Arbitrary large number of non trivial rescaling limits
AArbitrary large number of non trivial rescalinglimits
Matthieu Arfeux, Guizhen Cui
Abstract
We construct a family of rational map sequences providing an ar-bitrary large number of independent rescaling limits of non monomialtype. From this, we deduce the existence of a family of rational mapsproviding a non trivial dynamics on the Berkovich projective line overthe field of formal Puiseux series.
Let us denote by S := P ( C ) the Riemann sphere. In this paper we areinterested in the behavior of the elements of Rat d the set of rational maps f : S → S of exact degree d under the iteration by composition. Moreprecisely, we are interested in the phenomena of existence of rescaling limitsdetected in [Sti] but defined for the first time in [Kiw15] as follows. Definition.
For a sequence of rational maps ( f n ) n in Rat d , a rescaling is asequence of Moebius transformations ( M n ) n such that there exist k ∈ N anda rational map g of degree ≥ such that M n ◦ f kn ◦ M − n → g uniformly on compact subsets of S with finitely many points removed.If this k is minimal then it is called the rescaling period for ( f n ) n at ( M n ) n and g a rescaling limit for ( f n ) n . In that paper Jan Kiwi introduced the notions of independence of rescalingsand of dynamical dependence of rescalings below (these notions are not usedin this paper). 1 a r X i v : . [ m a t h . D S ] J un efinition (Independence of rescalings) . We say that two rescalings ( M n ) n and ( N n ) n of a sequence of rational maps ( f n ) n are independent and write N n ∼ M n if N n ◦ M − n → ∞ in Rat . That is, for every compact set K in Rat , the sequence N n ◦ M − n / ∈ K for n big enough. The rescalings are said to be equivalent if N n ◦ M − n → M in Rat . Definition (Dynamical dependence of rescalings) . Given a sequence ( f n ) n ∈ Rat d and given ( M n ) n and ( N n ) n of period dividing q . We say that ( M n ) n and ( N n ) n are dynamically dependent if, for some subsequences ( M n k ) n k and ( N n k ) n k , there exist ≤ m ≤ q , finite subsets S , S of S and non constantrational maps g , g such that L − n k ◦ f mn k ◦ M n k → g uniformly on compact subsets of S \ S and M − n k ◦ f q − mn k ◦ L n k → g uniformly on compact subsets of S \ S . Jan Kiwi proved the two following results.
Theorem A. [Kiw15] For every sequence in
Rat d for d ≥ , there are at most d − classes of dynamically independent rescalings with a non postcriticallyfinite rescaling limit. Theorem B. [Kiw15] For every sequence in
Rat , there are at most classesof dynamically independent rescalings. Following these result, J.Kiwi wrote in [Kiw15] the following natural ques-tion:
Question 1.1.
Is there a bound on the number of classes of dynamicallyindependent and non-monomial rescalings that a sequence in
Rat d can havethat would depend only on the degree d ? In this paper we prove that the answer is no. More precisely we prove thefollowing theorem: 2 heorem 1.
For every n ∈ N ∗ and d ≥ there exists a sequence of rationalmaps ( f k ) k in Rat d with n dynamically independent, post-critically finite andnon-monomial rescaling limits. Theorem A and Theorem B have been proven using non-archimedean dy-namics tools and later re-proven in [Arfa] and [Arfb] respectively, using adifferent approach based on the Deligne-Mumford compactification of themoduli space of stable punctured spheres. This compactification has beenrestated with the language of trees of spheres in [Arfc]. We suppose that thereader has a good knowledge of the definition provided in [Arfa]. With thisvocabulary, Theorem 1 can be written the following way:
Theorem 2.
