Modeling core parts of Zakeri slices I
Alexander Blokh, Lex Oversteegen, Anastasia Shepelevtseva, Vladlen Timorin
aa r X i v : . [ m a t h . D S ] F e b MODELING CORE PARTS OF ZAKERI SLICES I
ALEXANDER BLOKH, LEX OVERSTEEGEN,ANASTASIA SHEPELEVTSEVA, AND VLADLEN TIMORIN
To the memory of A.M. Stepin
Abstract.
The paper deals with cubic 1-variable polynomialswhose Julia sets are connected. Fixing a bounded type rotationnumber, we obtain a slice of such polynomials with the origin beinga fixed Siegel point of the specified rotation number. Such slices asparameter spaces were studied by S. Zakeri, so we call them
Zakerislices . We give a model of the central part of a slice (the subsetof the slice that can be approximated by hyperbolic polynomialswith Jordan curve Julia sets), and a continuous projection from thecentral part to the model. The projection is defined dynamicallyand is coherent with the dynamical-analytic parameterization ofthe Principal Hyperbolic Domain by Petersen and Tan Lei. Introduction
In this introduction we assume a certain level of familiarity withcomplex dynamics; detailed definitions will be given later on.For a polynomial P denote by r P s its affine conjugacy class. Bythe degree d polynomial parameter space one understands the space ofsuch classes of polynomials of degree d . Similarity between quadraticdynamical planes and slices of parameter spaces of higher degree poly-nomials is a recurring topic of research. A now standard mechanism(found in [BH01]) uses holomorphic renormalization. If, say, a cu-bic polynomial P is immediately renormalizable (i.e., has a connectedquadratic-like filled Julia set K ˚ p P q ), then one critical point of P be-longs to K ˚ p P q . The other critical point of P may eventually map to K ˚ p P q in which case P belongs to a quasiconformal copy of K ˚ p P q contained in the parameter space of cubic polynomials. A more gen-eral renormalization scheme established in [IK12] allows to find copies Date : February 15, 2021.2010
Mathematics Subject Classification.
Primary 37F45, 37F20; Secondary37F10, 37F50.
Key words and phrases.
Complex dynamics; Julia set; cubic polynomial; Siegeldisk; connectedness locus, external rays. of M ˆ M (where M is the quadratic Mandelbrot set) or MK (theset of pairs p c, z q , where c P M , and z belongs to the filled Julia set K p P c q of P c p w q “ w ` c ) in the cubic connectedness locus. In thenon-renormalizable case, things are much subtler.Suppose that a cubic polynomial P has a non-repelling fixed point a . It can always be arranged by a suitable affine conjugacy that a “ A p z q “ αz , where α P C zt u ,that leave 0 fixed. Much is known if P is renormalizable; this case,under the additional assumption that P tunes a hyperbolic polynomial,is considered in [IK12, SW20]. The remaining, non-renormalizable case,needs closer attention. Consider the set of all affine conjugacy classes r P s of cubic polynomials P with P p q “ | P p q| ď
1. A centralpart of this parameter space, analogous to the interior of the maincardioid, is the principal hyperbolic component consisting of classes r P s for all hyperbolic P with | P p q| ă r P s with | P p q| ă
1. This paper aimsat a similar description in the Siegel case under the assumption thatthe associated rotation number has bounded type.A powerful method of studying polynomials with non-repelling peri-odic points is based upon linearizations. Consider a polynomial f withattracting or neutral fixed point a (we discuss polynomials, but a lotof the results are in fact more general). A linearization is a holomor-phic map ψ of an open disk D p r q of radius r ą ψ p q “ a , and ψ p λz q “ f ˝ ψ p z q for all z P D p r q where λ “ f p a q .Assume that r ą ψ at 0. It is known that ψ : D p r q Ñ C is an embedding, cf. [Che20].Then ψ p D p r qq is called the linearization domain ∆ p f, a q of f around a . If | λ | ă
1, then ∆ p f, a q is compactly contained in the attractingbasin of a , and B ∆ p f, a q contains a critical point. In the case a “ p f, a q is denoted by ∆ p f q .Fix λ with | λ | ď
1. Let C λ be the space of complex linear conjugacyclasses of complex cubic polynomials with fixed point 0 of multiplier λ (alternatively, C λ consists of affine conjugacy classes of cubic polyno-mials with marked fixed point of multiplier λ ). For a cubic polynomial P p z q “ λz ` . . . , let r P s be its class in C λ . Write C λ Ă C λ for the connectedness locus in C λ . That is, r P s P C λ if the Julia set J p P q of P is connected. A central part of C λ is the set P λ of all r P s P C λ that ODELING CORE PARTS OF ZAKERI SLICES I 3 lie in the closure of the principal hyperbolic component. We are inter-ested in understanding the topology and combinatorics of P λ througha comparison with a suitable dynamical object.As the basis for comparison, consider the space of quadratic poly-nomials Q p z q “ Q λ p z q “ λz p ´ z { q . Then λ is the multiplier of thefixed point 0 of Q . Suppose that either | λ | ă λ “ e πiθ , where θ P R { Z is of bounded type. The finite critical point of Q is 1, thusthe linearizatiton domain ∆ p Q q around 0 contains 1 in its boundary[Pet96]. Let ψ “ ψ Q : D Ñ ∆ p Q q be the corresponding lineariza-tion (here D “ D p q ). By [Pet96], the set ∆ p Q q is locally connected.Therefore, ∆ p Q q is a Jordan disk (cf. [Mil06, Lemma 18.7]), and theRiemann map extends to a homeomorphism ψ : D Ñ ∆ p Q q . We nor-malize ψ so that ψ p q “
1. If | λ | “
1, then the map ψ conjugates therigid rotation by angle θ with the restriction of Q to ∆ p Q q . Considerthe quotient ˜ K p Q q of the set K p Q qz ∆ p Q q by the equivalence relation „ defined as follows. Two different points z , w are equivalent if bothbelong to B ∆ p Q q , and Re p ψ ´ p z qq “ Re p ψ ´ p w qq . Main Theorem.
