A family of rational maps with buried Julia components
AA family of rational mapswith buried Julia components
Sébastien GodillonNovember 24, 2013
Abstract
It is known that the disconnected Julia set of any polynomial map does not containburied Julia components. But such Julia components may arise for rational maps. Thefirst example is due to Curtis T. McMullen who provided a family of rational maps forwhich the Julia sets are Cantor of Jordan curves. However all known examples of buriedJulia components, up to now, are points or Jordan curves and comes from rational mapsof degree at least 5.This paper introduce a family of hyperbolic rational maps with disconnected Julia setwhose exchanging dynamics of postcritically separating Julia components is encoded bya weighted dynamical tree. Each of these Julia sets presents buried Julia components ofseveral types: points, Jordan curves, but also Julia components which are neither pointsnor Jordan curves. Moreover this family contains some rational maps of degree 3 withexplicit formula that answers a question McMullen raised. a r X i v : . [ m a t h . D S ] M a r ontents (cid:98) f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Cutting along a system of equipotentials . . . . . . . . . . . . . . . . . . . . . 133.3 Folding with an annulus-disk surgery . . . . . . . . . . . . . . . . . . . . . . . 163.4 Preimage of the branching part . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Achievement of the super-attracting cycle of period . . . . . . . . . . . . . . 183.6 Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 References 38 Introduction
For any rational map f of degree d (cid:62) on the Riemann sphere (cid:98) C , we denote by J ( f ) its Juliaset, namely the closure of the set of repelling periodic points. We recall that J ( f ) is a fullyinvariant non-empty perfect compact set which either is connected or has uncountably manyconnected components (see [Bea91], [CG93], [Mil06]). This paper focuses on the disconnectedcase. Every connected component of J ( f ) is called a Julia component and every connectedcomponent of the Fatou set (cid:98) C − J ( f ) is called a Fatou domain.A Julia component is said to be buried if it has no intersection with the boundary of anyFatou domain. In particular buried Julia components can not occur in the polynomial case(since the Julia set coincides with the boundary of the unbounded Fatou domain). The sameholds if the Julia set is a Cantor set, or more generally if the complementary of every Juliacomponent is connected (since the Fatou set is then connected). That suggests much moresophisticated topological structures for Julia sets with some buried Julia components thanthose encountered in the polynomial case.The first example of rational maps with buried Julia components is due to Curtis T.McMullen. Consider the family of rational maps given by g c,λ : z (cid:55)→ z d ∞ + c + λz d where d ∞ , d (cid:62) and c, λ ∈ C . The special case c = 0 has been studied in [McM88] (see also [DHL + d ∞ + 1 d < (H0)and if | λ | > is small enough then J ( g ,λ ) is a Cantor of Jordan curves, namely homeomorphicto the product of a Cantor set with a Jordan curve (see Figure 1.a). Recall that any Cantorset is homeomorphic to the no middle third set on a line segment which contains uncountablymany points which are not endpoints of any removing open segment. Each of these pointscorresponds to a buried Jordan curve in J ( g ,λ ) .In [PT00], the authors have provided another example by slightly modifying the map g − ,λ for d ∞ = 2 and d = 3 (that satisfies assumption (H0)) in a clever way: (cid:93) g − ,λ : z (cid:55)→ z ◦ ( z − ◦ z + λz = z − z + λz where λ ∈ C . If | λ | > is small enough then J ( (cid:93) g − ,λ ) has the same topological structure than J ( g ,λ ) exceptthat one fixed Julia component (which contains the boundary of the unbounded Fatou domainand hence is not buried) is quasiconformally homeomorphic to the Julia set of z (cid:55)→ z − .The uncountably many Julia components which are not eventually mapped under iterationonto this fixed Julia component are buried Jordan curves in J ( (cid:93) g − ,λ ) (see Figure 1.b).Examples of buried Jordan components which are not Jordan curves have appeared insome works. For instance in [BDGR08] (see also [DM08] and [GMR13]), the authors havestudied the family g c,λ for d ∞ = d (cid:62) (that satisfies assumption (H0)) and for a fixedparameter c chosen so that for the polynomial z (cid:55)→ z d ∞ + c the critical point 0 lies in acycle of period at least 2. In that case, if | λ | > is small enough then J ( g c,λ ) still hasuncountably many Jordan curves as buried Jordan components but also uncountably manypoints. The remaining Julia components are eventually mapped under iteration onto a fixedJulia component (which coincides with the boundary of the unbounded Fatou domain and3ence is not buried) quasiconformally homeomorphic to the Julia set of z (cid:55)→ z d ∞ + c . Each ofthese not buried Julia components has infinitely many “decorations” and every buried pointcomponent is accumulated by a nested sequence of such decorations (see Figure 1.c).Figure 1: a) J ( g ,λ ) for d ∞ = 2 , d = 3 and λ ≈ − . b) J ( (cid:93) g − ,λ ) for d ∞ = 2 , d = 3 and λ ≈ − . c) J ( g c,λ ) for d ∞ = d = 3 , c = − i and λ ≈ − .All the previous examples are rational maps of degree d ∞ + d at least 5 according toassumption (H0). The existence question of buried Julia components for rational maps ofdegree less than 5 has been raised in [McM88]. In the last decade, a number of papers haveappeared that deal with subfamilies of g c,λ or some slightly perturbations of it. Some of thempresent sophisticated Julia sets with buried Julia components, however the degree of theseexamples is always at least equal to 5. Furthermore the buried Julia components of theseexamples are points or Jordan curves.The aim of this paper is to answer the question Curtis T. McMullen has raised by providinga family of rational maps of degree which does not come from the family g c,λ and whoseJulia set presents buried Julia components of several types: points, Jordan curves but alsoJulia components which are neither points nor Jordan curves. One of our main result here isthe following. Theorem 1.
Consider the family of cubic rational maps given by f λ : z (cid:55)→ (1 − λ ) (cid:104) (1 − λ + 6 λ − λ ) z − λ (cid:105) ( z − (cid:104) (1 − λ − λ ) z − λ (1 − λ ) (cid:105) where λ ∈ C . If | λ | > is small enough then J ( f λ ) contains buried Julia components of several types: (point type) uncountably many points; (circle type) uncountably many Jordan curves; (complex type) countably many preimages of a fixed Julia component which is quasicon-formally homeomorphic to the connected Julia set of f : z (cid:55)→ z − . An example of such Julia set is depicted in Figure 2. J ( f λ ) is called a “Persian carpet”because of similarities with sophistications from carpet-weaving art: the Julia set of f : z (cid:55)→ z − appears as a watermark in the central motif of the carpet whose surface is covered byan elaborate pattern of Cantor of Jordan curves, and there are some small Julia components4igure 2: a) A Persian carpet: J ( f λ ) for λ ≈ − . b) J ( f ) which appears as a buried Julia component in J ( f λ ) . c) A magnification about a dust of the Persian carpet.everywhere that looks like dust. These small dusts contain nested sequences of finite coveringsof the Persian carpet which accumulate buried point components.The Persian carpet example is maximal among rational maps with buried Julia componentsin the sense that buried Julia components can not occur for rational maps of degree less than . Indeed, by a theorem in [Mil00], the Julia set of any quadratic rational maps is eitherconnected or a Cantor set.Furthermore, the Persian carpet example is maximal among geometrically finite rationalmaps (namely rational maps such that every critical point in the Julia set is preperiodic, inour case f λ is hyperbolic, namely it has no critical point in J ( f λ ) for | λ | > small enough)in the sense that every Julia component (not necessarily buried) of such a map is one of thethree types described in Theorem 1. That follows from two results. Firstly, by a theoremin [McM88], every periodic Julia component of a rational map is either a point or quasicon-formally homeomorphic to the connected Julia set of a rational map. Secondly, it has beenproved in [PT00] that every Julia component of a geometrically finite rational map which isnot eventually mapped under iteration onto a periodic Julia component is either a point or aJordan curve.The underlying idea in the construction of the Persian carpet example is that the sophis-ticated configuration on (cid:98) C of Julia components which are not points may be encoded by atree. Tree structures have appeared in various works on holomorphic dynamics (for instanceHubbard trees in [DH84] to classify postcritically finite polynomial maps). The tree consid-ered here is not embedded in (cid:98) C . It is seen as an abstract object which is very similar to, andactually inspired by, the trees introduced by Mitsuhiro Shishikura in [Shi89] which describethe configurations of Herman rings for rational maps.However, the purpose of this paper is only to introduce a family of rational maps comingfrom a particular tree which answers the question Curtis T. McMullen has raised. But notto discuss about the general existence question of rational maps whose configuration of Juliacomponents is encoded by any given tree (that will be the purpose of future works) even if ageneral construction may be suggested (especially statements and discussions in Section 2). Organization of the paper.
Section 2 deals with exchanging dynamics of postcriticallyseparating Julia components by weighted dynamical tree.5n Section 2.1, we specify the idea mentioned above by showing that, under assumption(H0), the exchanging dynamics of Julia components for the family g ,λ is encoded by a certainweighted dynamical tree ( H Q , w ) (see Theorem 2).The purpose of Section 2.2, is then to do the converse: starting from a particular dynamicaltree H P more sophisticated than H Q and a weight function w on its edges, Theorem 3 statesthe existence of rational maps with disconnected Julia set whose exchanging dynamics ofpostcritically separating Julia components is encoded by ( H P , w ) if (and, actually, only if)two conditions (H1) and (H2) hold. Theorem 4 shows that the Julia sets of these rationalmaps own buried Julia components of every expected type.The main part of the proofs of Theorem 3 and Theorem 4, that is the construction byquasiconformal surgery of the required rational maps, is detailed in Section 3.In Section 4, some properties of the rational maps constructed in the previous sectionare shown. The properties about exchanging dynamics (Section 4.1) conclude the proof ofTheorem 3 while the properties about topology of some Julia components (Section 4.2) givethe proof of Theorem 4.Section 5 deals with a particular choice of the weight function w for which the two as-sumptions (H1) and (H2) are satisfied and such that the rational maps in Theorem 3 andTheorem 4 have degree 3. In this case, an explicit formula is provided that concludes theproof of Theorem 1.Finally, some technical results used in the construction of Section 3 are collected in Section6 with proofs or references. Acknowledgment.
The author would like very much to thank Professor Tan Lei, the advisorof his thesis, together with Professor Cui Guizhen for all their useful comments and fruitfuldiscussions on this work. Finally, the author thanks the referees for several helpful suggestions.
For any rational map f : (cid:98) C → (cid:98) C , we denote by J ( f ) the set of Julia components and werecall that f induces a topological dynamical system on J ( f ) endowed with the usual distancebetween continua on (cid:98) C equipped with the spherical metric (notice that J ( f ) is closed for thisdistance, that is not true in general for the Hausdorff distance). This topological dynamicalsystem is called the exchanging dynamics of Julia components .We recall that the critical points of f are the points where f is not locally injective, and thepostcritical points of f are the points of the form f n ( c ) for some n (cid:62) and for some criticalpoint c . A Julia component J ∈ J ( f ) is said to be postcritically separating if J separates thepostcritical set of f , or equivalently if (cid:98) C − J has at least two connected components containingat least one postcritical point of f each. We denote by J crit ( f ) the subset of postcriticallyseparating Julia components in J ( f ) . Remark that J crit ( f ) is forward invariant, and thus f induces a topological dynamical system on J crit ( f ) . Consider the cubic polynomial Q : z (cid:55)→ z ( − z ) . It has two simple critical points: 0 whichis fixed, and 1 which is mapped on 0 after two iterations. (cid:57) (cid:57) (cid:47) (cid:47) (cid:1) (cid:1) H Q be its Hubbard tree, namely the smallest closed connected infinite union of internalrays which contains the postcritical set { , } (see [DH84]). In fact, H Q is the straight realsegment [0 , ] or more precisely the union of two edges [0 , ∪ [1 , ] while the vertices are , and . Both edges of H Q are homeomorphically mapped by Q onto the whole tree (see Figure3.b).Figure 3: a) The Julia set of the polynomial Q . b) The action of Q on the Hubbard tree H Q . c) The action of g ,λ on the set of Julia components J ( g ,λ ) .Denote by J ( H Q ) the intersection set between the Hubbard tree H Q and the Julia set J ( Q ) . Notice that J ( H Q ) is disconnected (actually a Cantor set) and Q induced a dynamicalsystem on it since the Hubbard tree H Q and the Julia set J ( Q ) are both invariant.Finally, let w be a weight function on the set of edges of H Q , say w ([0 , d ∞ and w ([1 , ]) = d where d ∞ , and d are positive integer.The result about the family g ,λ discussed in the introduction (see Section 1) may bereformulated as follows. Theorem 2.