For every n ∈ N ∗ and d ≥ there exists a dynamical systembetween trees of spheres of degree d limit of dynamically marked rationalmaps, with n cycles of spheres whose associated cover is post-critically finiteand non-monomial. In fact these two approaches are related and their relation has been sketchedin [Arfd]. The results of [Arfa] and [Arfb] are a translation of results in[Kiw15]. The authors use dynamical systems between trees of spheres in onecase or dynamics on Berkovich spaces in the other, in order to deduce resultsin holomorphic dynamics. The approach here is the exact reverse.We are inspired by the holomorphic dynamical notion of “Shishikura trees”.Those have been introduced by M. Shishikura in [Shi89] and [Shi02] for aspecial case and developed as a general idea during his talks. Following theseideas, [CP] define another notion of Shishikura trees that we will use for ourpurpose. Our result will be based on the constructions of self-graphting givenin [CP].Denote by D (cid:63) the punctured unit disc of C . In the analytic context, rescal-ings, rescaling limits, independence of rescalings, dynamical dependance ofrescaling limits are also defined by replacing n ∈ N tending to infinity by t ∈ D (cid:63) , n → ∞ by t → t (see [Arfd]). Jan Kiwi proved the following. Proposition 1.2 ([Kiw15]Proposition 6.1) . Consider a sequence of degree d rational maps f n . Let N ∈ N and assume that for all j = 1 , ...N the sequence ( M j,n ) n is a rescaling of period q j for ( f n ) n with rescaling limit g j . Thenthere exists a degree d holomorphic family ( f t ) t and, for each j = 1 , ...N , aholomorphic family of Moebius transformations ( M j,t ) t such that ( M j,t ) t is arescaling for ( f t ) t of period q j and limit g j . f ( M j,t ) t and ( M k,t ) t are dynamically dependent for ( f t ) t , then ( M j,n ) n and ( M k,n ) n are dynamically dependent for ( f n ) n . Hence Theorem 1 has the following consequence.
Corollary 1.
For every n ∈ N ∗ and d ≥ there exists a holomorphic family f t ∈ Rat d for t ∈ D (cid:63) with n dynamically independent, post-critically finiteand non-monomial rescaling limits. Using the bridge between non-archimedean dynamics and the dynamics ontrees of spheres made explicit in [Arfd], we deduce Corollary 2 below.Denote by L the completion of the field of formal Puiseux series over C equipped with its usual non-archimedean norm. Denote by P Berk theBerkovich projective line over L and recall the the type II points of P Berk arethe points separating at least three different branches.
Corollary 2.
For every n ∈ N ∗ and d ≥ there exists f ∈ L ( Z ) of degree d with n periodic type II points in disjoint cycles for the induced dynamics in P Berk whose reduction is post-critically finite and non-monomial.
We will not recall more details about the non-archimedean dynamics thatthe interested reader can find in [Kiw15] (or [Arfd] for a first reading).
Outline.
In Section 2, we recall the notion of Shishikura trees from [CP] with somesmall adaptations. Section 3 deals with the self-grafting construction. InSection 4, we explicit the relation between Shishikura trees and dynamicalsystems between trees of spheres. In Section 5, we prove Theorem 1 andTheorem 1. The article ends on a little appendix with some technical resultson stable trees.
Acknowledgments.
The authors want to thank Tan Lei for presenting theone to the other and for inviting us to discuss in Angers in LAREMA.
In this section we recall the constructions and results from [CP] with a fewchanges on some definitions. We point out some properties needed in thenext sections. 4 .1 Canonical multicurve
Here f denote a hyperbolic rational map. We denote by P f its post-criticalset and P (cid:48) f the set of accumulation points of P f .Suppose that E ⊂ S is a connected set which is neither open or closed. Wesay that E is disc-type if E is contained in a disk D with card D ∩ P f ≤ E annular-type if E is not disk-type and is contained in anannulus A with card A ∩ P f = 0. If E is neither disk-type nor annular-type,then we say that E is complex-type .Recall that given a set X ⊂ S , a multicurve on S \ X is a collection ofJordan curves γ in S \ X pairwise disjoint and non isotopic and non-peripheral(i.e. every connected component of S \ γ contains at least two points of P f ).A multicurve Γ f is totally stable (by f ) if each non-peripheral curve of f − ( γ ) for γ ∈ Γ f is isotopic rel P f to a curve in Γ f and if each curve γ ∈ Γ f is isotopic rel P f to a curve in f − ( γ (cid:48) ) for some curve γ (cid:48) ∈ Γ f . Theorem 2.1.