There is a natural continuous map Φ λ : P λ Ñ ˜ K p Q q . In this paper we do not address the issue of Φ λ being surjective ormonotone — a discussion of these properties is postponed to a laterpublication. The meaning of Φ λ being natural can be made precise: tocall it natural we require that Φ λ satisfies Property D stated below. Property D.
For any P P C λ with r P s P P λ , there exist a full P -invariant continuum X p P q containing a critical point c of P and acontinuous monotone map η P : X p P q Ñ K p Q q such that η P semi-conjugates f | X p P q with Q | η P p X p P qq , and Φ λ pr P s q is the image of η P p c q in ˜ K p Q q . In other words, quoting Douady, we “seed in the dynamical planeand reap the harvest in the parameter plane” . The letter D in PropertyD stands for “Dynamics” (or “Douady”).It can be observed that the Main Theorem is a direct (partial) ex-tension of [PT09]. According to [PT09], C. Petersen, P. Roesch andTan Lei planned a continuation that should have contained an ana-log of the above Main Theorem for parabolic slices. Apparently, thiscontinuation never appeared in print.2.
Background and a specification of the Main Theorem s:bcg
Take λ “ e πiθ , where θ P R z Q . Let p n { q n be the sequence of rationalapproximations of θ based on the continued fraction expansion. By the A. BLOKH, L. OVERSTEEGEN, A. SHEPELEVTSEVA, AND V. TIMORIN
Brjuno–Yoccoz theorem [Yoc95], a holomorphic germ f with f p q “ f p q “ λ is linearizable at 0 if and only if ÿ n “ log q n ` q n ă 8 . The latter condition is called the
Brjuno condition ; if θ satisfies it, θ issaid to be a Brjuno number . So, if θ is a Brjuno number, then a poly-nomial f with f p q “ f p q “ λ has a Siegel disk ∆ p f q around 0.Say that θ is bounded type if the continued fraction coefficients of θ arebounded. Any bounded type irrational number is Brjuno; the converseis not true (cf. [Yoc95]). From now on and throughout the paper, weset λ “ e πiθ and assume that θ is bounded type.Let us discuss the parameter slices that are of interest in this paper,and how to parameterize them. Start with the quadratic case. Recallthat a quadratic polynomial with a fixed point of multiplier λ is uniqueup to an affine conjugacy. We use the normalization Q λ p z q “ λz p ´ z { q with the property that 0 is the fixed point of multiplier λ and thefinite critical point of Q λ is 1. t:Pe Theorem 2.1 (Theorem A [Pet96]) . The Julia set J p Q λ q is locally con-nected and has zero Lebesgue measure. Moreover, the Siegel disk ∆ p Q λ q is a quasidisk. To parameterize cubic polynomials with fixed point at 0 and markedcritical points, we work with the space C ˚ λ of polynomials P c,λ p z q “ P c p z q “ λz ˆ ´ ˆ ` c ˙ z ` c z ˙ (we often fix λ and then omit it in the notation). The parameter c ischosen so that P c has critical points 1 and c , and λ is the multiplierof the fixed point 0. It is easy to verify that P c and P { c are linearlyconjugate and, hence, r P c s “ r P { c s . The map c ÞÑ P c establishes anisomorphism between C ˚ “ C zt u and C ˚ λ .If | λ | ă
1, then 0 is an attracting fixed point of P c . Let us nowfix λ “ e πiθ , where θ P R { Z is of bounded type, and describe theresults of [Zak99] where this case was studied in great detail. First,the disk ∆ p P c q is non-degenerate, at least one critical point of P c still belongs to B ∆ p P c q , and ∆ p P c q is a quasidisk. Let Z cλ be the set t P c P C ˚ λ | t c, u Ă B ∆ p P c qu . The set Z cλ , called the Zakeri curve , is aJordan curve. It divides the punctured plane C ˚ λ into two components, O ˚ λ p q and O ˚ λ p8q , each isomorphic to the punctured disk D zt u . Thecorresponding punctures are c “ c “ 8 , respectively. ODELING CORE PARTS OF ZAKERI SLICES I 5
Since P c and P { c are linearly conjugate, the involution c ÞÑ { c interchanges the punctured disks O ˚ λ p q and O ˚ λ p8q and maps Z cλ toitself. Observe that P and P ´ always belong to Z cλ . Moreover, thefollowing holds:(1) if c P O ˚ λ p q then c P B ∆ p P c q and 1 R B ∆ p P c q ,(2) if c P Z cλ then c, P B ∆ p P c q , and(3) if c P O ˚ λ p8q then c R B ∆ p P c q and 1 P B ∆ p P c q .Every class in C λ is represented by polynomials P c , P { c P C ˚ λ (withsuitable c ). Thus, the space C λ identifies with a quotient of C ˚ λ . Thecorresponding quotient map τ identifies P c with P { c . It restricts tohomeomorphisms on O ˚ λ p q and on O ˚ λ p8q and folds Z cλ to a simplearc Z λ . Moreover, the quotient projection is two-to-one on Z cλ excepttwo points P and P ´ . The space C λ can be described as τ p O ˚ p8q Y Z cλ q ; this description will be often used in the sequel. This topologicaldescription of C λ as a quotient surface can be upgraded to a complexanalytic description, in the spirit of [PT09].Recall that C λ Ă C λ is the connectedness locus in C λ ; write C cλ Ă C ˚ λ for the corresponding connectedness locus in O ˚ λ p8q Y Z cλ . In otherwords, C cλ consists of polynomials P P O ˚ λ p8q Y Z cλ such that K p P q isconnected. The superscript “c” in the notation C cλ means that c is thefree critical point (the other critical point 1 is associated with the Siegelpoint 0). More generally, this superscript appears in the notation ofa parameter space object if this object belongs to (or is contained in) C ˚ λ (in particular, critical points are marked), and c can be regardedas a free critical point. Note that Z cλ Ă C cλ as for c P Z cλ we have c, P B ∆ p P c q (and, hence, both critical points of P c are non-escaping).The set C λ coincides with the image of C cλ under the quotient map τ .We