This section describesthe 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 denoteby A ∗ λ ( ∞ ), and its boundary. The proof of Proposition 3 . Remark 2.
Along the paper, when we say that a compact set moves continuously with respectto parameters, we use the topology induced by the Hausdorff metric for compact sets.
Proposition 3.1.
Let S n,d,λ satisfying conditions (a), (b), and (c). Then, for | λ | smallenough, 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 criticalpoint of S n,d,λ in A ∗ λ ( ∞ ) is a continuous extension of a critical point of S n,a,Q in A ∗ ( ∞ ) .Proof. Observe that the three statements are trivially satisfied (by definition and Corollary2.2) for the unperturbed family. So this proposition says that this conditions are still trueif the perturbation is small enough. To prove the proposition we show the existence of ananalytic family of polynomial-like maps which ensures the continuous deformation of the keydynamical objects.Fix S n,a,Q and let U be the maximal domain of Bottcher coordinates around z = 0. Thecritical point ν lies on ∂U ⊂ A ∗ (0). Let γ be an analytic Jordan curve surrounding the originsuch that γ ∈ U \ S n,a,Q ( U ). We now show that the preimage of γ has a unique connectedcomponent. 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 . A − (under S n,a,Q ) isalso an annulus. Since ∂ A ∗ (0) is mapped by S n,a,Q onto itself with degree n + 1, A − is alsomapped onto A with degree n + 1. Let γ − be the connected component of ∂A − other than ∂ A ∗ (0). Since A − is mapped onto A with degree n + 1 under S n,a,Q , then γ − is mappedonto γ with degree n + 1 under S n,a,Q . Since S n,a,Q has (global) degree n + 1, we concludethat there is no other preimage of γ than γ − under the map S n,a,Q .Let V = Ext( γ ) and U = Ext( γ − ). It follows from the construction that U is compactlycontained in V and S n,a,Q | U : U (cid:55)→ V is a proper map of degree n + 1. Therefore, the triple( S n,a,Q | U , U , V ) is a degree n + 1 polynomial-like map (see [DH85], see also [BF14]). Wewant to extend (for | λ | small enough) this map to a J -stable analytic family of polynomiallike mappings. Observe that the map S n,d,λ depends analytically on λ for all z ∈ (cid:98) C \ D ε , CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 9 where D ε denotes the disk of radius ε centered at z = 0, for all λ ∈ C and all ε >
0. Recallthat S n,a,Q = S n,d, . Therefore, if | λ | is small enough, the continuous extensions of the n critical points (counting multiplicity) which lie in A ∗ ( ∞ ) = A ∗ ( ∞ ) for S n,a,Q lie in A ∗ λ ( ∞ )for S n,d,λ . Moreover, if | λ | is small enough then there exists a unique connected component γ − λ of γ under the map S − n,d,λ which is an analytic Jordan curve. In fact, γ − λ is a continuousdeformation of γ − and it is mapped with degree n +1 onto γ under S n,d,λ . Let U λ = Ext( γ − λ ).Decreasing | λ | if necessary, we can ensure the following. The set U λ is compactly containedin V and the only critical points of S n,d,λ in U λ are the ones which come from the continuousextension of the critical points in A ∗ ( ∞ ). Moreover, S n,d,λ | U λ : U λ (cid:55)→ V is a proper map ofdegree n + 1 and the triple ( S n,d,λ | U λ , U λ , 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 parametersfor which the previous conditions hold. Then, { ( S n,d,λ | U λ , U λ , V ) } λ ∈ Λ defines an analytic fam-ily of polynomial like mappings (see [DH85], see also [BF14]). Let K λ := { z ∈ U λ | S nn,d,λ ( z ) ∈U λ for all n ≥ } and J λ = ∂ K λ denote the filled Julia set and the Julia set of the polynomiallike map ( S n,d,λ , U λ , V ), respectively. Notice that K = A ∗ ( ∞ ). Notice also that all connectedcomponents of the interior of K λ are Fatou components of S n,d,λ . Therefore, since the point z = ∞ belongs to K λ for all λ ∈ Λ, we conclude that A ∗ λ ( ∞ ) ⊂ K λ .To finish the proof, we observe that since all critical points of S n,d,λ | U λ belong to A ∗ λ ( ∞ )it follows that the analytic family of { ( S n,d,λ , U λ , V ) } λ ∈ Λ is J -stable. In particular, the Juliasets J λ are quasicircles which are continuous deformations of J = ∂ A ∗ ( ∞ ) (see [DH85,Proposition 10]). Notice that, by Corollary 2 . ∂ A ∗ ( ∞ ) is a quasicircle. Since A ∗ λ ( ∞ ) ⊂ K λ ,we can conclude that ∂ A ∗ λ ( ∞ ) = J λ for all λ ∈ Λ. This proves (ii). Statements (i) and (iii)follow from the choice of the set of parameters Λ. (cid:3)
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 ina preimage of A ∗ λ ( ∞ ), different from it. Lemma 3.2.
Let S n,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 S n,d,λ also satisfies condition (d). For any constant K > , if | λ | is small enough, the disk (cid:110) | z | < K | λ | n + dnd (cid:111) belongs to a Fatou component T λ . TheFatou component T λ is mapped onto A ∗ λ ( ∞ ) and it is different from A ∗ λ ( ∞ ) .Proof. We begin with statement (i) . From condition (b) we know that, for fixed S n,a,Q ,there exists a constant K such that the set { z ∈ C | | z | > K } is compactly contained in theimmediate 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 K >
0. Assume that λ is such that (i) is satisfied for the constant K above. Let z ∈ C such that | z | < K | λ | nn + d . It follows that |S n,d,λ ( z ) | > (cid:12)(cid:12)(cid:12)(cid:12) λz d (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) z n ( z − a ) Q ( z ) (cid:12)(cid:12)(cid:12)(cid:12) > | λ | − ndn + d K d − | λ | n + d K n ( | a | + 1) M =: C ( λ ) + C ( λ ) . Notice that C ( λ ) tends to 0 as λ tends to 0. Because of assumption (d), C ( λ ) tends to ∞ as λ tends to 0. Shrinking | λ | if necessary, if | z | < | λ | nn + d K , then |S n,d,λ ( z ) | > K . Weconclude that the set (cid:110) | z | < K | λ | n + dnd (cid:111) belongs to a Fatou component. This Fatou componentcontains z = 0, which is mapped to ∞ with degree d . By continuity with respect to λ andProposition 3 . ∂ A ∗ λ ( ∞ ) is a quasicircle which surrounds z = 0. It follows that A ∗ λ ( ∞ ) doesnot contain the origin and z = 0 belongs to a preimage of A ∗ λ ( ∞ ), different from A ∗ λ ( ∞ ),which we denote by T λ . (cid:3) Recall that each map of the perturbed family has global degree n + d + 1. Hence, it has2( n + d ) critical points (counting multiplicity). By Proposition 3 . n + d − A ∗ λ ( ∞ ) ∪ { } . By continuity with respect to λ , there is a (simple) critical point ν λ which isthe continuous extension of the critical point ν of S n,a,Q in A ∗ (0). Each map has n + d + 1zeros, one of which, say w λ , corresponds to the continuous extension of w = a . We now givea description of the position of the remaining n + d critical points and the n + d preimages of z = 0 for S n,d,λ . Lemma 3.3.