There exist a totally stable multicurve Γ f in S \ P f such that ∀ D ∈ S \ Γ f , D contains a unique complex type Julia or is contained in acomplex-type Fatou domain, and conversely. These properties are stable under isotopy rel P f and the isotopy class ofsuch a curve is called the canonical multicurve of f . As a direct conse-quence of these definitions we have the following property: Lemma 2.2.
Let Γ f be a representative of the canonical multicurve and U bea connected component of S \ Γ f . Denote by cc∂U the collection of connectedcomponents of S \ ∂U . Then we have: • ∀ V ∈ cc∂U, card V ∩ P f ≥ and • card ( cc∂U ) + card ( U ∩ P f ) ≥ . Given a rational map f , Jordan curve is called a KB curve if it is a con-nected component of some Koenig or B¨ottcher coordinates level curve (cf[Mil06] for the definitions of the Koenigs and B¨ottcher coordinates). Simi-larly, a multicurve is KB if it consists of KB curves. By construction in [CP],we have: Proposition 2.3.
Every canonical multicurve has a representative consistingof a KB multicurve.
Such a representative is called a
KB canonical multicurve of f .5 .2 Shishikura trees In this section, Γ f denotes a KB canonical multicurve of f (cf Proposition2.3).Given a collection Γ of disjoint Jordan curves on S , we define the dual tree T Γ of Γ to be the tree whose vertices consist of the connected components of S \ Γ and whose edges are the elements of Γ and join two vertices if and onlyif it lies in both of their boundary.In [CP], the
Shishikura tree T f is defined to be the dual tree of Γ f . LetΓ − f := f − (Γ f ) and T − f be its associated dual tree. Define P − f := f − ( P f ).The map f induces a tree map τ f : T − f → T f (indeed, it maps adjacentvertices to adjacent vertices and edges to edges, cf Figure 1). Remark 2.4.
This map is slightly different than the one in [CP] where theauthors consider Γ the subset of Γ − f consisting of its non peripheral elementsin S \ P f to define a tree T and then a map τ : T → T f . There is a naturalmap π : T − f → T consisting in the inclusion. The relation between τ f and τ is provided by the following commutative diagram: T − f π (cid:47) (cid:47) τ f (cid:32) (cid:32) T τ (cid:0) (cid:0) T f and this change of definition, necessary here, does not affect the results citedfrom [CP]. With our definition, the map f induces also a degree function on the setof edges and vertices that we will denote by deg f . From the total stability ofΓ f we deduce a natural identification of T f in T − f and the map τ f inducesdynamics on T f . In [CP] the authors constructed a sequence of rational maps ( f n ) n using self-grafting . We recall here what this procedure allowed to construct.In [God], S´ebastien Godillon considered the following family of rationalmaps f ( z ) := (1 − λ )[(1 − λ + 6 λ − λ ) z − λ ]( z − [(1 − λ − λ ) z − λ (1 − λ )]6 x x Figure 1: The Shishikura trees T − f ( left) and T f (right) for the function f introduced in Section 3. The map τ f maps the vertices and edges to the onesof identical colors. The vertex x is fixed and the numbered vertices form acycle of period 4.for λ (cid:54) = 0 ,
1. We fix some parameter λ (cid:54) = 0 , f the corre-sponding element in this family. Figure 1 shows the Shishikura trees T − f and T f . In order to simplify the notations let us respectively denote T f and τ f by T and τ .The self-grafting is an induction procedure that from some Shishikura tree T n uses Thurston’s realization theorem to construct • a new tree T n +1 , • a map τ n +1 acting on it, and • a hyperbolic rational map f n +1 such that T n +1 = T f n +1 and τ n +1 = τ f n +1 .In addition we have T n (cid:67) T n +1 in the sense of the following definition. Definition 3.1.