want to describe the structure of the connectedness loci C cλ and C λ . t:Za-qs Theorem 2.2 ([Zak99]) . If P P C cλ , then P B ∆ p P q and B ∆ p P q is aquasicircle depending continuously on P P C cλ in the Hausdorff metric. Define the set P cλ as the subset of C cλ consisting of polynomials thatcan be approximated by sequences P n P C cλ n with | λ n | ă P n in the immediate basin of 0. This is the centralpart of C cλ which we want to model. It is easy to see that P cλ is acontinuum containing P (indeed, polynomials P rλ, converge to P as r Õ λ is fixed, it can be omitted from the notation of Q “ Q λ .We will define a continuous map Φ cλ from P cλ to the model space K p Q qz ∆ p Q q . Note that, in the case of marked critical points, the A. BLOKH, L. OVERSTEEGEN, A. SHEPELEVTSEVA, AND V. TIMORIN model space is simpler as we do not pass to a quotient. This mapis conjecturally a homeomorphism. As often happens in holomorphicdynamics, the definition of Φ cλ depends on a certain map between dy-namical planes. More precisely, we will define a P -invariant continuum X p P q Ă K p P q and a semi-conjugacy η P : X p P q Ñ K p Q q . The follow-ing theorem makes the Main Theorem more specific. t:main1 Theorem 2.3.
The map η P : X p P q Ñ K p Q q is monotone for every P P C cλ . For every P c P P cλ , the critical point c is in X p P c q . The map Φ cλ : P c ÞÑ η P c p c q is defined and continuous on P cλ . It takes values in K p Q qz ∆ p Q q . The Main Theorem follows directly from Theorem 2.3 by applyingthe quotient projection τ from C ˚ λ to C λ . Plan of the paper.
In Section 3, the principal tools of this paper aredeveloped. These include bubbles, legal arcs, and Siegel rays. To anextent, Siegel rays compensate for the absence of repelling cutpointsin the central part of K p P c q (here P c P C cλ ). They form a controllablecombinatorial structure, map forward in a regular way, and divide thecentral part of K p P q into smaller pieces. Section 4 discusses the issuesof stability. Roughly speaking, a dynamically defined set is stable if itmoves continuously as we change the parameters. Outside of the Zakericurve, stability can be defined in customary language of holomorphicmotions. However, since C cλ is not a Riemann surface at points ofthe boundary curve Z cλ , we need to consider a more general notionof an equicontinuous motion. The principal results of Section 4 claimthat Siegel rays are stable. Section 5 deals with the dynamical map η P : X p P q Ñ K p Q q defined on the central part X p P q of K p P q . Inparticular, property D is established for this map. Finally, Section 6concludes the proof of Theorem 2.3 and the Main Theorem.3. Bubbles s:bubss:petzak
An overview of [Zak99] . Consider generalized Blaschke products ,i.e., products of generalized Blaschke factors z ´ p ´ pz without assumingthat | p | ă p has to belong to D ). Thegeneralized Blaschke products have some common properties with theclassical Blaschke products. In particular, they have the inversion self-conjugacy: if B p z q is a generalized Blaschke product, then B p z q “ B p z q ,i.e., the inversion z ÞÑ z conjugates B with itself. It follows that criticalpoints of B are split in two groups: critical points inside D and theirinversions with respect to S that are located outside D . ODELING CORE PARTS OF ZAKERI SLICES I 7
In order to describe the structure of C cλ , Zakeri introduced an auxil-iary family of degree 5 generalized Blaschke products given by B p z q “ e πit z ˆ z ´ p ´ pz ˙ ˆ z ´ q ´ qz ˙ , where | p | ą , | q | ą
1. In addition to the inversion self-conjugacy (and,hence, the inversion symmetry of their critical points) the restrictionsof these maps on S (which is invariant) are homeomorphisms. Indeed,by the Argument Principle, the topological degree of B : S Ñ S is equal to the number of zeros minus the number of poles (countingmultiplicities) of B in D . The latter number is 3 ´ “ { p and 1 { q ).Zakeri chooses p and q so that B has a multiple critical point in S and two critical points c B , 1 { c B that may or may not belong to S . Theangle t P R { Z is adjusted so that B : S Ñ S has rotation number θ .Consider the Blaschke products B as above, with marked critical pointsand normalized (via conjugation by a rigid rotation) so that one of thecritical points is 1. Then the space B λ of all such B ’s is parameterizedby a single complex parameter µ P C z D (recall that λ “ e πiθ ; thus thedependence on θ is expressed through λ ) such that the critical pointsof B µ are µ and 1. Note also that B µ and B { µ are linearly conjugatefor µ P S but we distinguish them as elements of B λ .By a theorem of ´Swi¸atek and Herman [Swi98], the map B : S Ñ S is quasi-symmetrically (qs) conjugate to a rigid rotation. Consider aqc-extension H “ H B : D Ñ D of this quasi-symmetric conjugacy (takethe Douady–Earle extension [DE86] to make the construction unique),and define the modified Blaschke product ˜ B as B p z q for | z | ě H ´ ˝ Rot θ ˝ H p z q for | z | ă
1. Here Rot θ is the rigid rotation about 0by angle θ . Finally, ˜ B is shown to be qc conjugate to a cubic polynomial P P C cλ by finding a certain ˜ B -invariant conformal structure σ on C ,and straightening it. Here the critical point c of P corresponds to thecritical point µ of B . Define the non-escaping locus of B λ as the set of B µ P B λ for which the orbit of µ is bounded. Set P “ S p B q ; the map S from the non-escaping locus of B λ to C cλ is called the surgery map .The following proposition is Corollary 10.5 of [Zak99]. p:C10.5Za Proposition 3.1.