If the weighted dynamical tree ( H Q , w ) satisfies the following condition d ∞ + 1 d < (H0) then for every | λ | > small enough, the exchanging dynamics of Julia component of g ,λ isencoded by ( H Q , w ) in the following sense: (i) every critical orbit accumulates the super-attracting fixed point ∞ ; (ii) there exists a homeomorphism h : J ( g ,λ ) → J ( H Q ) such that the following diagramcommutes; J ( g ,λ ) g ,λ (cid:47) (cid:47) h (cid:15) (cid:15) J ( g ,λ ) h (cid:15) (cid:15) J ( H Q ) Q (cid:47) (cid:47) J ( H Q ) (iii) for every Julia component J ∈ J ( g ,λ ) , the restriction map g ,λ | J has degree w ( e ) where e is the edge of H Q which contains h ( J ) . Notice that J ( g ,λ ) = J crit ( g ,λ ) for | λ | > small enough since every Julia component is aJordan curve which separates the fixed critical point ∞ from some critical values close to .7 roof. We only sketch the proof since the main part is done in [McM88]. Indeed it is shownthat there exists a large annulus A centered at 0 and containing J ( g ,λ ) whose preimageconsists of two disjoint annuli A ∞ , A both nested in A and such that the restriction maps g ,λ | A ∞ : A ∞ → A and g ,λ | A : A → A are coverings of degree d ∞ and d , respectively.Using combinatorial reasoning from holomorphic dynamics, it is a classical exercise to provethat the set of connected components of J ( g ,λ ) = ∩ n (cid:62) g − n ,λ ( A ) is homeomorphic to the spaceof all sequences of two digits Σ = { , } N (equipped with the product topology making it aCantor set) and the exchanging dynamics is topologically conjugated to a 2-to-1 shift map σ : Σ → Σ defined by σ ( s , s , s , . . . ) = ( s , s , s , . . . ) . The same holds for the dynamicalsystem induced by Q on J ( H Q ) since for ε > small enough the real segment I = [ ε, − ε ] contains J ( H Q ) and its preimage consists of two disjoint real segment both included in I (onein each of the two edges of H Q ).Heuristically speaking, we may topologically think the Riemann sphere (cid:98) C as a smoothneighborhood’s boundary of the tree H Q embedded in the space R . The two points on thistopological sphere which correspond to ∞ and should be closed to the corresponding verticesof H Q which are and , respectively. If the neighborhood becomes smaller and smaller, everyJordan curves in J ( g ,λ ) is shrunk to a point in J ( H Q ) (see Figure 3.c). Consider a quadratic polynomial of the form P : z (cid:55)→ z + c where the parameter c ∈ C is chosen in order that the critical point 0 is periodic of period 4. There are exactly sixchoices of such a parameter. Let us fix c to be that one with the largest imaginary part,that is c ≈ − .
157 + 1 . i . The postcritical points are denoted by c k = P k (0) for every k ∈ { , , , } . c (cid:2) (cid:2) c (cid:47) (cid:47) α (cid:121) (cid:121) c (cid:87) (cid:87) c (cid:109) (cid:109) Let H P be the Hubbard tree of P (see Figure 4.b). As one-dimensional simplicial complex, H P may be described by a set of five vertices { c , c , c , c , α } where α is a fixed point of P and the following four edges: e = [ α, c ] H P ; e = [ α, c ] H P ; e = [ α, c ] H P ; e [ c , c ] H P .P homeomorphically acts on the edges as follows. P ( e ) = e P ( e ) = e P ( e ) = e ∪ e P ( e ) = e ∪ e Denote by J ( H P ) the intersection set between the Hubbard tree H P and the Julia set J ( P ) .Notice that J ( H P ) is disconnected (actually a Cantor set) and P induced a dynamical system8n it. Moreover the fixed branching point α belongs to J ( H P ) but not to the boundary ofany connected component of H P − J ( H P ) . Finally, let w be a weight function on the set ofedges of H P , say w ( e k ) = d k where d k is a positive integer for every k ∈ { , , , } .Figure 4: a) The Julia set of the polynomial P . b) The Hubbard tree H P . c) The action of P on a straightened copy of H P . Definition 1.
The transition matrix of the weighted dynamical tree ( H P , w ) is the -by- matrix M = ( m i,j ) i,j ∈{ , , , } whose entries are defined as follows. ∀ i, j ∈ { , , , } , m i,j = w ( e i ) if e j ⊂ P ( e i )0 otherwiseSince M is a non-negative matrix, it follows from Perron-Frobenius Theorem that the eigen-value with the largest modulus is real and non-negative. Let us call λ ( H P , w ) this leadingeigenvalue. The weighted dynamical tree ( H P , w ) is said to be unobstructed if λ ( H P , w ) < .Let us give some remarks about this definition.1. This definition is strongly related to obstructions which occur in Thurston characteri-zation of postcritically finite rational maps and all the theory behind (see [DH93])2. When ( H P , w ) is unobstructed, Perron-Frobenius Theorem and continuity of the spec-tral radius ensure the existence of a vector V ∈ R with positive entries such that M V < V . This remark will be useful later.3. Actually the transition matrix of ( H P , w ) is given by M = d d d d d d and an easy computation shows that λ ( H P , w ) is the largest root of X − (cid:18) d d d + 1 d d d (cid:19) X − d d d d . λ ( H P , w ) (cid:62) then λ ( H P , w ) (cid:54) d d d + d d d + d d d d , thus ( H P , w ) isunobstructed as soon as at least three of weights d , d , d , and d are (cid:62) . Conversely,if ( H P , w ) is unobstructed then one can show by exhaustion that at least two of weights d , d , d , and d are (cid:62) .4. For the McMullen’s example, the transition matrix of ( H Q , w ) may be defined as welland we get M = (cid:18) d ∞ d ∞ d d (cid:19) . An easy computation gives that λ ( H Q , w ) = d ∞ + d . Consequently the weighteddynamical tree ( H Q , w ) is unobstructed if and only if the assumption (H0) holds.The following result is analogous to Theorem 2. Theorem 3.
If the weighted dynamical tree ( H P , w ) satisfies the two following conditions (cid:98) d = 12 ( d + d + d − is an integer (cid:62) and max { d , d , d } (cid:54) (cid:98) d (H1) ( H P , w ) is unobstructed (H2) then there exists a rational map f of degree (cid:98) d + d such that the exchanging dynamics ofpostcritically separating Julia components of f is encoded by ( H P , w ) in the following sense: (i) every critical orbit accumulates a super-attracting cycle { z , z , z , z } of period ; (ii) there exists a homeomorphism h : J crit ( f ) → J ( H P ) such that the following diagramcommutes; J crit ( f ) f (cid:47) (cid:47) h (cid:15) (cid:15) J crit ( f ) h (cid:15) (cid:15) J ( H P ) P (cid:47) (cid:47) J ( H P ) (iii) for every Julia component J ∈ J crit ( f ) such that h ( J ) is not eventually mapped underiteration to the fixed branching point α , the restriction map f | J has degree w ( e k ) = d k where e k is the edge of H P which contains h ( J ) . The same heuristic as for Theorem 2 still holds: we may topologically think the Riemannsphere (cid:98) C as a smooth neighborhood’s boundary of the tree H P embedded in the space R . Theaction of f on this topological sphere follows that one of the dynamical tree H P . The pointson this topological sphere which correspond to the points in the super-attracting periodiccycle { z , z , z , z } should be closed to the corresponding vertices { c , c , c , c } of H P , andevery Julia component in J crit ( f ) closely surrounds a corresponding point in J ( H P ) . Theorem 4.
Under assumptions (H1) and (H2) there exists a rational map f satisfyingTheorem 3 and such that J ( f ) contains buried Julia components of several types: (point type) uncountably many points; (circle type) uncountably many Jordan curves; (complex type) countably many preimages of a fixed Julia component lying over the fixedbranching point α , say J α = h − ( α ) ∈ J ( f ) , which is quasiconformallyhomeomorphic to the connected Julia set of a rational map (cid:98) f .Moreover (cid:98) f has degree (cid:98) d and has only one critical orbit which is a super-attracting cycle { (cid:98) z , (cid:98) z , (cid:98) z } of period such that the local degree of (cid:98) f at (cid:98) z k is d k for every k ∈ { , , } . (cid:98) f corresponds to the dynamics of f on the fixed Julia component J α lying over the fixed branching point α . More precisely, there is a quasiconformal map ϕ from a neighborhood of J ( (cid:98) f ) onto a neighborhood of J α such that ϕ ◦ (cid:98) f = f ◦ ϕ (seethe construction of f in Section 3).2. The rational map (cid:98) f may also be seen as encoded by a weighted dynamical tree. Considerthe quadratic polynomial R : z (cid:55)→ z + (cid:98) c where (cid:98) c ∈ C is the parameter with thelargest imaginary part such that the critical point 0 is periodic of period 3, that is (cid:98) c ≈ − .
123 + 0 . i ( J ( R ) is known as the Douady’s rabbit). The Hubbard tree H R of R is described by a set of four vertices { (cid:98) c , (cid:98) c , (cid:98) c , (cid:98) α } where (cid:98) c k = R k (0) and (cid:98) α is a fixedpoint of R , and three edges of the form (cid:98) e k = [ (cid:98) α, (cid:98) c k ] H R for every k ∈ { , , } . Considerthe weight function w defined by w ( (cid:98) e k ) = d k for every k ∈ { , , } . Then the weighteddynamical tree ( H R , w ) encodes the action of (cid:98) f in the same setting as in Theorem 2and Theorem 3. Notice that the intersection set between H R and J ( R ) is reduced to J ( H R ) = { (cid:98) α } , that corresponds to the unique Julia component in J ( (cid:98) f ) = J crit ( (cid:98) f ) = { J ( (cid:98) f ) } . Finally, remark that the weighted dynamical tree ( H R , w ) is unobstructed assoon as assumption (H1) holds (actually λ ( H R , w ) = d d d ).3. The rational map (cid:98) f is unique up to conjugation by a Möbius map or equivalently it isunique as soon as its critical orbit { (cid:98) z , (cid:98) z , (cid:98) z } is fixed in (cid:98) C (see Lemma 1). However,the rational map f is not unique since the critical points which do not belong to thesuper-attracting periodic cycle { z , z , z , z } (but whose orbits accumulate it) may beperturbed in some neighborhoods without changing the exchanging dynamics and thetopology of Julia components.4. The rational map f is not postcritically finite since J ( f ) is disconnected (but it ishyperbolic from point (i) in Theorem 3). In particular Thurston characterization ofpostcritically finite rational maps (see [DH93]) can not be used to prove the existenceof f . However one could use the works of Tan Lei and Cui Guizhen about sub- hyper-bolic semi-rational maps in [CT11] but this paper presents a more explicit and moreconstructive method by quasiconformal surgery (see Section 3).5. The assumption (H1) is necessary. Indeed it is the smallest requirement such that thereexists a topological model for (cid:98) f , that is a branched covering combinatorially equivalentto (cid:98) f (see Lemma 16 and proof of Lemma 1).6. The assumption (H2) is necessary. Otherwise we can find a Thurston obstruction, thatis to say a multicurve Γ whose transition matrix is equal to M with leading eigenvalue λ (Γ) = λ ( H P , w ) (cid:62) . According to a result of Curtis T. McMullen in [McM94] itfollows that λ (Γ) = 1 and at least one curve in Γ is contained in an union of Fatoudomains where f is biholomorphically conjugated to a rotation. That is a contradictionsince every critical orbit of f accumulates a super-attracting periodic cycle. The aim of this section is to construct by quasiconformal surgery (we refer readers to [BF13]for a comprehensive treatment on this powerful method) a rational map f which sastifies11heorem 3 and Theorem 4. The strategy is to start from a rational map (cid:98) f whose Julia setcorresponds to the branching point α in H P (see Theorem 4) and then to modify this map inorder to create a folding corresponding to the critical point c . (cid:98) f The first step of the construction is to prove the existence of the rational map (cid:98) f which appearsin Theorem 4. This is done by Lemma 1 below. Lemma 1.