We say that a tree T X is compatible with a tree T Y andwrite T X (cid:67) T Y if • the vertices of T X are vertices of T Y and • for all vertices v , v , v and v of T X , the vertex v separates v , v and v in T X if and only if it does the same in T Y . T n between two vertices can be identified as the arc of ver-tices and edges separating the two corresponding vertices in T n +1 (cf Figure2). As a consequence T (cid:67) T n .Let us recall here some relations between the tree T n +1 and T n . We denoteby x and x the two vertices of the T represented on Figure 2. As T (cid:67) T n ,the vertices x and x are also vertices of T n . We will denote by [ x , x ] n thearc in T n joining those. The vertices of T n +1 lying in the edges of T n form acycle of period k >
2. Let v ∈ T n +1 be the vertex on [ x , x ] n +1 \ { x } whichis the closest to x . Denote by B the branch of T n +1 at v that containsthe vertex x (by the choice of v , this branch contains only vertices of T n ).The tree T n +1 is obtained from its elements identified with T n by attachingat each point v i +1 := τ in +1 ( v i ) for i = 0 ..k − B that we denoteby B i +1 .Denote by ι n +1 : T n +1 → T n +1 the involution that exchanges the twobranches B and B k via their natural identification. We extend the map τ n on T n +1 into a map f n +1 that maps B k to B and B i to B i +1 for i = 1 ..k − τ n +1 is defined to be ι n +1 ◦ f n +1 . Thedegree of τ n +1 is the one of τ on its elements identified with T and it is 1elsewhere. Remark 3.2.
With these properties we can point out in particular that • v ∈ [ x , x ] n +1 is k periodic for the map τ n +1 with some k > , • τ kn +1 maps B k with degree to B , and • τ n +1 maps B with degree to its image. In this section f denotes a hyperbolic rational map. We consider a Shishikuratree T f and denote by τ f : T − f → T f the associated Shishikura tree map. Weexplain how we associate to τ f a dynamical system between trees of sphereswhich is a limit of dynamically marked rational maps in the sense of [Arfa].8
23 4 501 32 0 B6B1 B2 x x x x x x B3B4 B5B0B0 vvv vv v
Figure 2: On the top left is the tree T f with its cycle as on Figure 1. Thetree lower left is the tree T f restricted to its elements identified with T f andin red is represented the branch B as labeled in Section 3. The v i form acycle. On the right is the tree T f and in red are the copies of the branch B from the previous tree. 9 .1 Notation for trees of spheres A tree of spheres marked by a finite set X is usually denoted by T X . It is acombinatorial tree that we denote by T X whose set of leaves is X togetherwith the data for each internal vertex v ∈ T X of a conformal sphere denotedby S v and a different attaching point i v ( e ) ∈ S v for every edge e adjacent to v . We denote by X v the set of attaching points of edges on S v .If T Y is another tree of spheres, we write T X (cid:67) T Y if T X (cid:67) T Y and if theidentified edges are attached at the same place.A cover F : T Y → T Z between two trees of spheres marked by finite sets Y and Z is the following data • a map F : T Y → T Z mapping leaves to leaves, internal vertices tointernal vertices, and edges to edges, • for each internal vertex v of T Y , an holomorphic ramified cover f v : S v → S F ( v ) that satisfies the following properties: – the restriction f v : S v − Y v → S F ( v ) − Z F ( v ) is a cover, – f v ◦ i v = i F ( v ) ◦ F , – if e is an edge between v and v (cid:48) , then the local degree of f v at i v ( e ) is the same as the local degree of f v (cid:48) at i v (cid:48) ( e ).For such a F , we say that ( F , T X ) is dynamical system between trees ofspheres if T X (cid:67) T Y and T X (cid:67) T Z . It follows from Lemma 2.2 that one can associate to every vertex v of T f a set X v ⊂ P f of cardinal 3 such that two points of X v lie in the same connectedcomponent U of S \ Γ f if and only if U = v . Define X to be the union (maybenot disjoint) of the X v and the set of critical values of f . Let Z := X ∪ f ( X )and let Y := f − ( Z ). Note that by choice we have X ⊂ Y ∩ Z .We define F to be the pair consisting of F := ( f | Y : Y → Z ) and thedegree function deg F : Y → N corresponding to the local degree of f | Y . This F is a portrait in the sense of [Arfa].10enote by i and j the respective inclusions of Y and Z into S . The map( f, i, j ) is a rational map dynamically marked by ( F , X ) in the sense of [Arfa],i.e. y and z are injective and we have the following commutative diagram : X (cid:47) (cid:47) (cid:32) (cid:32) Y i (cid:47) (cid:47) F (cid:15) (cid:15) S f (cid:15) (cid:15) Z j (cid:47) (cid:47) S with deg f ( i ( y )) = deg F ( y ) for all y ∈ Y and i | X = j | X . Recall that given a set X (cid:48) ⊂ S , any disjoint union of curves Γ (cid:48) ⊂ S \ X (cid:48) gives a natural partition of X (cid:48) that corresponds to the set of non emptyintersections of X (cid:48) with the different connected components of S \ Γ (cid:48) , wedenote it by P Γ (cid:48) ,X (cid:48) .For every γ ∈ Γ, let us fix an open annulus A γ in S \ X that retracts to γ and made of KB level curves. Let C r := { c ∈ C | (cid:60) ( c ) > } . Proposition 4.1.