There is an equicontinuous family of qc homeomor-phisms ϕ B : C Ñ C parameterized by B in the non-escaping locus of B λ such that S p B q “ ϕ B ˝ ˜ B ˝ ϕ ´ B , and normalized so that ϕ B p q “ . Set P “ S p B q . Note that the Siegel disk ∆ p P q of P equals ϕ B p D q ,and the Riemann map ψ ∆ p P q : D Ñ ∆ p P q coincides with ϕ B ˝ H ´ B . All H B are quasi-conformal with the same qc constant that depends only A. BLOKH, L. OVERSTEEGEN, A. SHEPELEVTSEVA, AND V. TIMORIN on θ . It follows (cf. Theorem 4.4.1 of [Hub06]) that H B and H ´ B forman equicontinuous family. We obtain the following corollary. c:equicont Corollary 3.2.
The extended Riemann maps ψ ∆ p P q (where P variesthrough C cλ ) form an equicontinuous family. Let C p D , C q be the space of all continuous maps from D to C withthe sup-norm. Corollary 3.2, in turn, implies the following. c:sup-conv Corollary 3.3.
The map from C cλ to C p D , C q taking P to ψ ∆ p P q iscontinuous.Proof. Suppose that a sequence P n P C cλ converges to P P C cλ . Wewant to prove that ψ n “ ψ ∆ p P n q converge uniformly to ψ “ ψ ∆ p P q .First note that ψ n p q “ ψ p q “ k and consider the point z “ P k p q P B ∆ p P q . Clearly, for fixed k and all large n , the points z n “ P kn p q P B ∆ p P n q are close to z . Since ψ n conjugate the rotationby angle θ with P n | ∆ p P n q , we necessarily have ψ n p e πikθ q “ z n . Similarly, ψ p e πikθ q “ z . Thus ψ n Ñ ψ point-wise on a dense subset of S . Byequicontinuity, it follows that ψ n Ñ ψ uniformly on S . Finally, by theMaximum Modulus Principle, ψ n Ñ ψ on D . (cid:3) Polar coordinates and bubbles.
Let U Ă C be an open topo-logical disk equipped with a distinguished center a P U , a certain radius r U P p , and a base point b P B U accessible from U . Anopen topological disk U equipped with these data is called a frameddomain . These data constitute a framing of U . For a framed domain U , consider the Riemann map ψ U : D p r U q Ñ U such that ψ p q “ a andlim u Ñ r U ψ U p u q “ b with u staying within a Stolz angle at r U . If U is aJordan disk, then ψ U extends to a homeomorphism ψ U : D p r U q Ñ U . d:polar Definition 3.4 (Polar coordinates, internal rays) . Let U be a frameddomain with center a P U , root point b , and radius r U . A point z P U has the form ψ U p ρ z e πiθ z q for some ρ z P r , r U q and θ z P R { Z . The polarradius function is by definition the function z ÞÑ ρ z on U . We alwaysextend this function (keeping the notation) to U by setting ρ z “ r U forall z P B U . The polar angle function is by definition the function z ÞÑ θ z on U zt a u . Note that this function is undefined when ρ z “
0. If U is aJordan disk (and only in this case), we extend the polar angle functionto U by continuity. Then, for z P B U , the angle θ z is determined by therelation z “ ψ U p r U e πiθ z q . Given any α P R { Z define the internal ray R U p α q as the set t z P U | θ z “ α u . Say that R U p α q lands at a point ODELING CORE PARTS OF ZAKERI SLICES I 9 w P B U if w is the only point in R U p α qz U . If U is a Jordan disk, thenevery internal ray R U p α q lands at the point ψ U p r U e πiα q .Assume now that either f : C Ñ C is in C cλ , or f “ Q λ . Recallthat λ “ e πiθ is fixed. Write ∆ p f q for the Siegel disk of f , and ψ f : D Ñ ∆ p f q for the Riemann map normalized so that ψ f p q “ ψ f p q “
1; recall that 1 is a critical point of f .Define a pullback of a connected set A Ă C under a polynomial f asa connected component of f ´ p A q . An iterated pullback of A under f is by definition an f n -pullback of A for some n ą d:bub-coor Definition 3.5 (Bubbles and polar coordinates on bubbles) . Bubbles of f are iterated pullbacks of ∆ p f q (thus, bubbles are open Jordandisks). Let A be a bubble of f , and let n be the smallest integer with f n p A q “ ∆ p f q . Such n is called the generation of A and denoted byGen p A q . For z P A z f ´ n p q , set θ z “ θ f n p z q ´ nθ and ρ z “ ρ f n p z q . Now,if z has polar coordinates ρ and α , then f p z q has polar coordinates ρ and α ` θ . Equivalently, the complex coordinate ρe πiα multipliesby λ under the action of f . Note that the polar radius extends as acontinuous function on the union of the closures of all bubbles.If a bubble A is a homeomorphic iterated pullback of ∆ p f q , thenwe define a framing of A as follows. The center o A of A is defined asthe only iterated preimage of 0 in A . If f n p A q “ ∆ p f q , then the basepoint of B A is defined as the point b A with f n p b A q “ f n p q . With thisframing, internal rays of A are defined. By definition, an internal rayof A consists of all points with a fixed value of polar angle. Then in A there is one internal ray of a given polar argument, and all internal raysconnect the center