If assumption (H1) holds then there exists a rational map (cid:98) f : (cid:98) C → (cid:98) C of degree (cid:98) d such that: (i) (cid:98) f has only one critical orbit which is a super-attracting cycle { (cid:98) z , (cid:98) z , (cid:98) z } of period 3 suchthat the local degree of (cid:98) f at (cid:98) z k is d k for every k ∈ { , , } ; (ii) J ( (cid:98) f ) is connected and the Fatou set (cid:98) C − J ( (cid:98) f ) has infinitely many connected componentswhich are simply connected.Moreover (cid:98) f is unique up to conjugation by a Möbius map. There are many ways to prove the existence of (cid:98) f (for instance by “blowing up” the edges ofsome triangle invariant by a Möbius map, see [PT98]). Here we give a simple proof provideda particular solution of the Hurwitz problem (see Section 6). Proof.
Up to conjugation by a Möbius map, we may fix three distinct points (cid:98) z , (cid:98) z , and (cid:98) z in (cid:98) C . Remark that if at least one of integers d , d , and d is equal to , says d = 1 , thenassumption (H1) leads to d = d = (cid:98) d and the rational map (cid:98) f = ϕ ◦ ( z (cid:55)→ z (cid:98) d ) ◦ (cid:101) ϕ − where ϕ and (cid:101) ϕ are two Möbius maps such that (cid:101) ϕ (1) = (cid:98) z , (cid:101) ϕ (0) = (cid:98) z , (cid:101) ϕ ( ∞ ) = (cid:98) z , and ϕ (1) = (cid:98) z , ϕ (0) = (cid:98) z , ϕ ( ∞ ) = (cid:98) z , satisfies (i) . Consequently we may assume that d , d , and d are (cid:62) .If follows that we may apply Lemma 16 since assumption (H1) easily implies condition(H1’) for the abstract branch data coming from d = (cid:98) d , and d i, = d i − for every i ∈ { , , } .We get a degree (cid:98) d branched covering H : S → S and three distinct points x , , x , , and x , in S such that the local degree of H at x i, is d i − for every i ∈ { , , } and H has nomore critical points than x , , x , , and x , . Let ϕ : S → (cid:98) C be any homeomorphism suchthat ϕ ( H ( x i, )) = (cid:98) z i for every i ∈ { , , } . Remark that the map ϕ ◦ H : S → (cid:98) C induces acomplex structure on S . In other words, the uniformization theorem gives a homeomorphism (cid:101) ϕ : S → (cid:98) C such that the map (cid:98) f = ϕ ◦ H ◦ (cid:101) ϕ − is holomorphic on (cid:98) C and thus a rationalmap of degree (cid:98) d . Moreover, up to postcomposition with a Möbius map, we may assume that (cid:101) ϕ ( x i, ) = (cid:100) z i − for every i ∈ { , , } so that (cid:98) f satisfies (i) .Now remark that for every k ∈ { , , } , the connected component containing (cid:98) z k of thesuper-attracting basin of (cid:98) f is simply connected since it contains at most one critical point.Moreover, any other Fatou component is eventually mapped by homeomorphisms onto one ofthese simply connected components. It follows that (cid:98) f satisfies (ii) .Finally let (cid:98) g be another rational map of degree (cid:98) d which satisfies (i) and (ii) for the samesuper-attracting periodic cycle { (cid:98) z , (cid:98) z , (cid:98) z } . Then z (cid:55)→ (cid:98) f ( z ) − (cid:98) g ( z ) is a rational map of degreeat most (cid:98) d for which 0 has at least d + d + d = 2 (cid:98) d + 1 preimages counted with multiplicity(every (cid:98) z k is a preimage of 0 with multiplicity d k ). Consequently this map is identically equalto 0, that is (cid:98) f = (cid:98) g . 12otice that the previous proof strongly uses the fact that the postcritical set contains onlythree points. Indeed if the postcritical set contains more than three points, there is still anuniformization map (cid:101) ϕ for S equipped with the complex structure coming from ϕ ◦ H , butthat may not be possible to postcompose (cid:101) ϕ with a Möbius map so that (cid:98) f satisfies (i) . In factwe would also need to check that the branched covering H has no Thurston obstructions (see[DH93]). Starting with the map (cid:98) f coming from Lemma 1, we need to divide (cid:98) C into several pieces onwhich the map f (or more precisely a quasiregular map F ) will be piecewisely defined. Thispartition comes from a certain system of equipotentials of (cid:98) f defined in Lemma 2 below.For every k ∈ { , , } , denote by B ( (cid:98) z k ) the connected component containing (cid:98) z k of thesuper-attracting basin of (cid:98) f . Recall that each B ( (cid:98) z k ) is a marked hyperbolic disk. Moreprecisely, Böttcher’s Theorem provides Riemann mappings φ k : D → B ( (cid:98) z k ) (namely biholo-morphic maps from the open unit disk D onto B ( (cid:98) z k ) such that φ k (0) = (cid:98) z k and the followingdiagram commutes. B ( (cid:98) z ) (cid:98) f (cid:15) (cid:15) D φ (cid:111) (cid:111) z (cid:55)→ z d (cid:15) (cid:15) B ( (cid:98) z ) (cid:98) f (cid:15) (cid:15) D φ (cid:111) (cid:111) z (cid:55)→ z d (cid:15) (cid:15) B ( (cid:98) z ) (cid:98) f (cid:15) (cid:15) D φ (cid:111) (cid:111) z (cid:55)→ z d (cid:15) (cid:15) B ( (cid:98) z ) D φ (cid:111) (cid:111) Recall that an equipotential β in any B ( (cid:98) z k ) is the image by φ k of an euclidean circle in D centered at . The radius of this circle is called the level of β and is denoted by L k ( β ) ∈ ]0 , ,in order that β = { z ∈ B ( (cid:98) z k ) / | φ − k ( z ) | = L k ( β ) } .Recall that any pair of disjoint continua β, β (cid:48) in (cid:98) C uniquely defines an open annulus in (cid:98) C denoted by A ( β, β (cid:48) ) . If β, β (cid:48) contain at least two points each, A ( β, β (cid:48) ) is biholomorphicto a round annulus of the form A r = { z ∈ C / r < | z | < } where r ∈ ]0 , only depends on A ( β, β (cid:48) ) . The modulus of A ( β, β (cid:48) ) is defined to be mod( A ( β, β (cid:48) )) = π log( r ) . In particular if β, β (cid:48) are two equipotentials in the same domain B ( (cid:98) z k ) of levels L k ( β ) > L k ( β (cid:48) ) then mod( A ( β, β (cid:48) )) = 12 π log (cid:18) L k ( β ) L k ( β (cid:48) ) (cid:19) . Finally for every k ∈ { , , } , denote by α k the compact connected subset of J ( (cid:98) f ) whichcorresponds to the boundary of B ( (cid:98) z k ) . Lemma 2.
If assumption (H2) holds then there exist three equipotentials β in B ( (cid:98) z ) , β in B ( (cid:98) z ) , and β in B ( (cid:98) z ) , together with two equipotentials β +3 and β − in B ( (cid:98) z ) such that L ( β ) > L ( β +3 ) > L ( β − ) nd the following linear system of inequalities holds. d mod( A ( α , β )) < mod( A ( α , β ))1 d mod( A ( α , β )) < mod( A ( α , β ))1 d mod( A ( α , β )) + 1 d mod( A ( β +3 , β − )) < mod( A ( α , β ))1 d mod( A ( β , β )) < mod( A ( β +3 , β − )) and mod( A ( β , β +3 )) > (1)Recall that the modulus is a conformal invariant, or more precisely if there is a holomorphiccovering of degree d from an open annulus A onto another one A (cid:48) then mod( A ) = d mod( A (cid:48) ) .Hence the first three inequalities in linear system (1) implies that the preimages under (cid:98) f ofthese equipotentials are arranged as shown in Figure 5. The fourth inequality will allow torealize the preimage of the branching point α in H P (see Lemma 4) while the last inequalityensures sufficient space to realize the folding corresponding to the critical point c (see Lemma3).Figure 5: The pattern of the equipotentials (and their preimages) coming from Lemma 2displayed on the Riemann sphere which is topologically distorted to emphasize thethree domains B ( (cid:98) z ) , B ( (cid:98) z ) , and B ( (cid:98) z ) (compare with Figure 4.c).The key point of the proof needs an inverse Grötzch’s inequality due to Cui Guizhen andTan Lei (see Section 6). Proof.
Let
C > be the constant coming from Lemma 17 for the marked hyperbolic disks B ( (cid:98) z ) , B ( (cid:98) z ) . Thus, for every pair of equipotentials β in B ( (cid:98) z ) and β in B ( (cid:98) z ) , we have d mod( A ( β , β )) (cid:54) d (mod( A ( α , β )) + mod( A ( α , β )) + C ) . x , x , x , x . d x < x d x < x d x + 1 d x < x d ( x + x + C ) < x (2)Using the transition matrix M coming from Definition 1, this system is equivalent to M X + Cd < X where X = x x x x . Recall that assumption (H2) states that the leading eigenvalue λ ( H P , w ) of M is less than .It follows from Perron-Frobenius Theorem and continuity of spectral radius the existence of avector V ∈ R with positive entries such that M V < V . Now taking µ > large enough (forinstance µ = ( Cd + 1)( v − d v − d v ) − ), the vector X = µV with positive entries solves thelinear system of inequations (2).The equipotentials β , β , β are uniquely defined by π log (cid:18) L k ( β k ) (cid:19) = mod( A ( α k , β k )) = x k for every k ∈ { , , } . For β +3 , choose an arbitrary equipotential in B ( (cid:98) z ) such that L ( β ) > L ( β +3 ) and π log (cid:18) L ( β ) L ( β +3 ) (cid:19) = mod( A ( β , β +3 )) > . Then β − is uniquely defined by L ( β +3 ) > L ( β − ) and π log (cid:18) L ( β +3 ) L ( β − ) (cid:19) = mod( A ( β +3 , β − )) = x . It follows from construction that β , β , β , β +3 , and β − satisfy all the requirements ofLemma 2, the fourth inequality in linear system (1) coming from the last inequality in linearsystem (2) and Lemma 17.It turns out in the proof above that the lower bound of the last inequality in linear system(1) may be changed for any positive constant (which depends only on the integers d , d , d , and d ). As we will see later in Lemma 3, the lower bound ensures sufficient space tomake the surgery in A ( β , β +3 ) . However, the author guesses that the last inequality in linearsystem (1) is not necessary (see discussion after the proof of Lemma 18).The system of equipotentials coming from Lemma 2 will be used to divide (cid:98) C into severalpieces on which a quasiregular map F will be piecewisely defined. This map F should becarefully defined in such a way that its dynamics is encoded by the weighted dynamical tree ( H P , w ) (see Theorem 3).For instance, the first step of the construction which corresponds to the dynamics on e ∪ e for H P is the following. Denote by β , the preimage of β in B ( (cid:98) z ) (see Figure 5).15rom the first inequality in linear system (1), β , is an equipotential of level L ( β , ) >L ( β ) . Denote by D ( β , ) the open disk bounded by β , and containing { (cid:98) z , (cid:98) z } (and hence J ( (cid:98) f ) ∪ B ( (cid:98) z ) ∪ B ( (cid:98) z ) as well). Then F is defined to be the rational map (cid:98) f on D ( β , ) . Remarkthat F | D ( β , ) continuously extends to β , by a degree d covering denoted by F | β , : β , → β . The aim of this part of the construction is to realize the folding corresponding to the criticalpoint c in H P . More precisely F should holomorphically maps a small annulus (correspondingto a neighborhood of c in H P ) onto a disk (corresponding to a neighborhood of c in H P )with respect to the degrees d , d .Let γ be an arbitrary equipotential in B ( (cid:98) z ) such that L ( γ ) < L ( β ) . Denote by D ( γ ) the open disk bounded by γ and containing (cid:98) z . In order to follow more easily the construction,we will slightly improve the notation. So let γ , be the equipotential β , keeping in mind that γ , will be mapped onto γ by a degree d covering. Notice that the first inequality in linearsystem (1) of Lemma 2 implies L ( β , ) > L ( γ , ) . Similarly let β , be the equipotential β +3 ,keeping in mind that β , will be mapped onto β by a degree d covering. Lemma 3.