There exists a holomorphic family of rational maps f t for t ∈ C r and a holomorphic motion Φ = (cid:12)(cid:12)(cid:12)(cid:12) C r × S → S ( t, z ) (cid:55)−→ Φ t ( z ) such that S f (cid:47) (cid:47) Φ t (cid:15) (cid:15) S Φ t (cid:15) (cid:15) S f t (cid:47) (cid:47) S , • P Φ t (Γ) , Φ t ( X ) = P Γ ,X , and • for every γ ∈ Γ f , Modulus(Φ t (A γ ) → ∞ when t → . Proof.
This construction uses the standard ”stretching” deformation de-scribed for example in [BF13]. For our purpose, we choose a fundamentalannulus in each of the grand orbit of the Fatou components that contain oneof the KB curve. By stretching the complex structure on these annuli by11aking them going to infinity in modulus when t approaches 0, we get sucha family of rational maps f t and a holomorphic motion:Φ = (cid:12)(cid:12)(cid:12)(cid:12) C r × S → S ( t, z ) (cid:55)−→ Φ t ( z )such that S f (cid:47) (cid:47) Φ t (cid:15) (cid:15) S Φ t (cid:15) (cid:15) S f t (cid:47) (cid:47) S . We choose for every γ ∈ Γ f , an open annulus A γ included in the Fatou setof f and that contain γ . By the stretching deformation gives that Φ t ( A γ )is still an annulus whose modulus tends to infinity as t →
0. Moreover,the f t are holomorphic so the postcritical set moves holomorphicaly with theparameter t so the sumption about the partitions follows. (cid:3) We define such a Φ and take a sequence t n ∈ C r converging to 0. Thenaccording to [Arfc](Theorem 3), after passing to a subsequence, we have thefollowing: Proposition 4.2.
The sequence ( f t n , Φ t n ◦ i, Φ t n ◦ j ) converges dynamicallyto a dynamical system between trees of spheres ( F : T Y → T Z , T X ) markedby F . Define T f to be the combinatorial tree of T f with the additional vertices X and edges between any element x ∈ X and the vertex corresponding tothe connected component of S \ Γ f containing x . Similarly, from T − f weconstruct T − f whose set of leaves is f − ( X ). It is clear from the definitionsthat we have T f (cid:67) T f and T − f (cid:67) T − f .In this section we prove the following statement: Theorem 4.3.
After changing the labeling of the internal vertices of T f wehave T f = T X and T − f (cid:67) T Y . The map F : T Y → T Z restricted to T f isthe map τ f and for every attaching point z of an edge e ∈ T X on a sphereassociated to a vertex v ∈ T X , the local degree of f | v at z is the degree of τ f at e . .3.1 Stable trees and compatibility A tree is stable if any internal vertex is adjacent to at least three edges.
Remark 4.4.
As a direct consequence of Lemma 2.2, the trees T f and T − f are stable. Also according to [Arfc], the combinatorial trees T X , T Y and T Z associated respectively to T X , T Y and T Z of Proposition 4.2 are stable. In this subsection we prove the following lemma.
Lemma 4.5. If T and T are two stable trees sharing the same set of leavesand such that T (cid:67) T , then T = T . For any choice of three different leaves in a tree, there exists a unique vertexof this tree separation them. Given an internal vertex v of a combinatorialtree T and a subset X of leaves, there is a natural partition of X associatedto v consisting of the collections of non empty intersections of X with thedifferent connected components of T \ { v } . Lemma 4.6.