Let S n,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, c λ,ξ ,and n + d zeros, z λ,ξ , given by c λ,ξ = ξ (cid:18) dQ (0) − na (cid:19) n + d λ n + d + o (cid:16) λ n + d (cid:17) ,z λ,ξ = ξ (cid:18) Q (0) a (cid:19) n + d λ n + d + o (cid:16) λ n + d (cid:17) . Proof.
Let us start with the zeros. Notice that all the zeros of S n,d,λ (except for w λ ) mustconverge to z = 0 when λ tends to 0. The zeros of S n,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 | smallenough. They are the fixed points of n + d operators T λ,ξ ( z ) = ξ (cid:18) Q ( z ) a − z (cid:19) n + d λ n + d = R ( z ) λ n + d , 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) (cid:54) = 0), so T λ,ξ ( z ) → λ → z λ,ξ by T λ,ξ (0) = ξ (cid:16) Q (0) a (cid:17) n + d λ n + d . Indeed, | z λ,ξ − T λ,ξ (0) | = | T λ,ξ ( z λ,ξ ) − T λ,ξ (0) | ≤ sup ω ∈ [0 ,z λ,ξ ] | T (cid:48) λ,ξ ( ω ) || z λ,ξ − | = | λ | n + d sup ω ∈ [0 ,z λ,ξ ] | R (cid:48) ( ω ) || z λ,ξ | . For | λ | small enough, there is no pole of R (cid:48) in a neighbourhood of z = 0 containing the linesegment [0 , z λ,ξ ], so it is bounded by a constant, say K . It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z λ,ξ − ξ (cid:16) Q (0) a (cid:17) n + d λ n + d λ n + d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ K | z λ,ξ | . CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 11
Finally, since lim λ → K | z λ,ξ | = 0, it follows that z λ,ξ = ξ (cid:18) Q (0) a (cid:19) n + d λ n + d + o (cid:16) λ n + d (cid:17) . It can be shown analogously that the n + d free critical points are solutions of the equation1 Q ( z ) (cid:2) ( n + 1) z n Q ( z ) − z n +1 Q (cid:48) ( z ) − anz n − Q ( z ) + az n Q (cid:48) ( z ) (cid:3) − λdz d +1 = 0 . As before, we write the operators S λ,ξ as S λ,ξ ( z ) = ξ (cid:18) dQ ( z )( n + 1) zQ ( z ) − anQ ( z ) − z Q (cid:48) ( z ) + azQ (cid:48) ( z ) (cid:19) n + d λ n + d , which have the critical points of S n,d,λ as fixed points. The argument made is identical since Q (0) (cid:54) = 0, so S λ,ξ are holomorphic in the neighbourhood of z = 0. Finally, the critical pointsof the perturbation map are of the form c λ,ξ = ξ (cid:18) dQ (0) − na (cid:19) n + d λ n + d + o (cid:16) λ n + d (cid:17) . (cid:3) Next we show that there exists a straight annulus (we will show later that it belongs to adoubly connected Fatou component) which is mapped into T λ under S n,d,λ . Let c = 12 min (cid:40)(cid:12)(cid:12)(cid:12)(cid:12) dQ (0) na (cid:12)(cid:12)(cid:12)(cid:12) n + d , (cid:12)(cid:12)(cid:12)(cid:12) Q (0) a (cid:12)(cid:12)(cid:12)(cid:12) n + d (cid:41) and c = 2 max (cid:40)(cid:12)(cid:12)(cid:12)(cid:12) dQ (0) na (cid:12)(cid:12)(cid:12)(cid:12) n + d , (cid:12)(cid:12)(cid:12)(cid:12) Q (0) a (cid:12)(cid:12)(cid:12)(cid:12) n + d (cid:41) . Lemma 3.4.
Let S n,d,λ satisfying conditions (a), (b), (c), and (d). Then, for | λ | smallenough, the straight annulus (4) Ω λ := A (cid:18) S c | λ | n + d , S c | λ | n + d (cid:19) contains the points c λ,ξ and z λ,ξ introduced in Lemma . and it is mapped into T λ under S n,d,λ .Proof. The first part of the statement follows directly from the algebraic expression of thepoints c λ,ξ and z λ,ξ in Lemma 3 .
3. The rest of the proof is devoted to show that S n,d,λ (Ω λ ) ⊂ T λ .Let m = min {| z | , Q ( z ) = 0 } and let M = min {| Q ( z ) | , | z | < m/ } (notice that M > z = 0 is not a root of Q ). Let z ∈ Ω λ . For | λ | small enough we have |S n,d,λ ( z ) | < c n | λ | nn + d ( | a | + 1) M + | λ | nn + d c d . We can rewrite this as |S n,d,λ ( z ) | < K | λ | nn + d , where K depends on Q , c and c , but it doesnot depend on z and λ . By Lemma 3 .
2, for | λ | small enough, the disk centered at z = 0 andof radius K | λ | nn + d lies in T λ , as desired. (cid:3) T λ A λ D λ ν λ A ∗ λ ( ∞ ) Figure 3.
Partition of the dynamical plane with respect to A ∗ λ ( ∞ ), A λ , T λ ,and D λ , described in Proposition 3 .
5. Blue and purple points denote zerosand critical points, respectively.In the next proposition we describe the skeleton of the dynamical plane for | λ | small (compare Figure 3). Recall that w λ is the zero of S n,d,λ which is the continuous extension of w = a and ν λ is the continuous extension of the critical point ν in A ∗ (0) of S n,a,Q . Proposition 3.5.