There exist an equipotential γ , in B ( (cid:98) z ) and a holomorphic branched covering F | A ( γ , ,γ , ) : A ( γ , , γ , ) → D ( γ ) such that: (i) L ( β , ) > L ( γ , ) > L ( γ , ) > L ( β , ) ; (ii) F | A ( γ , ,γ , ) has degree d + d and has d + d critical points counted with multiplicity,which one of them, denoted by c , satisfies F | A ( γ , ,γ , ) ( c ) = (cid:98) z ; (iii) F | A ( γ , ,γ , ) continuously extends to γ , ∪ γ , by a degree d covering F | γ , : γ , → γ and a degree d covering F | γ , : γ , → γ .Proof. Let G : A ( γ, γ (cid:48) ) → D be a holomorphic branched covering coming from Lemma 18 forthe integers n = d and n (cid:48) = d . Define the equipotential γ , by L ( γ , ) > L ( γ , ) and π log (cid:18) L ( γ , ) L ( γ , ) (cid:19) = mod( A ( γ , , γ , )) = mod( A ( γ, γ (cid:48) )) . Since mod( A ( γ , , β , )) = mod( A ( β , β +3 )) > (from the last inequality in linear system(1) of Lemma 2) and mod( A ( γ , , γ , )) = mod( A ( γ, γ (cid:48) )) (cid:54) (from the point (iii) in Lemma18), it follows that L ( γ , ) > L ( β , ) and the point (i) holds.Now let ψ be any biholomorphic map from A ( γ , , γ , ) onto A ( γ, γ (cid:48) ) . The existence of sucha biholomorphic map is ensured by the fact that these two open annuli have same modulus.Since A ( γ , , γ , ) and A ( γ, γ (cid:48) ) are bounded by quasicircles, ψ may be continuously extendedto γ , ∪ γ , by two homeomorphisms.Let c be the preimage under ψ of any critical point of G and let φ : D → D ( γ ) beany Riemann mapping of D ( γ ) such that φ ( G ( ψ ( c ))) = (cid:98) z . Since D ( γ ) is bounded by anequipotential, φ may be continuously extended to ∂ D by a homeomorphism.Then F | A ( γ , ,γ , ) = φ ◦ G ◦ ψ is holomorphic on A ( γ , , γ , ) and satisfies (ii) , and (iii) byconstruction.Figure 6 depicts the map F | A ( γ , ,γ , ) coming from Lemma 3.16igure 6: The map F | A ( γ , ,γ , ) coming from Lemma 3 displayed on the Riemann sphere whichis topologically distorted to emphasize the three domains B ( (cid:98) z ) , B ( (cid:98) z ) , and B ( (cid:98) z ) (compare with Figure 4.c). According to the last two sections, the map F is defined up to there on the union of the opendisk D ( β , ) containing { (cid:98) z , (cid:98) z } with the open annulus A ( γ , , γ , ) containing c . Moreover F maps c to (cid:98) z , (cid:98) z to (cid:98) z and (cid:98) z to (cid:98) z . Now we need to define F near (cid:98) z by sending (cid:98) z to c inorder to realize a cycle of period as required in Theorem 3. This should be done carefullyso that the quasiconformal surgery may be concluded.The first problem is that some preimage of J ( (cid:98) f ) (or more precisely of the open annulus A ( β , β ) containing J ( (cid:98) f ) ) must appear in B ( (cid:98) z ) (compare with Figure 4.c where the edge e = [ c , c ] H P contains a preimage of the branching point α ). This is done in Lemma 4 belowwhich essentially uses the fourth inequality in linear system (1) of Lemma 2. Lemma 4.
There exist an equipotential β , in B ( (cid:98) z ) and a holomorphic covering F | A ( β , ,β , ) : A ( β , , β , ) → A ( β , β ) such that: (i) L ( β , ) > L ( β , ) > L ( β − ) ; (ii) F | A ( β , ,β , ) has degree d and has no critical point; (iii) F | A ( β , ,β , ) continuously extends to β , ∪ β , by two degree d coverings F | β , : β , → β and F | β , : β , → β .Proof. Define the equipotential β , by L ( β , ) > L ( β , ) and π log (cid:18) L ( β , ) L ( β , ) (cid:19) = mod( A ( β , , β , )) = 1 d mod( A ( β , β )) . Since mod( A ( β , , β , )) = d mod( A ( β , β )) < mod( A ( β +3 , β − )) = mod( A ( β , , β − )) (from the fourth inequality in linear system (1) of Lemma 2), it follows that L ( β , ) > L ( β − ) and the point (i) holds.Now let ψ be any biholomorphic map from A ( β , , β , ) onto a round annulus of the form A r = { z ∈ C / r < | z | < } where r is defined by π log (cid:18) r (cid:19) = mod( A r ) = mod( A ( β , , β , )) . A ( β , , β , ) is bounded by equipotentials, ψ may be continuously extended to β , ∪ β , by two homeomorphisms which send β , onto { z ∈ C / | z | = 1 } and β , onto { z ∈ C / | z | = r } .Similarly, let Ψ be any biholomorphic map from the round annulus A r d onto A ( β , β ) .The existence of such a biholomorphic map is ensured by the fact that mod( A r d ) = 12 π log (cid:18) r d (cid:19) = d π log (cid:18) r (cid:19) = d mod( A ( β , , β , )) = mod( A ( β , β )) . Since A ( β , β ) is bounded by equipotentials, Ψ may be continuously extended to ∂A r d bytwo homeomorphisms which send { z ∈ C / | z | = 1 } onto β and { z ∈ C / | z | = r d } onto β .Then F | A ( β , ,β , ) = Ψ ◦ ( z (cid:55)→ z d ) ◦ ψ is holomorphic on A ( β , , β , ) and satisfies (ii) , and (iii) by construction.Figure 7 depicts the map F | A ( β , ,β , ) coming from Lemma 4.Figure 7: The map F | A ( β , ,β , ) coming from Lemma 4 displayed on the Riemann sphere whichis topologically distorted to emphasize the three domains B ( (cid:98) z ) , B ( (cid:98) z ) , and B ( (cid:98) z ) (compare with Figure 4.c). Now we achieve the definition of F near (cid:98) z . This is done in two parts. Firstly Lemma 5realizes a preimage of a neighborhood of (cid:98) z in B ( (cid:98) z ) . Then Lemma 6 defines F near (cid:98) z bysending a neighborhood of (cid:98) z onto a neighborhood of c (mapping (cid:98) z to c ).Let γ be an arbitrary equipotential in B ( (cid:98) z ) such that L ( β ) = L ( γ , ) > L ( γ ) >L ( γ , ) and A ( γ , γ , ) contains the critical point c . Lemma 5.
There exist two equipotentials γ , and δ +3 ,c in B ( (cid:98) z ) , a quasicircle δ + c in A ( γ , γ , ) which separates c from γ ∪ γ , , and a holomorphic covering F | A ( γ , ,δ +3 ,c ) : A ( γ , , δ +3 ,c ) → A ( γ , δ + c ) such that: (i) L ( β , ) > L ( γ , ) > L ( δ +3 ,c ) > L ( β − ) ; (ii) F | A ( γ , ,δ +3 ,c ) has degree d and has no critical point; (iii) F | A ( γ , ,δ +3 ,c ) continuously extends to γ , ∪ δ +3 ,c by two degree d coverings F | γ , : γ , → γ and F | δ +3 ,c : δ +3 ,c → δ + c . roof. Applying Lemma 19, we get a quasicircle δ + c in A ( γ , γ , ) which separates c from γ ∪ γ , and such that d mod( A ( γ , δ + c )) < mod( A ( β , , β − )) . Therefore we can find two equipotentials γ , and δ +3 ,c in B ( (cid:98) z ) so that L ( β , ) > L ( γ , ) > L ( δ +3 ,c ) > L ( β − ) and π log (cid:18) L ( γ , ) L ( δ +3 ,c ) (cid:19) = mod( A ( γ , , δ +3 ,c )) = 1 d mod( A ( γ , δ + c )) . The point (i) holds by definition. For the two other points, the proof may be achieved asthat one of Lemma 4.Figure 8 depicts the equipotentials involved in Lemma 5 and the map F | A ( γ , ,δ +3 ,c ) .Figure 8: The maps F | A ( γ , ,δ +3 ,c ) and F | D ( δ − ,c ) coming from Lemma 5 and Lemma 6 displayedon the Riemann sphere which is topologically distorted to emphasize the threedomains B ( (cid:98) z ) , B ( (cid:98) z ) , and B ( (cid:98) z ) (compare with Figure 4.c).It remains to define F near (cid:98) z . Let δ − c be an arbitrary quasicircle which separates c from δ + c . We slightly improve the notation by denoting δ − ,c the equipotential β − keeping in mindthat δ − ,c will be mapped onto δ − c by a degree d covering. Finally, denote by D ( δ − ,c ) the opendisk bounded by δ − ,c and containing (cid:98) z , and by D ( δ − c ) the open disk bounded by δ − c andcontaining c . Lemma 6.
There exists a holomorphic branched covering F | D ( δ − ,c ) : D ( δ − ,c ) → D ( δ − c ) suchthat: (i) F | D ( δ − ,c ) has degree d and has only one critical point which is (cid:98) z with F | D ( δ − ,c ) ( (cid:98) z ) = c ; (ii) F | D ( δ − ,c ) continuously extends to δ − ,c by a degree d covering F | δ − ,c : δ − ,c → δ − c .Proof. Let φ : D → D ( δ − ,c ) be any Riemann mapping of D ( δ − ,c ) such that φ (0) = (cid:98) z , andlet Φ : D → D ( δ − c ) be any Riemann mapping of D ( δ − c ) such that Φ(0) = c . Since D ( δ − ,c ) and D ( δ − c ) are bounded by quasicircles, φ and Φ may be continuously extended to ∂ D byhomeomorphisms.Then F | D ( δ − ,c ) = Φ ◦ ( z (cid:55)→ z d ) ◦ φ − gives the result.Figure 8 depicts the map F | D ( δ − ,c ) coming from Lemma 6.19 .6 Uniformization At first we sum up in the following table the definition of F up to there.domains images cont. extensionson boundaries critical pointswith multiplicity critical values D ( β , ) (cid:98) C β , d :1 −−−→ β (cid:98) z with mult. d − (cid:98) z with mult. d − F ( (cid:98) z ) = (cid:98) z F ( (cid:98) z ) = (cid:98) z A ( γ , , γ , ) D ( γ ) γ , d :1 −−−→ γ γ , d :1 −−−→ γ c ∈ { d + d crit. ptscounted with mult. } F ( c ) = (cid:98) z and others A ( β , , β , ) A ( β , β ) β , d :1 −−−→ γ β , d :1 −−−→ β ∅ ∅ A ( γ , , δ +3 ,c ) A ( γ , δ + c ) γ , d :1 −−−→ γ δ +3 ,c d :1 −−−→ δ + c ∅ ∅ D ( δ − ,c ) D ( δ − c ) δ − ,c d :1 −−−→ δ − c (cid:98) z with mult. d − F ( (cid:98) z ) = c So F is holomorphically defined on H = D ( β , ) ∪ A ( γ , , γ , ) ∪ A ( β , , β , ) ∪ A ( γ , , δ +3 ,c ) ∪ D ( δ − ,c ) with continuous extension on the boundary. It remains to define F on the complement Q = (cid:98) C − H = A ( β , , γ , ) ∪ A ( γ , , β , ) ∪ A ( β , , γ , ) ∪ A ( δ +3 ,c , δ − ,c ) . This is done in the followinglemma. Lemma 7.
The map F | H : H → (cid:98) C extends to a quasiregular map F : (cid:98) C → (cid:98) C by quasicon-formal coverings defined on each connected component of Q = (cid:98) C − H .Moreover there exists an open subset E ⊂ H such that F ( E ) ⊂ E and F ( Q ) ⊂ E . In particular, notice that the quasiregular map F : (cid:98) C → (cid:98) C has no more critical pointsthan those coming from the holomorphic restriction F | H : H → (cid:98) C . Proof.