of A with appropriate points on B A . In particular,this picture holds for all bubbles in the quadratic case.In the case of a cubic polynomial P c there might be a bubble B which contains c and, hence, maps forward two-to-one. In that casethe picture with polar angle function, the center and the internal raysis a bit different. More precisely, if c maps into the center of P c p B q ,then the pullbacks of an internal ray are two internal rays connecting c with appropriate points on B B . Now, suppose that P c p c q is not thecenter of P c p B q . Then the center x “ o P c p B q of P c p B q has two preimages x , x P B . Hence, with one exception, each internal ray of P c p B q pullsback to two internal rays, each connecting the appropriate pullback ofthe center of P c p B q with the appropriate point on B B . The exceptionis the internal ray R “ xy , of argument, say, α , passing through P c p c q ;its pullback is a “cross” with endpoints at x , x and at two preimagesof y , and with vertex at c . Given p ρ, α q , there is unique point z P ∆ p f q and a lot of other pointswith coordinates p ρ, α q . Any point with polar coordinates p ρ, α q mapsto f n p z q under f n , for some n depending on the point.The terms “bubbles” and “bubble rays” were introduced in the The-sis of J. Luo [Luo95] (cf. [AY09, Yan17] for a development of theseideas). However, the difference with our setup is that bubbles in thesense of Luo are Fatou components that are eventually mapped to a su-perattracting rather than a Siegel domain. Also, similar ideas are usedin [BBCO10] where some quadratic Cremer Julia sets were studied byapproximating them with Siegel Julia sets with specific properties.If f n : A Ñ ∆ p f q is a conformal isomorphism, then it also definesa framing of A so that the polar coordinates on A just defined areconsistent with this framing. By an oriented arc we mean an arc I whose one endpoint is marked as initial and the other is marked as terminal .Let A be a bubble of a cubic polynomial P c with r P c s P C cλ . Evi-dently, a point z P A can be connected with 0 by an arc I Ă K P withinitial point 0 and terminal point z . While such an arc is not unique,it is easy to see that for any bubble A the intersection I X A is a subarcof I . In what follows we will only consider arcs such that for all ofthe bubbles involved (except possibly for one) the intersection I X A is contained in the union of t o A u and two internal angles of A . Moreprecisely, let us now define legal arcs . d:reg-arc Definition 3.6 (Legal arcs) . Consider an oriented (directed) topolog-ical arc I Ă K p f q . Suppose that I ˝ is an open dense subset of I suchthat the following holds:(1) the set I z I ˝ can accumulate only at the terminal point of I ;(2) each component of I ˝ is contained in one bubble A and coincideswith a component of p A z Ť n ě f ´ n p qq X I .(3) the polar angle function is defined and constant on each com-ponent of I ˝ ;(4) P n p I q is not separated by c for n ě I is called a legal arc . Let α , . . . , α k , . . . be the values of thepolar angle on I ˝ taken in the order they appear on I ˝ . A linear orderof α i s is well defined since I is oriented. The finite or infinite sequence p α , . . . , α k , . . . q is called the (polar) multi-angle of I .Typically, we deal with legal arcs with initial point 0. In the multi-angle p α , . . . , α k , . . . q of I we will always have that α “ α , α “ α etc because these pairs of angles correspond to pairs of internal raysof adjacent bubbles that eventually map onto the same internal ray of ODELING CORE PARTS OF ZAKERI SLICES I 11 ∆ p f q and, hence, have the same polar argument (we make this moreprecise in Lemma 3.7). Legal arcs for polynomials, under the name ofregulated arcs, were introduced by Douady and Hubbard in [DH85a];they play a key role in the definition of a Hubbard tree for a post-critically finite polynomial. We use legal arcs in an essentially differentway.If z P K p f q is such that there is a legal arc I z from 0 to z , thenthe polar multi-angle (or just multi-angle ) of z is defined as the polarmulti-angle of I z . Note that if I z exists, then it is unique. l:polQ Lemma 3.7.
For any z P K p Q q , there is a legal arc I z from to z .Let p α , . . . , α k , . . . q be the multi-angle of z . Then each term α i , exceptpossibly the last term, has the form α i “ ´ m i θ . Here m i are nonneg-ative integers such that m i ` “ m i and m i ` ą m i ` . Moreover, z is uniquely determined by the multi-angle and (if the multi-angle isfinite) by ρ z . A sequence p α , . . . , α k , . . . q with the properties listed in Lemma 3.7is called a legal sequence of angles . We also define a legal angle as anangle of the form ´ mθ , where m is a nonnegative integer. Proof.