Let S n,d,λ satisfying conditions (a), (b), (c), and (d). Then, there exists aconstant C = C ( a, Q, n, d ) such that if λ (cid:54) = 0 and | λ | < C the following statements are satisfied:(i) The Fatou component T λ is simply connected and it is mapped with degree d onto A ∗ λ ( ∞ ) under S n,d,λ . There are no other preimages of A ∗ λ ( ∞ ) .(ii) There exists a Fatou component A λ which is doubly connected and contains exactly n + d simple critical points, given by c λ,ξ , and n + d zeros, given by z λ,ξ . Moreover, A λ is mapped with degree n + d onto T λ and surrounds the origin.(iii) Let A out be the annulus bounded by A λ and ∂ A ∗ λ ( ∞ ) . There exists a Fatou component D λ ⊂ A out which is simply connected, is mapped with degree onto T λ , and contains w λ .(iv) The critical point ν λ lies in A out \ D λ .(v) There are no preimages of T λ other than D λ and A λ .(vi) Let A in be the annulus bounded by ∂T λ and A λ . Then, A in is mapped onto the annulus A ( ∂T λ , ∂ A ∗ λ ( ∞ )) with degree d .Proof. Before proving the statements of the proposition we study the location and distributionof the critical points of S n,d,λ .By Proposition 3 . A ∗ λ ( ∞ ) is simply connected (in the Riemann sphere) and it is mappedonto itself with degree n + 1. Since the global degree of the map S n,d,λ is n + d + 1, thereexist exactly d preimages of ∞ outside A ∗ λ ( ∞ ), counting multiplicity. Since z = 0 is mappedto ∞ with degree d , there exist no other preimages of ∞ (different from the ones in A ∗ λ ( ∞ )and z = 0). Moreover T λ is mapped with degree d onto A ∗ λ ( ∞ ) (observe that up to this pointwe still do not know if T λ is simply connected). CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 13
Let Ω λ be the annulus defined in (4). By Lemma 3 .
4, we know that S n,d,λ (Ω λ ) ⊂ T λ and S n,d,λ ( T λ ) ⊂ A ∗ λ ( ∞ ). Thus, Ω λ ∩ T λ = ∅ and Ω λ is part of a multiply connected Fatoucomponent which is a preimage of T λ . We denote this Fatou component by A λ (observe thatup to this point we still do not know if A λ is doubly connected).We claim that w λ and ν λ do not belong to T λ ∪ A λ . To see the claim we will prove that,for sufficiently small values of | λ | , w λ and ν λ belong to the annulus bounded by ∂ A ∗ λ ( ∞ ) and A λ , denoted in what follows by A out . Let γ be a smooth Jordan curve which separates z = 0from ν and w , and such that S n,a,Q ( γ ) is a Jordan curve that surrounds z = 0 and lies inInt( γ ). Its existence follows from the B¨otcher coordinates of the fixed point z = 0 for theunperturbed map. Notice that, by construction, γ does not depend on λ and it has a definitedistance to z = 0. By continuity with respect to λ , for | λ | small enough, S n,d,λ ( γ ) ⊂ Int( γ ).Since S n,d,λ ( A λ ) ⊂ A ∗ λ ( ∞ ) we conclude that γ ∩ A λ = ∅ . Shrinking | λ | , if necessary, we claimthat γ lies outside Bdd (Ω λ ). Indeed, according to (4) the annulus Ω λ collapses to the originas λ → γ keeps in a definite distance to z = 0. Finally, notice that for | λ | small ν λ and w λ remain as close as we want to ν and w , respectively. Therefore, γ separates ν λ and w λ from T λ and A λ . Let C be a constant such that if | λ | < C all the above is true. Now weare ready to prove the statements.Since T λ contains only one critical point at z = 0 with multiplicity d − d onto the topological disk A ∗ λ ( ∞ ), it follows from the Riemann-Hurwitz formulathat T λ is simply connected. This proves (i) . Similarly, A λ contains exactly n + d simplecritical points and it is mapped with degree n + d onto the topological disk T λ . Thus, it isdoubly connected by Riemann-Hurwitz formula. This proves (ii) .The point w λ is a preimage of z = 0 which lies in A out , so it must belong to a Fatoucomponent, denoted by D λ , different from T λ and A λ . Moreover, S n,d,λ ( D λ ) = T λ . Since w λ is the only (simple) preimage of z = 0 in D λ we conclude that D λ is mapped with degree 1onto T λ and is a conformal copy of T λ . In particular, ν λ / ∈ D λ and all preimages of z = 0belong to either A λ and D λ . This proves (iii) , (iv) , and (v) .Finally, to prove statement (vi) we just notice that S n,d,λ | A in : A in → A ( ∂T λ , ∂ A ∗ λ ( ∞ )) isa proper map. Since its degree is accomplished on the boundaries and ∂T λ is mapped onto ∂ A ∗ λ ( ∞ ) with degree d , it follows that A in is mapped onto its image with degree d . (cid:3) We now prove that S n,d,λ is conjugate to a finite Blaschke product on the annulus A out introduced in the previous proposition. Proposition 3.6.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Then,there exist an analytic Jordan curve Γ ⊂ A λ which surrounds z = 0 , b ∈ D ∗ , θ ∈ [0 , , and aquasiconformal map ϕ : ˆ C → ˆ C such that ϕ ◦ R b,θ = ϕ ◦ S n,d,λ on A (Γ , ∂ A ∗ λ ( ∞ )) , where R b,θ = e πiθ z n z − b − bz is a Blaschke product.Proof. We claim that there exists an analytic Jordan curve Γ ⊂ A λ which surrounds z = 0and the n + d critical points c λ,ξ and which is mapped with degree n to its image under S n,d,λ .To see the claim let γ be an analytic Jordan curve in the interior of T λ surrounding z = 0and the n + d critical values, images of the critical points c λ,ξ . Clearly, the annulus A ( γ, ∂T λ )contains no critical values and, from the Riemann-Hurwitz formula (compare Corollary 2 . any connected component of its preimage is an annulus bounded by preimages of γ and ∂T λ .It follows from Proposition 3 . A λ , one associated to the internal boundary of A λ and another associatedto the external one. Denote them by G in and G out . By construction, S n,d,λ restricted to thosetwo preimages is a proper map. We know that S n,d,λ restricted to A λ is proper of degree n + d (see Proposition 3 . (ii) ) while S n,d,λ restricted to G in is proper of degree d (notice that thedegree is achieved in the boundary, compare with Proposition 3 . (vi) ). All together impliesthat S