Remark that every connected component of Q is an open annulus whose boundaryis the disjoint union of two quasicircles where F realizes two coverings of same degree (andsame orientation). By interpolation, F may be continuously extended to each connectedcomponent of Q by a covering of degree corresponding to that one on the boundary. Sinceall the connected components of the boundary of Q , together with their images by F , arequasicircles, each interpolation may be carefully done in such a way that the resulting map isactually quasiconformal on the Riemann sphere. In short, F quasiregularly extends to Q by • a degree d quasiconformal covering F | A ( β , ,γ , ) : A ( β , , γ , ) → A ( β , γ ) ; • a degree d quasiconformal covering F | A ( γ , ,β , ) : A ( γ , , β , ) → A ( γ , β ) ; • a degree d quasiconformal covering F | A ( β , ,γ , ) : A ( β , , γ , ) → A ( β , γ ) ; • a degree d quasiconformal covering F | A ( δ +3 ,c ,δ − ,c ) : A ( δ +3 ,c , δ − ,c ) → A ( δ + c , δ − c ) .In particular, we have F ( Q ) = A ( β , γ ) ∪ A ( β , γ ) ∪ A ( δ + c , δ − c ) (see figure 9 to follow thecontinuation of the proof).Now denote by β , the preimage of β in B ( (cid:98) z ) under F (thus under (cid:98) f ) and similarlyby β − , the preimage of β − in B ( (cid:98) z ) (see Figure 5). Moreover denote by D ( β , ) the open20isk bounded by β , and containing (cid:98) z , and by D ( β − , ) the open disk bounded by β − , andcontaining (cid:98) z . Finally, let E be the union D ( β , ) ∪ D ( β − , ) ∪ D ( δ − ,c ) ∪ A ( γ , , γ , ) .At first remark that E is an open subset of H = D ( β , ) ∪ A ( γ , , γ , ) ∪ A ( β , , β , ) ∪ A ( γ , , δ +3 ,c ) ∪ D ( δ − ,c ) . Indeed we have D ( β , ) ∪ D ( β − , ) ⊂ D ( β , ) from definition of D ( β , ) .Moreover, it follows from definition of F on H that F ( E ) = D ( β ) ∪ D ( β − ) ∪ D ( δ − c ) ∪ D ( γ ) where D ( β ) denotes the open disk bounded by β and containing (cid:98) z , and D ( β − ) = D ( δ − ,c ) is the open disk bounded by β − = δ − ,c and containing (cid:98) z .Furthermore, according to the whole construction, we have • from Lemma 2 and definition of γ : A ( β , γ ) ∪ D ( γ ) ⊂ D ( β , ) and D ( β ) ⊂ D ( β − , ) ; • from definition of γ and recalling β = γ , : A ( β , γ ) ⊂ A ( γ , , γ , ) ; • from definitions of δ − c , δ + c and γ : A ( δ + c , δ − c ) ∪ D ( δ − c ) ⊂ A ( γ , γ , ) ⊂ A ( γ , , γ , ) .Putting everything together gives the following diagram in which the arrows F −→ stand forimages under F , ⊂ −→ stand for inclusions, ⊂⊂ −→ stand for compact inclusions (namely A ⊂⊂ −→ B if and only if A ⊂ B ) and = −→ stands for equality. Q F (cid:15) (cid:15) = A ( β , , γ , ) F (cid:15) (cid:15) ∪ A ( γ , , β , ) F (cid:122) (cid:122) ∪ A ( β , , γ , ) F (cid:122) (cid:122) ∪ A ( δ +3 ,c , δ − ,c ) F (cid:122) (cid:122) F ( Q ) ⊂ (cid:15) (cid:15) = A ( β , γ ) ⊂⊂ (cid:15) (cid:15) ∪ A ( β , γ ) ⊂ (cid:42) (cid:42) ∪ A ( δ + c , δ − c ) ⊂⊂ (cid:36) (cid:36) E F (cid:15) (cid:15) = D ( β , ) F (cid:36) (cid:36) ∪ D ( β − , ) F (cid:36) (cid:36) ∪ D ( δ − ,c ) F (cid:36) (cid:36) ∪ A ( γ , , γ , ) F (cid:121) (cid:121) F ( E ) ⊂ (cid:15) (cid:15) = D ( γ ) ⊂⊂ (cid:15) (cid:15) ∪ D ( β ) ⊂⊂ (cid:15) (cid:15) ∪ D ( β − ) = (cid:15) (cid:15) ∪ D ( δ − c ) ⊂⊂ (cid:15) (cid:15) E ⊂ (cid:15) (cid:15) = D ( β , ) ⊂⊂ (cid:15) (cid:15) ∪ D ( β − , ) ⊂⊂ (cid:122) (cid:122) ∪ D ( δ − ,c ) = (cid:33) (cid:33) ∪ A ( γ , , γ , ) = (cid:95) (cid:95) H = D ( β , ) ∪ A ( γ , , γ , ) ∪ A ( β , , β , ) ∪ A ( γ , , δ +3 ,c ) ∪ D ( δ − ,c ) In particular, we deduce that F ( Q ) ⊂ E and F ( E ) ⊂ E ⊂ H . Furthermore, followingcompact inclusions, it turns out that F ( Q ) ⊂ E .Now we have a quasiregular map F from the Riemann sphere to itself whose dynamicsfollows that one of the weighted dynamical tree ( H P , w ) (see Figure 4.c). We need to find aholomorphic map f conjugated to F so that f follows the same dynamics as well ( f should sat-isfy the requirements of Theorem 3 and Theorem 4). To do so, we will apply the Shishikura’sfundamental lemma for quasiconformal surgery (stated for the first time in [Shi87]) that werecall below. 21igure 9: The map F coming from Lemma 7. On the left topological sphere, the black areastands for Q and the gray area stands for E . On the right topological sphere, theblack area stands for F ( Q ) and the gray area stands for F ( E ) . Lemma 8 (Shishikura’s fundamental lemma for quasiconformal surgery) . Let g : (cid:98) C → (cid:98) C bea quasiregular map. Assume there are an open set E ⊂ (cid:98) C and an integer N (cid:62) which satisfythe following conditions: • g ( E ) ⊂ E ; • g is holomorphic on E ; • g is holomorphic on an open set containing (cid:98) C − g − N ( E ) .Then there exists a quasiconformal map ϕ : (cid:98) C → (cid:98) C such that the map ϕ ◦ g ◦ ϕ − is holomorphic. The result stated in [Shi87] is a little more general but it easily implies the more explicitstatement of Lemma 8 (we refer readers to [Shi87] and [BF13] for a proof and more details).Here our map F satisfies the three assumptions (indeed F is holomorphic on H hence on E ⊂ H and Lemma 7 implies that (cid:98) C − F − ( E ) ⊂ (cid:98) C − Q = H ), so applying Lemma 8 gives aholomorphic map f : (cid:98) C → (cid:98) C quasiconformally conjugated to F : (cid:98) C → (cid:98) C as desired. Lemma 9.
The rational map f : (cid:98) C → (cid:98) C obtained above has degree (cid:98) d + d and has a super-attracting cycle { z , z , z , z } of period which is accumulated by every critical orbit. Inparticular, f is hyperbolic.Proof. Since f is quasiconformally conjugated to F , the critical points of f are images undera quasiconformal map ϕ of the critical points of F with same multiplicities. More precisely,the critical points of f are: • z = ϕ ( (cid:98) z ) ∈ ϕ ( D ( β , )) ⊂ ϕ ( E ) with multiplicity d − ; • z = ϕ ( (cid:98) z ) ∈ ϕ ( D ( β − , )) ⊂ ϕ ( E ) with multiplicity d − ; • d + d critical points counted with multiplicity in ϕ ( A ( γ , , γ , )) ⊂ ϕ ( E ) , which one ofthem is given by z = ϕ ( c ) ; • z = ϕ ( (cid:98) z ) ∈ ϕ ( D ( δ − ,c )) ⊂ ϕ ( E ) with multiplicity d − .22ccording to the Riemann-Hurwitz formula, it follows that the number of critical pointscounted with multiplicity is given by f ) − d −
1) + ( d −
1) + ( d + d ) + ( d − and hence deg( f ) = 12 ( d + d + d −
1) + d = (cid:98) d + d . Notice that { z , z , z , z } forms a super-attracting cycle of period . Moreover everycritical point of f lies in the forward invariant open set ϕ ( E ) , namely a disjoint union of fouropen subsets of (cid:98) C each containing one point of { z , z , z , z } . Consequently, every criticalorbit accumulates this super-attracting cycle. The aim of this section is to achieve the proofs of Theorem 3 and Theorem 4. More preciselywe are going to show that the rational map f constructed in the previous section satisfies allthe requirements of these two theorems. Section 4.1 focuses on the dynamical properties of f (stated in Theorem 3), and Section 4.2 deals with the topological properties of the Juliacomponent of f (stated in Theorem 4).In order to lighten notations, we forget the quasiconformal map ϕ provided by Lemma8 to denote the image under ϕ of any set introduced in the previous section (equivalentlyspeaking, we act as if the quasiregular map F constructed in the previous section is actuallyholomorphic). Consider the following pairwise disjoint open annuli (see Figure 10). A = A ( α , β ) , A = A ( α , β ) , A = A ( α , β ) , and A = A ( β +3 , β − ) . Then, consider the connected components of the preimage under f of A ∪ A ∪ A ∪ A whichare contained as essential subannulus in one of these open annuli, namely: • A , = A ( α , β , ) ; • A , = A ( α , β , ) ; • A , = A ( α , β , ) where β , is the preimage of β in B ( (cid:98) z ) (see Figure 5); • A , = A ( β +2 , , β − , ) where β +2 , is the preimage of β +3 in B ( (cid:98) z ) (see Figure 5); • A , = A ( α , , β , ) where α , is the preimage of α in A ( β , , β , ) (see Lemma 4); • A , = A ( β , , α , ) where α , is the preimage of α in A ( β , , β , ) (see Lemma 4).Notice that the notation is chosen so that each A i,j is contained as essential subannulus in A i , and f | A i,j : A i,j → A j is a degree d i covering. Remark that some connected componentsof f − ( A ) are included in A as well (from Lemma 5, see Figure 8), but none of them iscontained in A as essential subannulus. 23igure 10: The various annuli considered to encode the exchanging dynamics.Denote by A the collection of all connected components of the non-escaping set induced by f | U : U → A ∪ A ∪ A ∪ A on the union of subannuli U = A , ∪ A , ∪ A , ∪ A , ∪ A , ∪ A , . A = (cid:110) J connected component of { z ∈ U / ∀ n (cid:62) , f n ( z ) ∈ U } (cid:111) Let J α be the continuum in (cid:98) C which corresponds to the Julia set J ( (cid:98) f ) of (cid:98) f (more precisely, J α is the image of J ( (cid:98) f ) under the quasiconformal map ϕ provided by Lemma 8). Remarkthat J α is fixed under iteration of f and J α intersects U (along α ∪ α ∪ α ). Denote by A α the collection of all continua which are eventually mapped onto J α and whose every iterateintersects U . A α = (cid:110) J connected component of (cid:91) n (cid:62) f − n ( J α ) such that ∀ n (cid:62) , f n ( J ) ∩ U (cid:54) = ∅ (cid:111) Finally, denote by A (cid:63) the union A ∪ A α . As collection of pairwise disjoint continua, A (cid:63) isendowed with the topology coming from the usual distance between continua on the Riemannsphere (cid:98) C (equipped with the spherical metric). It turns out that f induced a topologicaldynamical system on A (cid:63) . This dynamical system may be encoded by the weighted dynamicaltree ( H P , w ) (see Section 2.2) as it is shown in the following lemma. Lemma 10.