Let T and T be combinatorial stable trees. Denote by X theset of leaves of T . Suppose that X is included in the set of leaves of T andthat for every x , x , x ∈ X distinct, the partitions of X associated to thevertices separating x , x , x are the same. Then, after relabeling the internalvertices of T or T , we have T (cid:67) T . Proof.
Take an internal vertex u of T . Then, by stability, this vertexseparate three distinct elements x , x , x of X . Consider the vertex v of T separating the same elements. Then, the vertices are considered modulorelabeling, we can suppose that u = v . According to these hypothesis, thisrelabeling can be made consistently on for all the triples of X , so for allthe vertices of T . It follows that we can consider that the vertices of T arevertices of T .Consider four vertices u, u , u , u of T . Suppose that u separates u , u and u in T . Chose an element x (resp. x , x ) in the branch on v containing v (resp. v , v ). Then, in the tree T , v is on the branch on v (resp. v ,then v ) containing x , x (resp. v , v , then v , v ). It follows that v separate v , v , v in T . The same argument prove the converse property. (cid:3) Proof. [Lemma 4.5] As T (cid:67) T , the set of vertices of T is included in the oneof T . Any internal vertex v of T separates three leaves so, because theseleaves are also leaves of T , there is a unique vertex v of T separating them13nd it follows that v = v . Hence T and T have the same set of vertices.From this we deduce that T (cid:67) T and it is easy to check that T = T . (cid:3) T f = T X and T − f (cid:67) T Y Lemma 4.7.
Up to relabeling the internal vertices of T X , we have T f = T X . Proof.
Consider a vertex v ∈ T f . We deduce from the definition of X that v ∈ T X separates three vertices x , x and x which are elements of X . Letus relabel the vertex in T X separating the same three elements in T X by v .Then consider a projective chart M t n : S → ˆ C that maps Φ t n ( x ) , Φ t n ( x )and Φ t n ( x ) to 0 , ∞ .Take x ∈ X and suppose that x and x are on the same branch on v ∈ T f ,then there is a curve γ ⊂ ∂v such that x and x are in the same componentof S \ γ . By construction of T X and according to Proposition 4.1, the points M t n ◦ Φ t n ( x ), M t n ◦ Φ t n ( x ) and the points M t n ◦ Φ t n ( x ), M t n ◦ Φ t n ( x ) liesin different components of ˆ C minus an annulus whose modulus converges toinfinity. Hence M t n ◦ Φ t n ( x ) → M t n ◦ Φ t n ( x ) , so x and x are in the same branch on v ∈ T X . As we said in Remark 4.4,the trees T f and T X are stable with the same set of leaves, hence Lemma 4.6and Lemma 4.5 conclude the proof. (cid:3) With the same reasoning we can also show the following:
Lemma 4.8.
After changing the labeling of the internal vertices of T − f wehave T − f (cid:67) T Y . We suppose for the rest of this paper that the labeling of the internalvertices of T X and T − f is such that T f = T X and T − f (cid:67) T Y . Every edge of T − f considered as a subset of T Y maps onto anedge of T f considered as a subset of T Z . All the elements of an edge of T − f considered as a subset of T Y have same degree. This lemma is proven in Annexe A.Let us recall the following result which is a direct consequence of the Ar-gument Principle: 14 emma 4.10.
Let g : S → S be a branched cover, X be a finite subset of S containing the critical values of g , Γ be a multicurve on S \ X , γ − be aconnected component of g − (Γ) and D be a connected component of S \ γ − .Then γ := g ( γ − ) is the unique curve of Γ such that there exist x, x (cid:48) ∈ X in the different connected components of S \ γ such that deg f ( γ − ) = | card D ∩ f − ( x ) − card D ∩ f − ( x (cid:48) ) | , where the cardinals are counted with multiplicity. Moreover this formulaworks for any couple ( x, x (cid:48) ) chosen in different connected components of S \ γ . In [Arfc](proof of Proposition 3.14), the lemma below is proven by passingto the limit the previous lemma.
Lemma 4.11.