Suppose first that z is in the closure of a bubble A of Q . Theargument will use induction on Gen p A q . If A “ ∆ p Q q , then I z is asegment or the closure of some internal ray of ∆ p Q q , the multi-angle of z is p θ z q , and z is determined by θ z and ρ z . Thus we now assume that A ‰ ∆ p Q q . Then I z intersects B ∆ p Q q at a point x that is eventually mappedto 1. Let m be the non-negative integer with Q m p x q “
1, then α “´ m θ . The arc Q m p I z q “ I Q m p z q contains R ∆ p Q q p q , and Q m ` p I z q “ I Q m ` p z q Y R ∆ p Q q p θ q . The multi-angle of Q m p z q starts with 0, 0 sinceboth initial components of I ˝ Q m p z q map onto R ∆ p Q q p θ q . We may assumeby induction that the multi-angle of Q m ` p z q is p ˜ α , . . . , ˜ α k q , where˜ α i “ ´ ˜ m i θ for 2 ď i ă k , and ˜ m i satisfy the desired properties. In thiscase the multi-angle of z is p α , . . . , α k q , where α “ α “ ´ m θ and α i “ ˜ α i ´ p m ` q θ “ ´ m i θ with m i “ ˜ m i ` m ` i ě ď i ă k . Thus, we proved that every point from the closure of everybubble of Q has a multi-angle. Moreover, the latter is a legal sequenceof angles.Suppose now that z P K p Q q is not in the closure of a bubble. Thenthere is a sequence of pairwise different bubbles A , . . . , A k , . . . suchthat A k Ñ t z u in the Hausdorff metric. Moreover, we can assume that A “ ∆ p Q q and A i X A i ` “ t z i u , where z i is eventually mapped to1. Clearly, I z i is an initial segment of I z j with j ą i . Set I z to be theclosure of the union of all I z i ; then I z is a legal arc from 0 to z . It follows that there is an infinite legal sequence of angles such that themulti-angle z i is an initial segment of this sequence. This infinite legalsequence is then the multi-angle of z . Thus, all points of K p Q q havewell-defined multi-angles.Given a legal sequence of angles p α , . . . , α k , . . . q , there is a uniquesequence of bubbles A , . . . , A i , . . . such that the point z i P A i X A i ` has multi-angle p α , . . . , α i q . If the sequence p α i q is infinite, thenthe corresponding sequence of bubbles converges to a unique point z determined by the infinite multi-angle p α i q . If the sequence p α i q isfinite, then it defines a unique last bubble A n in the correspondingsequence of bubbles and an internal ray R “ R A n p α n q or R A n p α n ´ q in A n . All points of R together with the landing point of R (and no otherpoints) have the given multi-angle. These points are then determinedby the legal sequence p α , . . . , α k , . . . q and the polar radius. (cid:3) Sequences of bubbles defined in the proof of Lemma 3.7 for points z P K p Q q , are called bubble rays . ss:bubblerays Bubble rays and bubble chains.
Take P “ P c P C cλ and set Q “ Q λ . Define Y p P q as the set of all points z P K p P q for which thereis a legal arc I z from 0 to z . Note that, by definition, Y p P q includes∆ p P q and is forward invariant: P p Y p P qq Ă Y p P q . However, in general, Y p P q does not have to be closed.Every z P Y p P q has a multi-angle and, if the latter is finite, the polarradius ρ z . Moreover, it is not hard to see that the multi-angle of z is alegal sequence of angles. Set ρ z “ 8 if the multi-angle of z is infinite.Similarly, we set ρ w “ 8 for points w P K p Q q not on the boundaryof a bubble of Q . The map η P : Y p P q Ñ K p Q q takes z P Y p P q to aunique point w “ η P p z q with the same multi-angle and polar radius.By definition of Y p P q and properties of multi-angles and polar radii, η P ˝ P “ Q ˝ η P on Y p P q .Let A be a bubble of generation n . If P n : A Ñ ∆ p P q is one-to-one,then A is called off-critical . If c P A , then A is called critical . Finally, if A is a pullback of a critical bubble, it is said to be precritical . For anybubble A , one can define its root point r p A q . When A is off-critical, theroot point is uniquely defined by the formula P n ´ p r p A qq “
1. When A is critical or precritical, there are legal paths from 0 to some pointsin A . All these paths intersect the boundary of A at the same point;this point is by definition the root point r p A q . There are two points z , z P B A such that P n ´ p z q “ P n ´ p z q “
1, and the point r p A q is oneof them. ODELING CORE PARTS OF ZAKERI SLICES I 13 d:leg