n,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 originas well as all critical points c λ,ξ , as desired.The remaining part of the proof is analogous to the one of [Can18, Proposition 3.1], so weprovide the main idea and leave the details to the reader. The strategy is to use a similarconstruction 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 thecurve γ , keep S n,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 = 0as superattracting fixed point of local degree n ( F is actually holomorphic around z = 0).The map F coincides with S n,d,λ on Ext(Γ), all points in Int( ∂ A ∗ λ ( ∞ )) converge to z = 0under iteration of F , and it maps Int( ∂ A ∗ λ ( ∞ )) onto itself with degree n + 1 (since we havethat z = 0 maps to itself with degree n and w λ 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 basinof 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 withdegree n + 1, the Blaschke product has the form R b,θ = e πiθ z n z − b − bz , where b ∈ D ∗ and θ satisfies | e πiθ | = 1. Since F coincides with S n,d,λ in A (Γ , ∂ A ∗ λ ( ∞ )), it follows that S n,d,λ isconjugate to R b,θ in A (Γ , ∂ A ∗ λ ( ∞ )). (cid:3) It will be crucial in what follows to have a deep understanding of the preimages of curveswhich surround the origin z = 0 (as well as Fatou components). The following propositiondescribes this in a precise way. Proposition 3.7.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Let γ ⊂ A ( ∂T λ , ∂ A ∗ λ ( ∞ )) be a Jordan curve which surrounds z = 0 . Then, S − n,d,λ ( γ ) contains asingle connected component in A in , which surrounds z = 0 and is mapped with degree d onto γ . The other components of S − n,d,λ ( γ ) lie in A out and, depending on the location of the freecritical value, i.e. S n,d,λ ( ν λ ) , one of the following holds:(i) If S n,d,λ ( ν λ ) ∈ Int ( γ ) , then S − n,d,λ ( γ ) has a single connected component in A out . Indeed,it is a Jordan curve which surrounds z = 0 and it is mapped with degree n + 1 onto γ under S n,d,λ .(ii) If S n,d,λ ( ν λ ) ∈ γ , then S − n,d,λ ( γ ) has a single connected component in A out consistingof 2 Jordan curves intersecting precisely at ν λ . One is a Jordan curve γ − whichsurrounds z = 0 , but not w λ , and it is mapped with degree n onto γ . The other is aJordan curve γ − w which surrounds w λ , but not z = 0 , and it is mapped with degree onto γ . The curve γ − does not surround γ − w . CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 15 (iii) If S n,d,λ ( ν λ ) ∈ Ext ( γ ) , then S − n,d,λ ( γ ) has 2 disjoint components in A out . One is aJordan curve γ − which surrounds z = 0 , but not w λ , and it is mapped with degree n onto γ . The other is a Jordan curve γ − w which surrounds w λ , but not z = 0 , and itis mapped with degree onto γ . The curve γ − does not surround γ − w .Proof. We first notice that given any Jordan curve in A ( ∂T λ , ∂ A ∗ λ ( ∞ )) all preimages shouldbe located either in A in or A out since T λ , A λ and A ∗ λ ( ∞ ) are mapped outside A ( ∂T λ , ∂ A ∗ λ ( ∞ )).Moreover, by Proposition 3 . (vi) any Jordan curve in A ( ∂T λ , ∂ A ∗ λ ( ∞ )) should have preim-age(s) in A in as well as in A out .Let γ ∈ A ( ∂T λ , ∂ A ∗ λ ( ∞ )) be a Jordan curve surrounding the origin. First we study thetopology of the preimage(s) of γ in A in . By Proposition 3 . γ has exactly d preimages in A in .Let γ be one of the preimages of γ in A in . The goal is to show that in fact γ is mapped by T λ with degree d , so there are no other preimages whatsoever. Observe that I nt ( γ ) shouldcontain either a pole, a zero, or z = 0, otherwise it cannot be mapped to γ which surrounds z = 0. Therefore, γ surrounds z = 0. Take the annulus A ( γ, ∂ A ∗ λ ( ∞ )) and consider itspreimage in A in . Since the only preimage of ∂ A ∗ λ ( ∞ ) in A in is ∂T λ and A in contains nocritical point, the preimage should be the annulus A ( ∂T λ , γ ). The map S n,d,λ | A ( ∂T λ ,γ ) isproper of degree d since S n,d,λ maps ∂T λ onto ∂ A ∗ λ ( ∞ ) with degree d . We conclude that γ 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 A out isanalogous to the one of [Can18, Proposition 3.3] by using that S n,d,λ is conjugate to theBlaschke product R θ,b in A out (see Proposition 3 . (cid:3) Remark 3.
It follows from Proposition . that each Fatou component different from T λ 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 A in and the otherlies in A out . The following lemma shows that Fatou components which do not surround the origin donot have preimages which surround it.
Lemma 3.8.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Let U be amultiply connected Fatou component. If U does not surround z = 0 , then no component of itspreimage S − n,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 (cid:48) = Bdd( U ) and V (cid:48) the preimage of U (cid:48) which contains V . Observe that U (cid:48) ⊂ A ( ∂T λ , ∂ A ∗ λ ( ∞ )). Since V (cid:48) ⊂ A in ∪ A out , it can contain at most one critical point.Since S n,d,λ | V (cid:48) : V (cid:48) → U (cid:48) is proper and V’ contains at most one critical point, it follows fromthe Riemann Hurwitz formula that V (cid:48) is simply connected (compare Corollary 2 . V surrounds the origin, then z = 0 lies in V (cid:48) . However, this is impossible since z = 0 is mappedto ∞ and U (cid:48) is bounded. (cid:3) Proposition 3 . | λ | small enough the map S n,d,λ is essentially uni-criticalsince all critical points except ν λ belong to A λ ( ∞ ). Up to now, however, we have not imposedany particular dynamical behaviour for ν λ . With the aim of proving the main results of thispaper from now on we restrict ourselves to parameters for which the free critical point ν λ belongs to A λ ( ∞ ) (sometimes the term captured parameters is used). (a) The partition with respect to U ν and A λ . T λ U ν A λ U A ∗ λ ( ∞ ) U n +1 U n U d (b) The partition with respect to S n,d,λ ( U ν ). T λ S n,d,λ ( U ν ) A ∗ λ ( ∞ ) V n +1 V n Figure 4.