Let F : T Y → T Z be a cover between trees of spheres limit ofmarked rational maps. Let e be an edge of T Y adjacent to a vertex v and D denote a connected component of T Y \ { e } . Then the edge F ( e ) is the uniqueedge of T Z satisfying for a couple ( z, z (cid:48) ) ∈ Z with z and z (cid:48) lie in differentcomponents of T Z \ { F ( e ) } deg f v ( i v ( e )) = | card D ∩ F − ( z ) − card D ∩ F − ( z (cid:48) ) | , where deg f v ( i v ( e )) denotes the local degree of f v at the attaching point of v and the cardinals are counting the critical points with multiplicity. Moreoverthis equality holds also for any such couple ( z, z (cid:48) ) . Proof. [Theorem 4.3] Lemma 4.7 proved the equality T f = T X .First consider a vertex v of T − f . Take any edge e − between v and v (cid:48) ∈ T − f .As an edge of T − f , e − is a closed curve in S . Denote by D a connectedcomponent of S \ e − . According to Lemma 4.10, for any z, z (cid:48) ∈ Z lyingin the distinct connected components of S \ f ( e − ), we have deg f ( e − ) = | card D ∩ f − ( z ) − card D ∩ f − ( z (cid:48) ) | .According to Lemma 4.8, e − can be considered as a subset of T Y andLemma 4.9 assures that F ( e − ) is the arc between F ( v ) and F ( v (cid:48) ) in T Z which corresponds to an edge e of T X . Comparing the formulas given inLemma 4.11 and Lemma 4.10, as f and F are equal on Y , we deduce that e = f ( e − ). Hence we proved that F ( e − ) = F ([ v, v (cid:48) ]) = [ F ( v ) , F ( v (cid:48) )] = e = f ( e − ) . And the assumptions about the local degrees follow the same way. (cid:3) Trees and rescaling limits
In this section we prove Theorem 1 and Theorem 2.Let f be an element of the sequence described in Section 3. According toRemark 3.2, the map τ f has a periodic vertex v of period k >
1, the degreealong this cycle is 4 and a non critical branch of T − f maps to a critical oneof T f by τ kf so the corresponding f kv is not monomial. Hence Theorem 4.3proves Theorem 2.Theorem 1 is a translation of Theorem 2 via the following theorem. Theorem C ([Arfa]Theorem 2) . Let F be a portrait, let ( f n , y n , z n ) n ∈ Rat F ,X and let ( F , T X ) be a dynamical system of trees of spheres. Supposethat f n (cid:67) −→ φ Yn ,φ Zn F . If v is a periodic internal vertex in a critical cycle with exact period k , then f kv : S v → S v is a rescaling limit corresponding to the rescaling ( φ Yn,v ) n .In addition, for every v (cid:48) in the cycle, ( φ Yn,v (cid:48) ) n and ( φ Yn,v ) n are dynamicallydependent rescalings. Appendix A: Compatibility of covers
Here we prove a more general result than Lemma 4.9.
Definition A.1.
We say that a cover between trees of spheres F (cid:48) : T Y (cid:48) → T Z (cid:48) is compatible with a cover between trees of spheres F : T Y → T Z of samedegree and write F (cid:48) (cid:67) F if • T Y (cid:48) (cid:67) T Y , • T Z (cid:48) (cid:67) T Z , and • F (cid:48) = F | Y . For two vertices v, v (cid:48) of a combinatorial tree T , we define [ v, v (cid:48) ] to be thearc between v and v (cid:48) and ] v, v (cid:48) [:= [ v, v (cid:48) ] \ { v, v (cid:48) } . We define the annulus] | v, v (cid:48) | [ to be the connected component of T \ { v, v (cid:48) } containing ] v, v (cid:48) [. Fora cover between trees of spheres F we denote by V F the set of leaves whichare the images of the critical leaves of F .16 heorem A.2. Let F : T Y → T Z be a cover between trees of spheres and Z (cid:48) be a set with at least three elements containing V F . There exists a uniquecover between trees of spheres F (cid:48) : T F − ( Z (cid:48) ) → T Z (cid:48) such that F (cid:48) (cid:67) F . Proof.