Definition 3.8 (Legal bubbles and bubble correspondence) . A bubble A of P with A X Y p P q ‰ ∅ is called legal . Thus, A is legal if and onlyif r p A q P Y p P q , and P i p r p A qq ‰ c for i ă Gen p A q . If a legal bubble A is off-critical, then A Ă Y p P q . Clearly, η P p A X Y p P qq lies in a uniquebubble A Q of Q . Say that A and A Q correspond to each other. Thiscorrespondence between some bubbles of P and all bubbles of Q iscalled the bubble correspondence . By definition, if A is a legal bubbleof P , then P p A q is also a legal bubble of P . Moreover, if A correspondsto A Q , then P p A q corresponds to Q p A Q q .Define p R { Z q ‹ as the set of finite sequences of angles and p R { Z q N asthe set of infinite sequences of angles. The mapΠ : p R { Z q ‹ ztp qu Ñ p R { Z q ‹ acts as follows. Take ~α “ p α , α , α , . . . q P p R { Z q ‹ . If α “ α “ p ~α q “ p α ` θ, . . . q , otherwise Π p ~α q “ p α ` θ, α ` θ, α ` θ, . . . q .Then any element of p R { Z q ‹ of length ą p q under Π. On p q , the map Π is undefined. Clearly, Π can be alsodefined as a self-map of p R { Z q N , by the same rule.Let A be a legal bubble of P of generation n . If A is off-critical,recall that the center of A is by definition the preimage of 0 under P n : A Ñ ∆ p P q . The incoming radius of any off-critical bubble A isits radius R p A q connecting r p A q to its center. Now, if c P A then P | A is two-to-one. To define the center of A , recall the earlier analysis ofpullbacks of radii into critical bubbles. It follows from that analysisthat there are two cases depending on the mutual location of R p P p A qq and P p c q . If R p P p A qq does not contain P p c q , or P p c q is the centerof P p A q , then there is a unique pullback of R p P p A qq that connects r p A q with a preimage of the center of P p A q , and this preimage of thecenter of P p A q is said to the center of A . The remaining case is when R p P p A qq contains P p c q but P p c q is not the center of P p A q . In thatcase the center of A is not defined.Finally, let A be precritical; then P | A is one-to-one. If the center of P p A q is defined, set the center of A to be the pullback of the center of P p A q into A ; if the center of P p A q is not defined, then the center of A is not defined either. If the center of A is defined, it is denoted o A .The multi-angle of A is defined as the multi-angle of the root pointof A . If A has multi-angle ~α , then P p A q has multi-angle Π p ~α q . We cannow describe multi-angles of legal bubbles. p:ma-bub Proposition 3.9.
Let ~α be a finite legal sequence of angles of oddlength starting with ´ mθ for a nonnegative m P Z . Then ~α is a multi-angle of some legal bubble if and only if no eventual Π -image of aninitial subsequence of ~α is the multi-angle of c . The assumption that ~α starts with ´ mθ is essential if ~α has length1 (otherwise it follows from the definition of a legal sequence). Proof.
By Definition 3.8, a bubble A is legal if and only if no image P i p I r p A q q with 0 ď i ă Gen p A q contains c . On the other hand, c P P i p I r p A q q if and only if the Π i -image of an initial subsequence of ~α isthe multi-angle of c . (cid:3) The concept of a bubble ray was used in the proof of Lemma 3.7. d:bubray
Definition 3.10 (Bubble rays, bubble chains, core curves) . Take a le-gal bubble A of P and a point z P A X Y p P q , z ‰ r p A q . A legal arc I z from 0 to z passes through bubbles A “ ∆ p P q , . . . , A n “ A in thisorder and through no other bubbles. The sequence A , . . . , A n is calleda bubble chain (to z ) . A bubble ray is a sequence A “ p A , A , . . . q oflegal bubbles A i such that A , . . . , A n is a bubble chain, for every finite n . Set Ť A “ Ť i ě A i . Bubble chains and bubble rays for Q “ Q λ aredefined similarly. The core curve of A is defined as the union of I z i ,where z i P A i X A i ` . (Note that I z i Ă I z j for i ă j .) Definition 3.11 (Landing bubble rays) . Consider a bubble ray A “p A i q for P . We say that A lands at a point z if t z u is the upper limitof the sequence A i , that is t z u “ č i A i Y A i ` Y . . .. In general, the right hand side is denoted by lim A and is called the limit set of A . If A lands at z , then we also say that A is a bubble ray to z . Similar definitions apply to the dynamical plane of Q .Consider a bubble ray A “ p A i q for P . If P p A q ‰ A , then wedefine P p A q as p A , P p A q , P p A q , . . . q . Otherwise, P p A q “ A , andwe define P p A q as p A , P p A q , P p A q , . . . q . If P m p Ť A q “ Ť A , then A is said to be periodic of period m . Let I be the core curve of A .If m is the minimal period of A under P , then P m p I q “ I . However, P m : I Ñ I is not one-to-one; I is folded at critical points of P m . l:ubub Lemma 3.12.
For z in the dynamical plane of P , there is at most onebubble chain or a bubble ray to z . ODELING CORE PARTS OF ZAKERI SLICES I 15
Proof.
Suppose that A and A are different bubble rays or bubblechains to z . If A is a bubble ray, then set I to be its core curve; other-wise set I “ I z . The arc I is defined similarly, with A replaced by A .If A ‰ A , then there is a bounded open set U in C whose boundary iscontained in I Y I . By the Maximum Modulus Principle, the sequence P n is bounded on U , hence equicontinuous. We conclude that U is inthe Fatou set, that is, U is in a single bubble, a contradiction. (cid:3) ss:landbubrays Landing of bubble rays.