Partitions of the dynamical plane introduced in Theorem 3 . . ν λ ∈ A λ ( ∞ ) \ ( A ∗ λ ( ∞ ) ∪ T λ ∪ D λ ∪ A λ ).We further assume that ν λ belongs to a Fatou component which is an eventual preimage of A λ that surrounds z = 0. The following result gives relevant notation and determines a partitionof the dynamical plane (compare Figure 4) that will be extremely useful to study achievableconnectivities of Fatou components. Theorem 3.9.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Assume that ν λ ∈ U ν , where U ν is a Fatou component which is eventually mapped onto A λ and surrounds z = 0 . Then, U ν is triply connected and U ν ⊂ A out . Moreover, the following statements hold.(i) The set U ν bounds an open disk U which is mapped with degree onto the opendisk V n ∪ T λ , where V n is the annulus bounded by ∂T λ and S n,d,λ ( U ν ) . In particular, w λ ∈ U .(ii) The annulus U n +1 bounded by U ν and ∂ A ∗ λ ( ∞ ) is mapped with degree n + 1 onto theannulus V n +1 bounded by S n,d,λ ( U ν ) and ∂ A ∗ λ ( ∞ ) .(iii) The annulus U n bounded by A λ and U ν is mapped with degree n onto the annulus V n bounded by ∂T λ and S n,d,λ ( U ν ) .(iv) The annulus U d bounded by ∂T λ and A λ (i.e., the annulus A in ) is mapped with degree d onto the annulus A ( ∂T λ , ∂ A ∗ λ ( ∞ )) .(v) The Fatou component U ν lies in V n +1 and it is mapped under S n,d,λ with degree n + 1 onto its image. In particular, U ν surrounds S n,d,λ ( U ν ) . Remark 4.
Statement (iv) in Theorem . coincides with statement (vi) of Proposition . .We use this double naming ( U d and A in ) in order to uniformize notation in what follows.Notice that every set U i , i = d, n, n + 1 , is mapped onto its image with degree i . Thisnotation is particularly useful in Section 4. Also, notice that in order to simplify notation we CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 17 avoid indicating the dependence of U ν and U i , i = 1 , d, n, n + 1 , with respect to the parameter λ .Proof. By the Riemann-Hurwitz formula, the iterated preimages of A λ are doubly connectedunless they contain a critical point. Under our hypothesis, this occurs precisely at U ν (sincethe only critical point eventually mapped in A λ is ν λ ). A direct application of the Riemann-Hurwitz formula implies that, since ν λ is a simple critical point, U ν is triply connected.Moreover, U ν ⊂ A out since ν λ ∈ A out . This proves the first part of the statement.From above ∂ U ν has three components. Since U ν separates z = 0 from z = ∞ , there shouldbe exactly two components of ∂ U ν surrounding z = 0. We denote them by γ in c and γ out c , where γ in c ⊂ Int( γ out c ). The other component of ∂ U ν , denoted by γ c , does not surround z = 0.Set U = Int( γ c ). Since γ c is mapped onto a component of ∂ S n,d,λ ( U ν ), U is mappedeither to the bounded or the unbounded component of the complement of S n,d,λ ( U ν ) (whichis an annulus by hypothesis). However, since all poles are in A ∗ λ ( ∞ ) ∪ T λ , then U should bemapped onto the bounded component of S n,d,λ ( U ν ). Therefore, U contains the zero w λ (andno other preimages of z = 0). We conclude that S n,d,λ | U has degree 1. In particular, γ ismapped onto its image with degree 1. This proves (i) .Let U n +1 be the annulus bounded by U ν and ∂ A ∗ λ ( ∞ ), and let V n +1 = S n,d,λ ( U n +1 ). Byconstruction, V n +1 is the annulus bounded by S n,d,λ ( U ν ) and ∂ A ∗ λ ( ∞ ). It is immediate thatthe map S n,d,λ | U n +1 : U n +1 → V n +1 is proper. Since the degree is accomplished on theboundaries and ∂ A ∗ λ ( ∞ ) is mapped onto itself with degree n + 1, U n +1 is mapped onto V n +1 with degree n + 1. This proves (ii) . The proof of statement (iii) is similar and statement (iv) was already proven in Proposition 3 . (v) by contradiction. Assume that U ν does not lie in V n +1 .Then, either U ν maps onto itself (which is impossible) or V n +1 ⊂ U n +1 . This would imply that U n +1 is mapped under S n,d,λ on itself and, hence, there exists a periodic Fatou componentdifferent from A λ ( ∞ ). This is impossible since, by assumption, the orbits of all critical pointsconverge to z = ∞ . (cid:3) Remark 5.
Under the assumptions of Theorem . , if U is a Fatou component which sur-rounds z = 0 and lies in U n +1 or U n , then it follows from Proposition . that its image liesin Bdd ( U ) . Indeed, U n and U n +1 are contained in A out (see Figure 4), and A out belongs tothe 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 ontaking special iterated preimages of U ν . With this in mind we end the section by stating thefollowing corollary of Proposition 3 . . Corollary 3.10.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Assume that ν λ ∈ U ν , where U ν is an iterated preimage of A λ which surrounds z = 0 . Let W be a Fatoucomponent which surrounds z = 0 , different from T λ and A λ . Then, the following statementshold.(i) If W ⊂ V n +1 , then it has a unique preimage in U d , which surrounds z = 0 and ismapped under S n,d,λ onto W with degree d , and a unique preimage in U n +1 , whichsurrounds z = 0 and is mapped under S n,d,λ onto W with degree n + 1 .(ii) If W ⊂ V n , then it has a unique preimage in U d , which surrounds z = 0 and ismapped under S n,d,λ onto W with degree d , and two further preimages. One lies in U n , surrounds z = 0 and is mapped under S n,d,λ onto W with degree n . The other one lies in U , does not surround z = 0 , and is mapped under S n,d,λ onto W with degree . Proofs of theorems A and B
In this section we prove Theorem A and Theorem B. We first show that Fatou componentswhich do not surround z = 0 cannot be used to achieve higher connectivities. Lemma 4.1.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Assume alsothat ν λ ∈ U ν . Let V be a Fatou component which does not surround z = 0 . Then V and allof 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 preimageof it. By Lemma 3 . U does neither surround z = 0. It follows that Bdd( U ) does not containany critical point. Therefore, the map S n,d,λ | Bdd( U ) : Bdd( U ) → Bdd( V ) is a proper map ofdegree 1. We can conclude that S n,d,λ | U : U → V is conformal, so U and V have the sameconnectivity. (cid:3) Next we give the form of the connectivities of Fatou components of S n,d,λ . We want toremark that not all these connectivities are achievable (see Theorem A). Proposition 4.2.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Assumealso that ν λ ∈ U ν , where U ν is an iterated preimage of A λ which surrounds z = 0 . All Fatoucomponents have connectivity 1, 2, or κ = ( n + 1) i n j d (cid:96) + 2 , where i, j, (cid:96) ∈ N .Proof. By Proposition 3 . D λ and all its eventual preimages have connectivity 1 since none ofthem can contain critical points. Analogously, all eventual preimages of A λ other than U ν andits preimages have connectivity 2 since none of them contain critical points (see Corollary 2 . U ν . By Lemma 4 .