There exists a homeomorphism h : A (cid:63) → J ( H P ) such that the following diagramcommutes. A (cid:63) f (cid:47) (cid:47) h (cid:15) (cid:15) A (cid:63) h (cid:15) (cid:15) J ( H P ) P (cid:47) (cid:47) J ( H P ) Moreover for every J ∈ A , the restriction map f | J has degree w ( e k ) = d k where e k is the edgeof H P which contains h ( J ) .Proof. At first, remark there is a subannulus A i,j for some i, j ∈ { , , , } if and only ifthe ( i, j ) -entry of the transition matrix M = ( m i,j ) i,j ∈{ , , , } is non-zero (see Definition 1).Indeed, recall that the transition matrix is M = d d d d d d . (Σ , σ ) associated to thetransition matrix M , namely the restriction of the 4-to-1 shift map on the subset of allinfinite sequences of digits in { , , , } such that every adjacent pair of entries lies in { (0 , , (1 , , (2 , , (2 , , (3 , , (3 , } . Σ = (cid:110) s = ( s , s , s , . . . ) ∈ { , , , } N / ∀ k (cid:62) , m s k ,s k +1 (cid:54) = 0 (cid:111) σ : Σ → Σ , s = ( s , s , s , . . . ) (cid:55)→ σ ( s ) = ( s , s , s , . . . )Σ is endowed with the topology coming from the following distance, making it a Cantor set. ∀ s, s (cid:48) ∈ Σ , d ( s, s (cid:48) ) = (cid:88) k (cid:62) | s k − s (cid:48) k | k Let S α be the subset of Σ of three infinite sequences of repeating , , digits. S α = (cid:110) (0 , , , , , , , , , . . . ) , (1 , , , , , , , , , . . . ) , (2 , , , , , , , , , . . . ) (cid:111) We shall identify these three sequences in Σ , and similarly every subset of sequences whichare eventually mapped in S α after the same itinerary under σ . More precisely, let ∼ be theequivalence relation on Σ defined by ∀ s, s (cid:48) ∈ Σ , s ∼ s (cid:48) ⇐⇒ ∃ n (cid:62) / (cid:26) ∀ k ∈ { , , . . . , n } , s k = s (cid:48) k σ n ( s ) , σ n ( s (cid:48) ) ∈ S α and let Σ (cid:63) be the topological quotient space Σ / ∼ . Remark that Σ (cid:63) is a Cantor set as well forthe quotient topology induced by ∼ . Abusing notations, every equivalence class containingonly one infinite sequence s ∈ Σ which is not eventually mapped in S α is still denoted by s ∈ Σ (cid:63) , and the map induced by the shift map on Σ (cid:63) is still denoted by σ .We are going to show that ( A (cid:63) , f ) is topologically conjugated to (Σ (cid:63) , σ ) . To do so, considerthe itinerary map h : A → Σ (cid:63) defined by ∀ J ∈ A , h ( J ) = ( s , s , s , . . . ) with f k ( J ) ⊂ A s k for every k (cid:62) . This map is well defined and injective by definition of A .To prove that h extends to a homeomorphism from A (cid:63) to Σ (cid:63) , we first define by inductionfor every s = ( s , s , s , . . . ) ∈ Σ an infinite sequence of subannuli ( A s ,s ,...,s n ) n (cid:62) such thatfor every n (cid:62) , A s ,s ,...,s n is contained in A s as essential subannulus, and f | A s ,s ,...,sn : A s ,s ,...,s n → A s ,s ,...,s n is a degree d s covering. Denote by A s = A s ,s ,s ,... the limit set (cid:84) n (cid:62) A s ,s ,...,s n which is a continuum.If s is not eventually mapped in S α , then A s ,s ,...,s n is contained in U = A , ∪ A , ∪ A , ∪ A , ∪ A , ∪ A , for every n (cid:62) large enough and thus A s is a connected component of thenon-escaping set, that is an element of A . Moreover, h ( A s ) = s holds from definition of theitinerary map h .On the contrary, if s is in S α , then A s is either α , α or α , and in particular A s iscontained in J α . More generally, if s is eventually mapped in S α , then A s is contained in acontinuum J which is eventually mapped onto J α , that is an element of A α . Moreover, forevery s (cid:48) ∈ Σ such that s (cid:48) ∼ s , A s (cid:48) is contained in the same continuum J ∈ A α .Therefore h extends to a bijective map from A (cid:63) to Σ (cid:63) , by associating to J ∈ A α theequivalence class h ( J ) ∈ Σ ∗ of the itinerary s = ( s s , s , . . . ) ∈ Σ of any subcontinuum in25 which is eventually mapped into α ∪ α ∪ α . Furthermore, this extension is actually aconjugation between f and σ . ∀ J ∈ A (cid:63) , h ( f ( J )) = σ ( h ( J )) It remains to prove the continuity. Fix J ∈ A (cid:63) and let s = ( s , s , s , . . . ) ∈ Σ be a classrepresentative of h ( J ) . Let J (cid:48) be another element of A (cid:63) such that some class representative s (cid:48) = ( s (cid:48) , s (cid:48) , s (cid:48) , . . . ) ∈ Σ of h ( J (cid:48) ) is arbitrary close to s . That implies the first n digits of s and s (cid:48) coincide for arbitrary large n (cid:62) . In particular, A s and A s (cid:48) are contained in A s ,s ,...,s n .Remark that f n | A s ,s ,...,sn : A s ,s ,...,s n → A s n is a covering of degree d s d s . . . d s n − tendingto infinity with n (since assumption (H2) implies that at least two of weights d , d , d , and d are (cid:62) , see Definition 1). Therefore A s and A s (cid:48) are contained in an open annulus ofarbitrary small modulus. Then, using extremal length (see [Ahl73]), it follows that A s ⊂ J and A s (cid:48) ⊂ J (cid:48) are arbitrary close, hence J and J (cid:48) are arbitrary close in A (cid:63) . Consequently h − is continuous. The continuity of h follows from a similar argument.Similarly, we can show that ( J ( H P ) , P ) is topologically conjugated to (Σ (cid:63) , σ ) by a home-omorphism h : J ( H P ) → Σ (cid:63) . Indeed recall that the dynamical tree H P is described by a setof four edges e , e , e , e where P acts as follows (see Section 2.2). P ( e ) = e P ( e ) = e P ( e ) = e ∪ e P ( e ) = e ∪ e Thus, we may find four connected open subsets I , I , I , and I respectively included in e , e , e , and e together with six connected open subsets I , , I , , I , , I , , I , , and I , suchthat: • each I i,j is contained in I i and P | I i,j : I i,j → I j is a homeomorphism; • and J ( H P ) = (cid:110) z ∈ V / ∀ n (cid:62) , P n ( z ) ∈ V (cid:111) ∪ (cid:110) z point in (cid:83) n (cid:62) P − n ( α ) ∩ V (cid:111) where V = I , ∪ I , ∪ I , ∪ I , ∪ I , ∪ I , .Consequently, we can show as above that the itinerary map h : { z ∈ V / ∀ n (cid:62) , P n ( z ) ∈ V } → Σ (cid:63) extends to a homeomorphism from J ( H P ) to Σ (cid:63) which conjugates the dynamics of P and σ .Finally, taking h = h − ◦ h concludes the proof.Remark that the proof of Theorem 3 is almost completed. Indeed point (i) comes fromLemma 9 while points (ii) and (iii) follows from Lemma 10 (since A is, by definition, theset of continua J in A (cid:63) such that J is not eventually mapped under iteration to the fixedcontinuum J α , or equivalently, such that h ( J ) is not eventually mapped under iteration tothe fixed branching point α ). It only remains to prove that A (cid:63) is actually the set J crit ( f ) ofall postcritically separating Julia components of f . Lemma 11.
The following equality of sets holds. A (cid:63) = J crit ( f ) Proof.
Recall that the postcritical set is contained in the forward invariant set E = D ( β , ) ∪ D ( β − , ) ∪ D ( δ − ,c ) ∪ A ( γ , , γ , ) (see Lemma 7 and Figure 9) and each point of the super-attracting cycle { z , z , z , z } lies in a different connected component of E . In particular26 ( f ) is the set of all points whose orbit remains in (cid:98) C − E = A ∪ A ∪ A ∪ A ∪ K α where K α is the complement in (cid:98) C of B ( (cid:98) z ) ∪ B ( (cid:98) z ) ∪ B ( (cid:98) z ) (see Figure 10).It follows that every element J in A is a Julia component. Moreover J is postcriticallyseparating as limit set of nested essential subannuli which separate each the super-attractingcycle { z , z , z , z } (see proof of Lemma 10). Therefore A ⊂ J crit ( f ) .Similarly, every element J in A α is a Julia component. Moreover recall that J intersects U along a limit set of nested essential subannuli which separate each the super-attracting cycle { z , z , z , z } (see proof of Lemma 10). Therefore A α ⊂ J crit ( f ) and A (cid:63) = A ∪ A α ⊂ J crit ( f ) .Conversely, let J be a postcritically separating Julia component of f . Remark that J isnot contained in K α − J α . Indeed, recall that every connected component of (cid:98) C − J α is simplyconnected (see Lemma 1) and that ∂K α = α ∪ α ∪ α ⊂ J α , therefore every connectedcompact subset of any connected component of K α − J α does not separate the postcriticalpoints. Consequently either J is J α ∈ A α ⊂ A (cid:63) or f n ( J ) stays in A ∪ A ∪ A ∪ A for every n (cid:62) . Assume that J is not J α .Recall that every connected component of the preimage under f of A ∪ A ∪ A ∪ A whichis contained in this compact union, is contained either in U or in some connected componentsof f − ( A ) included in A (from Lemma 5, see Figure 8), says A (cid:48) , . However every A (cid:48) , is notcontained in A as essential subannulus, and hence does not separate the postcritical points.In particular J is not contained in any A (cid:48) , . Furthermore, J can not eventually fall in some A (cid:48) , after some iterations of f , otherwise f n ( J ) would not be postcritically separating forsome n (cid:62) contradicting the fact that J is postcritically separating. It follows that f n ( J ) stays in U for every n (cid:62) and hence J ∈ A (cid:63) that concludes the proof. Existence of each of the three types of buried Julia components which occurs in J ( f ) is shownin this section, that proofs Theorem 4. Lemma 12 (Point type buried Julia components) . There exist uncountably many buried Juliacomponents in J ( f ) which are points.Proof. Let A (cid:48) , = A ( β +3 , , β − , ) be a connected component of f − ( A ) contained in A = A ( β +3 , β − ) (from Lemma 5, see Figure 8) where β +3 , and β − , are preimages of β +3 and β − ,respectively. Recall that A (cid:48) , is not contained in A as essential subannulus. In particular,the connected component of (cid:98) C − β +3 , containing A (cid:48) , is an open disk D ( β +3 , ) contained in A and such that f | D ( β , ) : D ( β +3 , ) → D is a homeomorphism where D = D ( β +3 ) is the opendisk bounded by β +3 and containing A .Using notations coming from the proof of Lemma 10, consider the subannulus A , , , , contained in A as essential subannulus and such that f | A , , , , : A , , , , → A is a degree d d d d covering. Since assumption (H2) implies that at least two of weights d , d , d , and d are (cid:62) (see Definition 1), it follows that this degree is (cid:62) and hence, there are at least 2disjoint preimages under f | A , , , , of D ( β +3 , ) in A , , , , ⊂ A ⊂ D , says D and D .Finally we have two disjoint open disks D and D in D such that f | D : D → D and f | D : D → D are homeomorphisms. It is then a classical exercise to prove that thenon-escaping set D = { z ∈ D ∪ D / ∀ n (cid:62) , ( f ) n ∈ D ∪ D } is a Cantor set homeomorphic to the space of all sequences of two digits Σ = { , } N . In par-ticular, D contains uncountably many points. Furthermore every point in D is a buried pointin J ( f ) since A ⊂ D contains infinitely many postcritically separating Julia components.27 emma 13 (Circle type buried Julia components) . There exist uncountably many buriedJulia components in J ( f ) which are wandering Jordan curves.Proof. This is mostly a consequence of the main result in [PT00] claiming that every wander-ing Julia component of a geometrically finite rational map is either a point or a Jordan curve.Here our map f is hyperbolic (from Lemma 9) therefore every wandering Julia componentin J crit ( f ) must be a Jordan curve (since a point is obviously not postcritically separating).Moreover, according to the proof of Lemma 10, the set of wandering Julia components in J crit ( f ) exactly corresponds to the set of all the infinite sequences in Σ (cid:63) which are not even-tually periodic. In particular, there are uncountably many such Julia components. Finally,uncountably many of them must be buried since the Fatou set only has countably many Fatoudomains and each of them only has countably many Jordan curves as connected componentsof its boundary. Lemma 14 (Complex type buried Julia components) . J α and all its countably many preim-ages, are buried Julia components in J ( f ) .Proof. Coming back to the proof of Lemma 10, remark that every infinite sequence in S α isnot isolated in Σ . Therefore, α k has no intersection with the boundary of any Fatou domaincontained in B ( (cid:98) z k ) for every k ∈ { , , } . It remains to show that J α has no intersection withthe boundary of any Fatou domain in K α = (cid:98) C − ( B ( (cid:98) z ) ∪ B ( (cid:98) z ) ∪ B ( (cid:98) z )) . Recall that everyconnected component of K α − J α , that is a connected component of (cid:98) C − J α , is eventuallymapped under iteration onto B ( (cid:98) z k ) for some k ∈ { , , } (since f is defined to be (cid:98) f on K α ⊂ D ( β , ) ). By continuity of f , it follows that J α has no intersection with the boundaryof any Fatou domain contained in any connected component of K α − J α . Consequently J α isburied. The same holds as well for every preimage of J α by continuity of f . In this section, we proof Theorem 1 stated in the introduction (see Section 1). Firstly weshow that a particular choice of the weight function w gives a rational map of degree (inLemma 15). Then we compute an explicit formula for this particular example. Lemma 15.