We define T Z (cid:48) to be the tree of sphere whose set of leaves is Z (cid:48) , whose internal vertices are the vertices of T Z separating at least threeelements of Z (cid:48) and whose edges and their attaching points are such that T Z (cid:48) (cid:67) T Z . We define Y (cid:48) := F − ( Z (cid:48) ) and T Y (cid:48) whose vertices are the preimageby F of the one of T Z (cid:48) . Clearly the internal vertices of T Y (cid:48) are the verticesof T Y who separate at least three elements of Y (cid:48) . We define again the edgesand their attaching points of T Y (cid:48) to be such that and T Y (cid:48) (cid:67) T Y . We justhave to prove that F (cid:48) := F | T Y is a combinatorial tree map and that thecorresponding F (cid:48) := F | T Y is a cover between trees of spheres.Consider an edge of T Y (cid:48) between two vertices v and v (cid:48) . As V F ⊂ Z (cid:48) , theset Y (cid:48) contains all of the critical leaves of F and the annulus ] | v, v (cid:48) | [ does notcontain any critical leave. Then the result follows from Lemma A.3 below. (cid:3) Lemma A.3.
If Suppose that [ v, v (cid:48) ] is an arc and that ] | v, v (cid:48) | [ does not con-tain any critical leaf, then all of the elements of [ v, v (cid:48) ] have same degree and F ([ v, v (cid:48) ]) = [ F ( v ) , F ( v (cid:48) )] . Proof.
The critical internal vertices form paths between critical leaves (see[Arfb]). Hence, as ] | v, v (cid:48) | [ does not contain critical leaves, its only criticalvertices can be on ] v, v (cid:48) [. It follows that for any v (cid:48)(cid:48) ∈ ] v, v (cid:48) [, the map f v (cid:48)(cid:48) hasexactly two critical points, so these ones have same degree and all the f v (cid:48)(cid:48) have same degree.The image of [ v, v (cid:48) ] is a connected graph that contains F ( v ) and F ( v (cid:48) ) so[ F ( v ) , F ( v (cid:48) )] ⊂ F ([ v, v (cid:48) ]). Suppose that F ([ v, v (cid:48) ]) has an end F ( w ) differentthan F ( v ) and F ( v (cid:48) ). Then F ( w ) has an unique adjacent edge in F ([ v, v (cid:48) ])whose attaching point is thus the image of the two only critical points of f v (cid:48)(cid:48) which is no possible, so we have a contradiction. (cid:3) For every cover F , let us denote by Π F , F (cid:48) ( F ) the cover F (cid:48) associated to F by the previous theorem. In fact we have proven the following result: Theorem A.4.
The application Π F , F (cid:48) : rev F → rev F (cid:48) is continuous. eferences [Arfa] Matthieu Arfeux. Dynamics on trees of spheres. Submitted.[Arfb] Matthieu Arfeux. Approximability of dynamical systems betweentrees of spheres. Submitted.[Arfc] Matthieu Arfeux. Compactification and trees of spheres covers. Sub-mitted.[Arfd] Matthieu Arfeux. Berkovich spaces and Deligne-Mumford compact-ification. on Arxiv.[BF13] Bodil Branner and N´uria Fagella. Quasiconformal Surgery in Holo-morphic Dynamics , volume 141 of
Cambridge studies in advancedmathematics . Cambridge, first edition, 2013.[CP] Guizhen Cui and Wenjuan Peng. On the cycles of components ofdisconnected julia sets. In preparation.[God] S´ebastien Godillon. A family of rational maps with buried Juliacomponents.
Ergodic Theory Dynam. Systems
35, no. 6, 1846-1879,2015.[Kiw15] Jan Kiwi. Rescaling limits of complex rational maps.
Duke Math.J. , 164(7):1437-1470, 2015.[Mil06] John Milnor.
Dynamics in one complex variable , volume 160 of
An-nals of Mathematics Studies . Princeton University Press, Princeton,NJ, third edition, 2006.[Shi89] Mitsuhiro Shishikura. Trees associated with the configuration ofHerman rings.
Ergodic Theory Dynam. Systems , 9(3):543–560, 1989.[Shi02] Mitsuhiro Shishikura. A new tree associated with Herman rings.