1, it suffices to studypreimages of U ν 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 U ν which surround z = 0 mapto their images with degree d, n , or n + 1. Since U ν has connectivity 3 = 1 + 2, the possibleconnectivities of the preimages of U ν surrounding z = 0 are of the form κ = ( n + 1) i n j d (cid:96) + 2,where i, j, (cid:96) ∈ N . (cid:3) According to Corollary 3 .
10, if U is a iterative preimage of U ν which surrounds z = 0,its preimages which surround z = 0 may be located in U d , U n or U n +1 . Next lemma showsthat there are achievable upper bounds for the itineraries of iterated preimages of U ν . Recallfrom Remark 5 that if U ⊂ U n is an iterated preimage of U ν , then either S n,d,λ ( U ) ⊂ U n or S n,d,λ ( U ) ⊂ U d . Let k ≥ S kn,d,λ ( U ν ) ⊂ Bdd( A λ ). The first halfof Lemma 4 . U ⊂ U n is a preimage of U ν which surrounds z = 0, then theitinerary of U intersects U d in p ≤ k iterations. The second half of Lemma 4 . U ⊂ U d is a preimage of U ν which surrounds z = 0, then there exist at least k − U which lie in U n . Lemma 4.3.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Assume alsothat ν λ ∈ U ν . Let k ≥ such that S kn,d,λ ( U ν ) ⊂ Bdd ( A λ ) and S jn,d,λ ( U c ) ⊂ U n for ≤ j < k . CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 19 U ν U T λ W B B o ut B i n W A λ Figure 5.
Description of the situation in the proof of Lemma 4 .
3, where k = 2and W ⊂ U d . In this case, B = B out2 ∪ A λ ∪ B in2 . (i) Let U ⊂ U n be an iterated preimage of U ν which surrounds z = 0 . Then, there exists p ≥ such that S p ( U ) ⊂ U d and S p (cid:48) ( U ) ∈ U n for ≤ p (cid:48) < p . Moreover, p satisfies p ≤ k .(ii) Let U ⊂ U d be a preimage of U ν which surrounds z = 0 . Then, there exists U (cid:48) ⊂ U n such that S jn,d,λ ( U (cid:48) ) ⊂ U n for ≤ j < k − and S k − n,d,λ ( U (cid:48) ) = U .Proof. Set W i = S in,d,λ ( U ν ) , i = 0 , . . . , k . By Remark 5, W i surrounds W i +1 , i = 0 , . . . , k − B i +1 be the annulus bounded by W i and W i +1 , i = 0 , . . . , k −
1. It follows that S n,d,λ ( B i ) = B i +1 , i = 1 , . . . , k −
1. Observe that if W k ⊂ U d , then B k = B o utk ∪ A λ ∪ B i nk , where B o utk = B k ∩ U n and B i nk = B k ∩ U d (see Figure 5). Along the proof we distinguish the cases W k = A λ and W k ⊂ U d .We now prove statement (i) . Assume first that W k = A λ . Let U ⊂ B i , i = 1 , . . . , k , bean eventual preimage of U ν which surrounds z = 0. Then S k +1 − in,d,λ ( U ) ⊂ U d . We can concludethat if U ⊂ U n is a preimage of U ν which surrounds z = 0, then there exists p ≤ k suchthat S pn,d,λ ( U ) ⊂ U d . In fact, p = k − i if U ⊂ B i . Now assume W k ⊂ U d . For k = 1, S n,d,λ ( U n ) ⊂ U d and the conclusion follows. For k ≥ W ⊂ U n (so B k − and B k exist). Let U ⊂ B i , i = 1 , . . . , k −
1, be a preimage of U ν which surrounds z = 0 and let V = S k − − in,d,λ ( U ). Observe that V ⊂ B k − is an eventual preimage of U ν which surrounds z = 0 and U d is the disjoint union of B i nk , W k , and S n,d,λ ( B o utk ). Then, S n,d,λ ( V ) ⊂ B o utk (and S n,d,λ ( V ) ⊂ U d ) or S n,d,λ ( V ) ⊂ B i nk (and S n,d,λ ( V ) ⊂ U d ). Since S k − − in,d,λ ( B i ) = B k − , i = 0 , . . . , k −
1, this concludes the proof of statement (i) .Now we prove (ii) . Let U ⊂ U d be a preimage of U ν which surrounds z = 0. Assumefirst that W k = A λ . Then U d = S n,d,λ ( B k ) = S kn,d,λ ( B ). So there exists U (cid:48) ⊂ U such that S jn,d,λ ( U (cid:48) ) ⊂ B j +1 ⊂ U n for 0 ≤ j < k and S kn,d,λ ( U (cid:48) ) = U . Now let W k ⊂ U d . For k = 1there is nothing to prove. For k ≥ W ⊂ U n (so B k − and B k exist). Moreover, U d = B i nk ∪ W k ∪ S n,d,λ ( B o utk ). Since U is a preimage of U ν we have that U (cid:54) = W k . We distinguish 2 cases. If U ⊂ B i nk , then U ⊂ S n,d,λ ( B k − ) and we can take preimages throughthe sets B i so that there exists U (cid:48) ⊂ B such that S jn,d,λ ( U (cid:48) ) ⊂ B j +1 ⊂ U n for 0 ≤ j < k − S k − n,d,λ ( U (cid:48) ) = U . Finally, if U ⊂ S n,d,λ ( B o utk ), then U ⊂ S n,d,λ ( B k ) and there exists U (cid:48) ⊂ B such that S jn,d,λ ( U (cid:48) ) ⊂ B j +1 ⊂ U n for 0 ≤ j < k and S kn,d,λ ( U (cid:48) ) = U . This concludesthe proof of statement (ii) . (cid:3) 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 connectivityof a Fatou component is different from 1 or 2, then it has to be of the form κ = ( n +1) i d j n (cid:96) +2where i, j, (cid:96) ∈ N . It follows from the Riemann-Hurwitz formula that if f : U → V is proper ofdegree q without critical points and V has connectivity p + 2, then U has connectivity qp + 2.Moreover, these connectivities are achieved through preimages of U ν . In order to increase theconnectivity we have to take preimages of U ν , which has connectivity 3 = 1 + 2. It followsfrom Remark 5 that if U ⊂ U s , 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 U n . By Remark 5,every Fatou component U ⊂ U n is eventually mapped to U d , without passing through U n +1 .By Lemma 4 . (i) , for every backwards iteration through U d there are at most k backwardsiterations in U n . Since U ν (cid:54)⊂ U d ∪ U n , it follows that (cid:96) ≤ jk . (cid:3) The final part of this section is devoted to the proof of Theorem B. The following lemmashows that there are no restrictions to the exponents of n + 1 and d in respect to achievableconnectivities. Lemma 4.4.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Assume alsothat ν λ ∈ U ν , where U ν is an iterated preimage of A λ which surrounds z = 0 . Then, thefollowing hold:(i) There exists an eventual preimage of U ν which lies in U n +1 , surrounds z = 0 , and hasconnectivity κ = ( n + 1) i + 2 , ∀ i ∈ N .(ii) Let V be an eventual preimage of U ν which surrounds z = 0 and let κ be the con-nectivity of V . Then, there exists a Fatou component, which surrounds z = 0 , ofconnectivity d ( κ −
2) + 2 . In particular, there exists a Fatou component which surrounds z = 0 and has connectivity κ = ( n + 1) i d j + 2 , ∀ i, j ∈ N .Proof. First we prove (i) . Recall that, by Theorem 3 . U ν ⊂ V n +1 . By Corollary 3 .