The following weight function on the set of edges of H P ( d , d , d , d ) = (1 , , , satisfies assumptions (H1) and (H2) from Theorem 3 and Theorem 4. In particular there aresome rational maps of degree whose Julia set contains buried Julia components of severaltypes: (point type) uncountably many points; (circle type) uncountably many Jordan curves; (complex type) countably many preimages of a fixed Julia component which is quasicon-formally homeomorphic to the connected Julia set of (cid:98) f : z (cid:55)→ z − .Proof. Assumption (H1) is obviously satisfied, indeed (cid:98) d = 12 ( d + d + d −
1) = 12 (1 + 2 + 2 −
1) = 2 = max { d , d , d } . M = and an easy computation shows that λ ( H P , w ) is the largest root of X − X − that is λ ( H P , w ) ≈ . < .Applying Theorem 3 and Theorem 4 gives a rational map of degree (cid:98) d + d = 2 + 1 = 3 .Furthermore, recall that the rational map (cid:98) f which appears in Theorem 4 has degree (cid:98) d = 2 and has only one critical orbit which is a super-attracting cycle { (cid:98) z , (cid:98) z , (cid:98) z } of period suchthat the local degrees of (cid:98) f at (cid:98) z , (cid:98) z and (cid:98) z are d = 1 , d = 2 and d = 2 , respectively. Up toconjugation by a Möbius map, we may assume that (cid:98) z = 0 , (cid:98) z = 1 and (cid:98) z = ∞ . It turns outthat there is then only one such quadratic rational map which is (cid:98) f : z (cid:55)→ z − . (cid:98) z = 0 (cid:47) (cid:47) (cid:98) z = 1 (cid:47) (cid:47) (cid:98) z = ∞ (cid:118) (cid:118) Remark that this choice of weight function is the only one which gives a degree 3 andwhich satisfies assumptions (H1) and (H2).The construction by quasiconformal surgery detailed in Section 3 does not provide analgebraic formula for the rational map f in Theorem 3 and Theorem 4. Furthermore thedegree (cid:98) d + d of f increases quickly with the weight function w so the algebraic relationsbehind are complicated to study. However the particular rational map of degree 3 comingfrom Lemma 15 is simple enough to allow a computation by hand of an algebraic formula.Let f be a rational map coming from the construction detailed in Section 3 for the partic-ular choice of weight function in Lemma 15 . Recall that the local degrees of f at z , z and z are d = 2 , d = 2 and d = 1 , respectively. In particular, z and z are simple critical points.It remains d + d = 1 + 1 = 2 critical points counted with multiplicity coming from definitionof f near z (see Lemma 3), namely two simple critical points, one is z by construction andthe orbit of the other one accumulates the super-attracting cycle { z , z , z , z } .Up to conjugation by a Möbius map, we assume that z = 1 , z = ∞ and z = 0 . So 1and ∞ are critical points whereas 0 is a singular point. In order to simplify notations, denoteby λ the critical point z ( λ will be the parameter of our family) and by λ (cid:48) the last criticalpoint. z = λ (cid:47) (cid:47) z = 1 (cid:47) (cid:47) z = ∞ (cid:47) (cid:47) z = 0 (cid:118) (cid:118) λ (cid:48) (cid:47) (cid:47) . . . Since f has degree 3, it is of the form f : z (cid:55)→ a z + a z + a z + a b z + b z + b z + b . z = 1 is mapped to z = ∞ with a local degree 2, the denominator may factor as f : z (cid:55)→ a z + a z + a z + a ( z − ( b (cid:48) z + b (cid:48) ) . We do likewise for z = ∞ which is mapped to z = 0 with a local degree 2. f : z (cid:55)→ a z + a ( z − ( b (cid:48) z + b (cid:48) ) . Now use the fact that z = 0 is mapped to z = λ to get f : z (cid:55)→ a z + λ ( z − ( b (cid:48) z + 1) . (3)It remains two informations coming from the fact that z = λ is mapped to z = 1 with alocal degree 2. Namely f ( λ ) = 1 and f (cid:48) ( λ ) = 0 which lead to the two following equationssatisfied by a and b (cid:48) . (cid:40) ( λ − ( λb (cid:48) + 1) = λ ( a + 1) a ( λ − ( λb (cid:48) + 1) = λ ( a + 1) (cid:104) (3 λ − λ + 1) b (cid:48) + 2( λ − (cid:105) Remark that we may easily simplify the second equation by using the first one (luckily) (cid:26) ( λ − ( λb (cid:48) + 1) = λ ( a + 1) a = (3 λ − λ + 1) b (cid:48) + 2( λ − or equivalently (cid:26) λa − λ (1 − λ ) b (cid:48) = 1 − λ + λ a − (1 − λ )(1 − λ ) b (cid:48) = − λ and solving this linear system of two equations gives a = (1 − λ )(1 − λ + λ ) − λ (1 − λ )( − λ ) λ (1 − λ ) − λ (1 − λ ) = 1 − λ + 6 λ − λ − λ b (cid:48) = (1 − λ + λ ) − λ ( − λ ) − λ (1 − λ ) + λ (1 − λ )(1 − λ ) = 1 − λ − λ − λ (1 − λ ) . Finally, putting these expressions in expression (3) leads to the following formula for f whichdepends on the parameter λ . f λ : z (cid:55)→ (1 − λ ) (cid:104) (1 − λ + 6 λ − λ ) z − λ (cid:105) ( z − (cid:104) (1 − λ − λ ) z − λ (1 − λ ) (cid:105) Remark that f λ ( z ) = z − (1 − λ + O λ → ( λ )) for every complex number z , thus f λ isactually a particular perturbation of f = (cid:98) f : z (cid:55)→ z − .Some more computations provide an algebraic formula for the critical point λ (cid:48) , namely λ (cid:48) = − λ (1 − λ + 11 λ − λ + 5 λ )(1 − λ − λ )(1 − λ + 6 λ − λ ) = − λ + O λ → ( λ ) . According to the construction detailed in Section 3, there exist some choices of λ suchthat f λ satisfies Theorem 1. Recall that the two critical points z = λ and λ (cid:48) ∼ λ → − λ shouldlie in B ( (cid:98) z ) (see Section 3), and hence near (cid:98) z which corresponds to z = 0 . Indeed, we canroughly prove for every | λ | > small enough that30 f λ ( λ (cid:48) ) lies in a disk centered at z = 1 and of radius of order | λ | ; • the image under f λ of a disk centered at z = 1 and of radius of order | λ | is containedin the complement of a disk centered at (thus containing z = ∞ ) and of radius oforder | λ | − ; • the image under f λ of the complement of a disk centered at (thus containing z = ∞ )and of radius of order | λ | − is contained in a disk centered at z = 0 and of radius oforder | λ | ; • the image under f λ of a disk centered at z = 0 and of radius of order | λ | is containedin a disk centered at z = λ and of radius of order | λ | ; • the image under f λ of a disk centered at z = λ and of radius of order | λ | is containedin a disk centered at z = 1 and of radius of order | λ | .It turns out that the orbit of the critical point λ (cid:48) accumulates the super-attracting cycle { z , z , z , z } for every | λ | > small enough. Consequently, we may encode the exchangingdynamics of Julia components of f λ as it is explained in Section 4, proving that f λ satisfiesTheorem 1 for every | λ | > small enough.Numerically, picking any parameter λ in the big hyperbolic component surrounding ofthe parameter space of the family f λ (see Figure 11.a) provides a Persian Carpet example inthe dynamical plane (see Figure 11.b).Figure 11: a) The parameter plane of f λ for | λ | (cid:47) − , that includes the bifurcation locus(in black) and hyperbolic parameters (in white). is at the center of the pictureand the big hyperbolic component around corresponds to the Persian carpets. b) The dynamical plane for λ ≈ − , that includes the Persian carpet J ( f λ ) (inblack) and the Fatou set (in white).31 Appendix
In this section, we collect some technical results used in the construction of Section 3.
The first result of this section deals with the Hurwitz problem on the topological sphere S .Namely given an abstract branch data of degree d (cid:62) , that is a table of positive integers D = ( d i,j ) ( i,j ) ∈I where I = { ( i, j ) / i ∈ { , , . . . , n } and j ∈ { , , . . . , k i }} for some positiveintegers n , k , k , . . . , k n and such that for every i ∈ { , , . . . , n } : d i,j (cid:62) for some j ∈ { , , . . . , k i } , and k i (cid:88) j =1 d i,j = d, (4)we consider the question on realizability of this abstract branch data by a branched coveringon S , that is the existence of a degree d branched covering H : S → S and a finite collectionof distinct points X = { x i,j / ( i, j ) ∈ I} in S such that: • ∀ ( i, j ) ∈ I , H ( x i,j ) = y i for some y i ∈ S ; • H | S − X : S − X → S − { y i / i ∈ { , , . . . , n }} is a degree d covering; • ∀ ( i, j ) ∈ I , the local degree of H at x i,j is d i,j .Adolf Hurwitz has proved (see [Hur91]) that the solution is as follows. Let S d be thesymmetric group of all permutations of { , , . . . , d } . Then D is realizable if and only if thereexist permutations σ , σ , . . . , σ n in S d such that: (i) ∀ i ∈ { , , . . . , n } , σ i is a product of k i disjoint cycles of length d i, , d i, , . . . , d i,k i ; (ii) σ σ . . . σ n = 1 in S d ; (iii) (cid:104) σ , σ , . . . , σ n (cid:105) transitively acts on { , , . . . , d } .However, such algebraic conditions are rather difficult to verify for an arbitrary abstractbranch data D . Easier sufficient and necessary conditions have been provided in some specificcases (see for instance [EKS84] or [Bar01]). The following lemma gives the solution in a veryspecial case involved in Lemma 1. Lemma 16 (An Hurwitz solution) . Let D be an abstract branch data of degree d (cid:62) suchthat n = 3 and d i,j = 1 for every i ∈ { , , } and j (cid:62) . Then D is realizable if and only ifthe following condition is satisfied. d = 12 ( d , + d , + d , − (H1’)Remark that in this special case, the abstract branch data D is uniquely determined by adegree d (cid:62) together with three positive integers d , , d , , and d , such that (cid:54) d i, (cid:54) d for every i ∈ { , , } . 32 roof. The necessity of condition (H1’) comes from the Riemann-Hurwitz formula. Indeed ifthere exists a degree d branched covering h : S → S which realizes D then the number ofcritical points counted with multiplicity is given by d − d , −
1) + ( d , −
1) + ( d , − since there are no more critical points than x , , x , , and x , from assumption.For the sufficiency of condition (H1’), consider the two following cycles in S d σ = (1 , , . . . , d , ) and σ = ( d, d − , . . . , d − d , + 1) of length d , and d , respectively. Notice that d , = ( d − d , + 1) + ( d − d , ) from condition(H1’), and d , (cid:54) d from assumption (4). Therefore d − d , + 1 (cid:54) d , and in particular (cid:104) σ , σ (cid:105) transitively acts on { , , . . . , d } .A simple computation shows that the composition of σ and σ (with σ performed first)is given by σ σ = (1 , , . . . , d − d , , d, d − , . . . , d , ) which is a cycle of length ( d − d , ) + ( d − d , + 1) = d , from condition (H1’). Denote by σ the inverse permutation ( σ σ ) − which is a cycle of length d , as well.Finally we get three permutations σ , σ , and σ in S d which satisfy the Hurwitz conditions.Indeed (i) holds since σ i is a cycle of length d i, for every i ∈ { , , } , (ii) directly follows fromdefinition of σ , and (iii) holds since (cid:104) σ , σ (cid:105) = (cid:104) σ , σ , σ (cid:105) transitively acts on { , , . . . , d } . The following useful result is due to Cui Guizhen and Tan Lei [CT11]. It is the key ingredientof the proof of Lemma 2.