10 andRemark 5, for any i ≥ U which surrounds z = 0 such that S jn,d,λ ( U ) ⊂ U n +1 , for j = 0 , . . . i −
1, and S in,d,λ ( U ) = U ν . Since U ν has connectivity 3 andno eventual preimage of U ν contains a critical point, it follows by succesively applying theRiemann-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 U ν which surrounds z = 0, of connectivity κ . By Corollary 3 . V has a preimage in U d which surrounds z = 0 and which is mappedonto it with degree d . Since V cannot contain any critical value, by the Riemann-Hurwitzformula the connectivity of this preimage of V is d ( κ −
2) + 2. This concludes the proof of (ii) . (cid:3) We can now proceed with proof of Theorem B.
CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 21
Proof of Theorem B.
Fix i ≥ , j ≥ , (cid:96) ≥ (cid:96) ≤ j ( k − U (which will be an iterated preimage of U ν ) of connectivity κ = ( n + 1) i d j n (cid:96) + 2.If k = 1, then by Lemma 4 . S n,d,λ ( U ν ) ⊂ U n and U d ⊂ V n .By Lemma 4 . (i) , there exists a Fatou component V which is an iterated preimage of U ν ,surrounds z = 0, lies in U n +1 , and has connectivity ( n + 1) i + 2. This concludes the proof for j = 0.Assume that j (cid:54) = 0 (remember that k > . (i) , there exists a preimageof V in U d which surrounds z = 0. By Lemma 4 . (ii) , there exists V (1) ⊂ U n which surrounds z = 0 such that S rn,d,λ ( V (1) ) ⊂ U n f or ≤ r < k − , S k − n,d,λ ( V (1) ) ⊂ U d a nd S kn,d,λ ( V (1) ) = V. Recall that no iterated preimage of U ν contains a critical point and so, by Corollary 3 .
10, ifthey lie in U i , i ∈ { n + 1 , n, d } , they map forward with degree i . Applying this criteria tothe iterated preimages of V up to V (1) , we get from the Riemann-Hurwitz formula that κ (cid:16) S rn,d,λ ( V (1) ) (cid:17) = ( n + 1) i dn k − − r + 2 , r = 0 , . . . k − . Starting the process with V (1) instead of V we can take a preimage of V (1) in U d and thenwe can take up to k − U n to land on, say, V (2) . As above, we get that κ (cid:16) S rn,d,λ ( V (2) ) (cid:17) = ( n + 1) i d n k − − r + 2 , r = 0 , . . . k − . Repeating the same process j -times we conclude that there exist Fatou components withconnectivity κ (cid:16) S rn,d,λ ( V ( s ) ) (cid:17) = ( n + 1) i d s n s ( k − − r + 2 , s = 1 , . . . j, r = 0 , . . . k − . If ( j − k − < (cid:96) ≤ j ( k −
1) we are done (take s = j in the previous formula). If (cid:96) ≤ t ( k − t ≤ ( j − t and take j − t preimages in U d to get thedesired connectivity. (cid:3) Proof of theorem C
In this section we prove Theorem C. We first show that there is a sequence of preimages of A λ which surround z = 0 and accumulate on ∂ A ∗ λ ( ∞ ). We want to remark that the set A out depends on λ even if we do not indicate it in its notation. Lemma 5.1.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Let A ,λ := A λ .Then there exist { A m,λ } m ≥ iterated preimages of A ,λ , S n,d,λ ( A m +1 ,λ ) = A m,λ , such that thefollowing properties are satisfied:(i) Each Fatou component A m,λ is surrounded by A m +1 ,λ , that is, A m,λ ⊂ Bdd ( A m +1 ,λ ) .In particular, all { A m,λ } m ≥ lie in A o ut .(ii) The sequence of Fatou components { A m,λ } m ≥ accumulate on ∂ A ∗ λ ( ∞ ) as m → ∞ .Proof. Every Fatou component which surrounds z = 0 has exactly 2 boundary componentswhich surround z = 0. It follows from Proposition 3 . A out which surrounds z = 0 has exactly a preimage in A out which surrounds z = 0. Let { A m,λ } m ≥ be the sequence of Fatou components obtained by taking consecutive preimages of A ,λ in A out which surround z = 0. Since S n,d,λ is conjugated to a Blaschke product on A out , by Proposition 3 .
6, the Fatou components A m,λ accumulate on ∂ A ∗ λ ( ∞ ) as m goes to ∞ . It alsofollows from the conjugation with the Blaschke product that A m,λ ⊂ Bdd( A m +1 ,λ ) for all m ≥ (cid:3) The multiply connected sets A m,λ surround z = 0. Therefore, there are exactly 2 boundarycomponents of A m,λ which surround z = 0. We denote them by ∂ Int A m,λ and ∂ Ext A m,λ ,where ∂ Int A m,λ ⊂ Int( ∂ Ext A m,λ ). Next lemma tells as that if m is large enough then thereare parameters λ such that ν λ ∈ A m,λ . Lemma 5.2.
Let S n,d,λ satisfying (a), (b), (c), and (d). Let λ (cid:54) = 0 , | λ | < C . Then, if m ∈ N ∗ is big enough, there exists a parameter λ such that ν λ ∈ A m,λ .Proof. The idea of the proof is to show that, if m is big enough, we can find a parameter λ such that ν λ ∈ Bdd( A m,λ ) and a parameter λ m such that ν λ m belongs to the unboundedcomponent of C ∗ \ A m,λ m . We will then conclude that there exists a parameter λ (cid:48) m such that ν λ (cid:48) m ∈ A m,λ (cid:48) m .Fix λ such that all hypothesis hold. Then A ,λ is well defined, and so are A m,λ , m > m be such that ν λ ∈ Bdd( A m ,λ ). Then, for all m ≥ m we have ν λ ∈ Bdd( A m,λ ).For fixed m ≥ m , we want to find the parameter λ m . If λ = 0, the critical point ν belongsto the boundary of the maximum domain of definition of the B¨ottcher coordinate of A ∗ (0)under S n,a,Q . Therefore, the orbit of ν under S n,a,Q accumulates on z = 0 but never mapsonto it. Observe that S n,d,λ converges uniformly to S n,a,Q on compact subsets of C ∗ \ D (cid:15) as λ →
0, where (cid:15) > D (cid:15) denotes the disk of radius (cid:15) centered at z = 0.Consequently, for fixed m ≥ (cid:15) >
0, if | λ | is small enough then A m,λ ⊂ D (cid:15) . Since ν λ → ν as λ →
0, it follows that if | λ | is small enough then ν λ belongs to the unbounded componentof C ∗ \ A m,λ . It is enough to take λ m to be any such λ .To finish the proof we need to show that when we move continuously the parameter from λ to λ m we need to find intermediate parameters λ (cid:48) m such that ν λ (cid:48) m ∈ A m,λ (cid:48) m . By Proposition 3 . ∂ A ∗∞ moves continuously with respect to λ . Since ∂ A ∗∞ and ∂T λ cannot containcritical values, it follows that both boundary components of A ,λ move continuously withrespect to λ . For fixed λ (cid:48) , the set ∂A m,λ , m ≥
1, moves continuously with respect to λ in aneighbourhood of λ (cid:48) unless ∂A m,λ (cid:48) (or an iterated image of ∂A m,λ (cid:48) ) contains a critical point.Here by moving continuously we mean that every connected component of ∂A m,λ is a Jordancurve that moves continuously with respect to the Hausdorff metric (in particular, it doesnot pinch itself or split in several connected components). Notice that since there is only afree critical point, at most one of the 2 components of ∂A m,λ which surround z = 0 may notmove continuously for λ in a neighbourhood of λ (cid:48) . Using Proposition 3 . S n,d,λ ( ∂ Ext A m,λ ) = ∂ Ext A m − ,λ and S n,d,λ ( ∂ Int A m,λ ) = ∂ Int A m − ,λ . Assume that for λ (cid:48) we have ν λ (cid:48) ∈ ∂A m,λ (cid:48) . Then A m − ,λ is an annulus that moves continuously with respectto λ for all λ in a neighbourhood of λ (cid:48) . If ν λ (cid:48) ∈ ∂ Int A m,λ (cid:48) then S n,d,λ ( ν λ (cid:48) ) ∈ ∂ Int A m − ,λ (cid:48) . ByProposition 3 .
7, for λ in a neighbourhood of λ (cid:48) exactly one of the following holds (see thethree upper figures in Figure 6): • If S n,d,λ ( ν λ ) ∈ Int( ∂ Int A m − ,λ ) then ν λ ∈ Int( ∂ Int A m,λ ) and A m,λ is doubly connected. • If S n,d,λ ( ν λ ) ∈ ∂ Int A m − ,λ then ν λ ∈ ∂ Int A m,λ . Then, A m,λ is doubly connected and ∂ Int A m,λ consists of the union of 2 Jordan curves. • If S n,d,λ ( ν λ ) ∈ A m − ,λ then ν λ ∈ A m,λ and A m,λ is triply connected. CHIEVABLE CONNECTIVITIES OF FATOU COMPONENTS FOR A FAMILY OF RATIONAL MAPS 23 ν λ S n,d,λ ( ν λ ) A k − ,λ A k,λ ν λ S n,d,λ ( ν λ ) A k − ,λ A k,λ ν λ S n,d,λ ( ν λ ) A k − ,λ A k,λ ν λ S n,d,λ ( ν λ ) A k − ,λ A k,λ ν λ S n,d,λ ( ν λ ) A k − ,λ A k,λ A (cid:48) ν λ S n,d,λ ( ν λ ) A k − ,λ A k,λ Figure 6.
The top figures correspond to the possible cases of ν λ lying in aneighbourhood of ∂ Int A m,λ . The top figures correspond to the possible casesof ν λ lying in a neighbourhood of ∂ Ext A m,λ .On the other hand, if ν λ (cid:48) ∈ ∂ Ext A m,λ (cid:48) then S n,d,λ ( ν λ (cid:48) ) ∈ ∂ Ext A m − ,λ (cid:48) . By Proposition 3 . λ in a neighbourhood of λ (cid:48) exactly one of the following holds (see the three lower figuresin Figure 6): • If S n,d,λ ( ν λ ) ∈ A m − ,λ then ν λ ∈ A m,λ and A m,λ is triply connected. • If S n,d,λ ( ν λ ) ∈ ∂ Ext A m − ,λ then ν λ ∈ ∂ Ext A m,λ . Then, A m,λ is doubly connected, ∂ Ext A m,λ is a Jordan curve, and there is an extra preimage A (cid:48) of A m − ,λ such that ∂A m,λ ∩ ∂A (cid:48) = ν λ . • If S n,d,λ ( ν λ ) ∈ Ext( ∂ Int A m − ,λ ) then ν λ ∈ Ext( ∂ Int A m,λ ) and A m,λ is doubly con-nected.It follows from the previous configurations that if we move continuously the parameter λ from λ until λ m we can find parameters λ (cid:48) m such that ν λ (cid:48) m ∈ A m,λ (cid:48) m . This finishes the proofof the result. (cid:3) We can now proceed with proof of Theorem C.
Proof of Theorem C.
Fix i, j, (cid:96) . We have to prove that there exists λ for which there is aFatou component of connectivity κ = ( n + 1) i d j n (cid:96) + 2 and a Fatou component of connectivity κ = ( n + 1) i + 2. Recall that the results in Section 4 required the free critical point ν λ to lie in a preimage of A λ which surrounds z = 0. By Lemma 5 .
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Departament de Matem`atiques, Universitat Jaume I, 12071 Castell´o de la Plana, Spain
Email address : [email protected] Departament de Matem`atiques i Inform`atica at Universitat de Barcelona and BarcelonaGraduate School of Mathematics, 08007 Barcelona, Catalonia.
Email address : [email protected] Departament de Matem`atiques i Inform`atica at Universitat de Barcelona and BarcelonaGraduate School of Mathematics, 08007 Barcelona, Catalonia.
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