Lemma 17 (Inverse Grötzsch’s inequality) . Let
D, D (cid:48) be two disjoint marked hyperbolic disksin (cid:98) C whose boundaries (not necessarily disjoint) are respectively denoted by α, α (cid:48) . Then thereexists a positive constant C > such that for every pair of equipotentials β in D and β (cid:48) in D (cid:48) the following inequalities hold: mod( A ( α, β )) + mod( A ( α (cid:48) , β (cid:48) )) (cid:54) mod( A ( β, β (cid:48) )) (cid:54) mod( A ( α, β )) + mod( A ( α (cid:48) , β (cid:48) )) + C. The left hand side is the classical Grötzsch’s inequality. The right hand side is a conse-quence of Koebe / Theorem. We refer readers to [CT11] for a complete proof.
The following lemma is a technical ingredient in the construction of Section 3 needed toholomorphically map an annulus onto a disk (see Lemma 3). It is very similar to the keylemma in [PT99] (see also [BF13]) about an annulus-disk branched covering. However, ourannulus-disk map here requires to be holomorphic (see Lemma 7 and Lemma 8).
Lemma 18 (Annulus-disk holomorphic map) . Let n, n (cid:48) be two positive integers. Then thereexists a holomorphic branched covering G : A ( γ, γ (cid:48) ) → D from an open annulus in (cid:98) C boundedby a pair of disjoint quasicircles γ, γ (cid:48) onto the open unit disk D such that: (i) G has degree n + n (cid:48) and has n + n (cid:48) critical points counted with multiplicity; ii) G continuously extends to γ ∪ γ (cid:48) by a degree n covering G | γ : γ → ∂ D and a degree n (cid:48) covering G | γ (cid:48) : γ (cid:48) → ∂ D ; (iii) mod( A ( γ, γ (cid:48) )) (cid:54) .Proof. There are many ways to prove the existence of such a map. Here this proof uses theproperties of the McMullen’s family g ,λ : z (cid:55)→ z n + λz n (cid:48) for | λ | > small enough (see [McM88] and [DHL +
08] for a complete study of this family).Recall that g ,λ has degree n + n (cid:48) , and its critical set contains n + n (cid:48) simple critical points ofthe form c k = (cid:18) n (cid:48) n (cid:19) / ( n + n (cid:48) ) | λ | / ( n + n (cid:48) ) e kiπ/ ( n + n (cid:48) ) where k ∈ { , , . . . , n + n (cid:48) } (the other critical points are ∞ of multiplicity n − if n > and of multiplicity n (cid:48) − if n (cid:48) > ). Moreover, the preimages of are of the form g − ,λ (0) = {| λ | / ( n + n (cid:48) ) e kiπ/ ( n + n (cid:48) ) / k ∈ { , , . . . , n + n (cid:48) }} . Let A be the preimage of the open unit disk D , namely A = g − ,λ ( D ) . We are going to provethat for every | λ | > small enough A is connected and actually an open annulus separating and ∞ . Indeed remark that for every z ∈ C with modulus | z | = | λ | / ( n + n (cid:48) ) we have | g ,λ ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12) z n + λz n (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) | λ | n/ ( n + n (cid:48) ) + | λ || λ | n (cid:48) / ( n + n (cid:48) ) = 2 | λ | n/ ( n + n (cid:48) ) . Similarly for every k ∈ { , , . . . , n + n (cid:48) } we have | g ,λ ( c k ) | (cid:54) (cid:18) n (cid:48) n (cid:19) n/ ( n + n (cid:48) ) | λ | n/ ( n + n (cid:48) ) + (cid:16) nn (cid:48) (cid:17) n (cid:48) / ( n + n (cid:48) ) | λ || λ | n (cid:48) / ( n + n (cid:48) ) = C | λ | n/ ( n + n (cid:48) ) with C = (cid:18) n (cid:48) n (cid:19) n/ ( n + n (cid:48) ) + (cid:16) nn (cid:48) (cid:17) n (cid:48) / ( n + n (cid:48) ) = (cid:18) n + n (cid:48) n (cid:19) n/ ( n + n (cid:48) ) × (cid:18) n + n (cid:48) n (cid:48) (cid:19) n (cid:48) / ( n + n (cid:48) ) (cid:54) (by using the arithmetic-geometric mean inequality x n/ ( n + n (cid:48) ) × y n (cid:48) / ( n + n (cid:48) ) (cid:54) nn + n (cid:48) x + n (cid:48) n + n (cid:48) y ).So if λ is such that < | λ | n/ ( n + n (cid:48) ) < then A contains the circle centered at and ofradius | λ | / ( n + n (cid:48) ) where all the preimages of lie, together with n + n (cid:48) simple critical pointsof g ,λ . In particular, A is a connected set which separates and ∞ and it follows from theRiemann-Hurwitz formula applied to the degree n + n (cid:48) branched covering g ,λ | A : A → D that A is an open annulus.Now let γ be the outer boundary of A , namely the boundary of the connected component of (cid:98) C − A containing ∞ , and γ (cid:48) be the inner boundary of A , namely the boundary of the connectedcomponent of (cid:98) C − A containing . It turns out that A = A ( γ, γ (cid:48) ) and G = g ,λ | A : A ( γ, γ (cid:48) ) → D satisfy (i) . The point (ii) follows from the fact that g ,λ realizes a degree n (respectively n (cid:48) )branched covering on the the connected component of (cid:98) C − A containing ∞ (respectively )34ith no critical points on the boundary. Moreover γ and γ (cid:48) are quasicircles as preimages ofthe unit circle ∂ D by conformal maps.For the point (iii) , remark that for every R (cid:62) we have: | z | (cid:54) R /n (cid:48) | λ | / ( n + n (cid:48) ) ⇒ | g ,λ ( z ) | (cid:62) | λ || z | n (cid:48) − | z | n (cid:62) | λ | n/ ( n + n (cid:48) ) (cid:0) R − R n (cid:1) , and | z | (cid:62) R /n | λ | / ( n + n (cid:48) ) ⇒ | g ,λ ( z ) | (cid:62) | z | n − | λ || z | n (cid:48) (cid:62) | λ | n/ ( n + n (cid:48) ) (cid:0) R − R n (cid:48) (cid:1) . In particular if R = 2 | λ | − n/ ( n + n (cid:48) ) , then max { R n , R n (cid:48) } (cid:54) R < R (since λ was chosen so that < | λ | n/ ( n + n (cid:48) ) < that implies R > ) and hence | z | (cid:54) R /n (cid:48) | λ | / ( n + n (cid:48) ) or | z | (cid:62) R /n | λ | / ( n + n (cid:48) ) ⇒ | g ,λ ( z ) | > | λ | n/ ( n + n (cid:48) ) R = 1 . Consequently the preimage A = A ( γ, γ (cid:48) ) of the unit disk is contained as essential subannulusin the round annulus { z ∈ C / R /n (cid:48) | λ | / ( n + n (cid:48) ) < | z | < R /n | λ | / ( n + n (cid:48) ) } and the Grötzsch’sinequality gives mod( A ( γ, γ (cid:48) )) (cid:54) π log (cid:32) R /n | λ | / ( n + n (cid:48) )1 R /n (cid:48) | λ | / ( n + n (cid:48) ) (cid:33) = 12 π (cid:18) n + 1 n (cid:48) (cid:19) log( R ) . In particular, if λ is fixed so that | λ | n/ ( n + n (cid:48) ) = e π < then R = 2 | λ | − n/ ( n + n (cid:48) ) = e π and mod( A ( γ, γ (cid:48) )) (cid:54) (cid:18) n + 1 n (cid:48) (cid:19) (cid:54) . In the proof above, is obviously not the optimal upper bound for mod( A ( γ, γ (cid:48) )) . Theauthor guesses that this modulus is arbitrary small when λ is close to . But one can provethat the modulus of the smallest round annulus containing A ( γ, γ (cid:48) ) as essential subannulusis bounded by below by a positive constant which does not depend on λ . The same happensif the open unit disk D is replaced by any euclidean open disk centered at and containingthe critical values. However we do not need a sharper estimation than (iii) in this paper (seeLemma 2 and the proof of Lemma 3). The following lemma is used to define the quasicircle δ + c in the construction of Section 3 (seethe proof of Lemma 5). Lemma 19 (Separating quasicircle) . Let A ( γ, γ (cid:48) ) be an open annulus in (cid:98) C bounded by a pairof disjoint quasicircles γ, γ (cid:48) , and let a be a point in A ( γ, γ (cid:48) ) . Then there exists a quasicircle δ in A ( γ, γ (cid:48) ) which separates a from γ ∪ γ (cid:48) such that mod( A ( γ, δ )) is arbitrary small. The main idea is merely to define a quasicircle δ close enough to the boundary γ . Wethough provide an explicit proof which uses the definition of the modulus by extremal length(see [Ahl73]). Proof.
Up to a biholomorphic change of coordinates, we may assume that γ = { z ∈ C / | z | = 1 } , γ (cid:48) = { z ∈ C / | z | = e − π mod( A ( γ,γ (cid:48) )) } , and a ∈ ] e − π mod( A ( γ,γ (cid:48) )) , . x to be the positive real number (1 + e − π mod( A ( γ,γ (cid:48) )) ) / . For every ε > small enough,define δ ε to be the euclidean circle centered at x and of radius − x − ε . Notice that δ ε isincluded in A ( γ, γ (cid:48) ) and that δ ε separates a from γ ∪ γ (cid:48) for every ε > small enough.For every angle θ (small enough), consider the path (cid:96) θ joining δ ε to γ of the form (cid:96) θ = { z = re iθ / R θ (cid:54) r (cid:54) } with R θ > maximal so that R θ e iθ ∈ δ ε . By classical results fromeuclidean geometry and trigonometry, we get R θ = x cos( θ ) + (cid:113) (1 − x − ε ) − x sin ( θ ) . Since θ (cid:55)→ R θ is an even function with a local maximum at θ = 0 , it follows for every ε > small enough that θ ∈ [ −√ ε, √ ε ] = ⇒ R θ (cid:62) R √ ε = x cos( √ ε ) + (cid:113) (1 − x − ε ) − x sin ( √ ε )= 1 − − x − x ) ε + O ε → ( ε ) (cid:62) − Cε (5)where C is a positive constant fixed so that C > − x − x ) .Now recall that the modulus of A ( γ, δ ε ) is given by the extremal length of the collection L of rectifiable paths connecting δ ε and γ , namely mod( A ( γ, δ ε )) = sup ρ (cid:0) inf (cid:96) ∈ L (cid:82) (cid:96) ρ | dz | (cid:1) (cid:82) A ( γ,δ ε ) ρ dxdy where the supremum is over all measurable functions ρ : A ( γ, δ ε ) → [0 , + ∞ ] such that (cid:82) A ( γ,δ ε ) ρ dxdy < + ∞ . Let ρ be such a measurable function. For every θ (small enough),we have inf (cid:96) ∈ L (cid:90) (cid:96) ρ | dz | (cid:54) (cid:90) (cid:96) θ ρ | dz | = (cid:90) R θ ρ ( re iθ ) dr that leads, integrating over θ ∈ [ −√ ε, √ ε ] and applying the Cauchy-Schwarz inequality, to √ ε inf (cid:96) ∈ L (cid:90) (cid:96) ρ | dz | (cid:54) (cid:32)(cid:90) √ ε −√ ε (cid:90) R θ ρ ( re iθ ) rdrdθ (cid:33) / (cid:32)(cid:90) √ ε −√ ε (cid:90) R θ r drdθ (cid:33) / (cid:54) (cid:18)(cid:90) A ( γ,δ ε ) ρ dxdy (cid:19) / (cid:32)(cid:90) √ ε −√ ε log (cid:18) R θ (cid:19) dθ (cid:33) / . Therefore it follows from the inequality (5) that (cid:0) inf (cid:96) ∈ L (cid:82) (cid:96) ρ | dz | (cid:1) (cid:82) A ( γ,δ ε ) ρ dxdy (cid:54) ε (cid:90) √ ε −√ ε log (cid:18) R θ (cid:19) dθ (cid:54) ε (cid:90) √ ε −√ ε log (cid:18) − Cε (cid:19) dθ = 12 √ ε log (cid:18) − Cε (cid:19) . Finally we take the supremum over all measurable functions ρ to get mod( A ( γ, δ ε )) (cid:54) √ ε log (cid:18) − Cε (cid:19) ∼ ε → C √ ε → ε → that concludes the proof. 36 eferences [Ahl73] Lars V. Ahlfors. Conformal invariants: topics in geometric function theory .McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York,NY, 1973.[Bar01] Krzysztof Barański. On realizability of branched coverings of the sphere.
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