Recall that P P C cλ . If A “ p A n q is aperiodic bubble ray for P of minimal period m , then, clearly, P m takesseveral first bubbles A , . . . , A k to A , and A k ` to A . In this casewe say that P m shifts bubbles of A by k . We always have k ě t:752 Theorem 3.13 (Theorem 7.5.2 [BFMOT13]) . Let f be a polynomial,let K p f q be connected, and let X Ă J p f q be an invariant continuum.Suppose that X is not a singleton. Then TH p X q contains a rotationalfixed point or an invariant parabolic domain. Here a rotational fixed point means one of the following: ‚ an attracting fixed point; ‚ a repelling or parabolic fixed point where no invariant periodicexternal ray lands; ‚ a Siegel point; ‚ a Cremer point.Theorem 3.13 is related to the following result of [GM93]. Let f bea polynomial of any degree ą
1. Consider the union Σ f of all invariantexternal f -rays with the set Fix f of their landing points. By [GM93], every complementary component A of Σ f contains a unique invariantrotational object (i.e., either a rotational fixed point of f or an invariantparabolic domain) . A subset of Σ f consisting of two rays landing at thesame point and their common landing point is called a cut . The resultof [GM93] just cited can be restated as follows: any pair of differentinvariant rotational objects for f is separated by a cut from Σ f . t:land Theorem 3.14.
Let A be a periodic bubble ray for P . Then A landsat a periodic repelling or parabolic non-rotational point of P .Proof. Let L be the limit set of A . It is easy to see that L and ∆ p P q are disjoint as otherwise some boundary points of some bubbles from A will be shielded from infinity, a contradiction.Let L be of minimal period m , and consider the map f “ P m .It suffices to prove that L is degenerate. Suppose otherwise. Then by Theorem 3.13, the set L contains an f -invariant rotational object T (rotational f -fixed point or an f -invariant parabolic domain). Asabove, construct the set Σ f ; by [GM93], one of its cuts separates T and 0. Evidently, A cannot intersect this cut which implies that L must be located on one side of the cut while T is located on the otherside. A contradiction. Hence L is an f -fixed point. Since it belongsto J p P q , it is not attracting. If it is Cremer or Siegel, then, againrelying on [GM93], we separate L from 0 with a rational cut, again acontradiction. Hence L is an f -fixed repelling or parabolic point a . If a is rotational, then periodic rays landing at a undergo a nontrivialcombinatorial rotation under f . Let W Ą Ť A be a wedge boundedby two consecutive f -rays landing at a . Locally near a , the wedge W is mapped to some other wedge disjoint from W . A contradiction with Ť A Ă W . (cid:3) Stability s:stab
We start with a very general continuity property. Let Rat d be thespace of all degree d rational self-maps of C with the topology of uni-form convergence. We also write Comp for the space of all compactsubsets of C with the Hausdorff metric. Note that the Hausdorff met-ric on Comp as well as the uniform convergence on Rat d are associatedwith the spherical metric on C . The following lemma is basically aconsequence of the Open Mapping property of holomorphic functions. l:Hcont Lemma 4.1.
Consider the map from
Rat d ˆ Comp Ñ Comp given by p f, X q ÞÑ f ´ p X q . This map is continuous.Proof.
Fix p f, X q P Rat d ˆ Comp. Choose ε ą
0. We need to showthat, if δ “ δ p ε q ą p g, Y q is δ -close to p f, X q ,then g ´ p Y q is ε -close to f ´ p X q . Here p g, Y q being δ -close to p f, X q means that g is δ -close to f and Y is δ -close to X . By definition, g ´ p Y q being ε -close to f ´ p X q means that for every point x P f ´ p X q , thereis y P g ´ p Y q that is ε -close to x , and vice versa: for every y with g p y q P Y , there is x P f ´ p X q that is ε -close to y .First, take x P f ´ p X q . Then g p x q is δ -close to f p x q . There is a point y ˚ P Y that is δ -close to f p x q , since Y is δ -close to X . Finally, y ˚ being2 δ -close to g p x q implies the existence of y P g ´ p y ˚ q that is ε -close to x . Moreover, the corresponding choice of δ can be made independentof g . Indeed, by the Open Mapping property, the f -image of the ε -neighborhood of x includes the 4 δ -neighborhood of f p x q . Hence, the ODELING CORE PARTS OF ZAKERI SLICES I 17 g -image of the ε -neighborhood of x incudes the 3 δ -neighborhood of f p x q , and the latter includes the 2 δ -neighborhood of g p x q .Now take y P g ´ p Y q ; the argument is similar to the above. By theassumption, g p y q is δ -close to f p y q , and there is a point x ˚ P X thatis δ -close to g p y q . Since x ˚ is 2 δ -close to f p y q , there is a point x suchthat f p x q “ x ˚ and x is ε -close to y . (cid:3) We will need the following corollary of Lemma 4.1. c:comp-cont
Corollary 4.2.
Suppose that p f, X q P Rat d ˆ Comp , that X is con-nected, and that A is a component of f ´ p X q . If there are no criticalpoints of f in A , then for all p g, Y q close to p f, X q there is a componentof g ´ p Y q close to A and not containing critical points of g .Proof. In what follows, “ ε -close” means “at distance at most ε ”. Forsome ε ą
0, the 5 ε -neighborhood of A maps homeomorphically ontoa neighborhood of X . Take p g, Y q so close to p f, X q that g ´ p Y q is ε -close to f ´ p X q . This is possible by Lemma 4.1. Moreover, we mayassume that g is injective on the 4 ε -neighborhood of A . Let B be acomponent of g ´ p Y q intersecting the 2 ε -neighborhood of A . It followsthat B lies entirely in the 2 ε -neighborhood of A . Indeed, if a pointof B is at distance 2 ε from A , then it cannot be ε -close to f ´ p X q bythe assumption that f is injective on the 5 ε -neighborhood of A . Inparticular, the closest point to any b P B is in A , and this closest pointis ε -close to b . Observe also that B is the only component of g ´ p Y q in the 2 ε -neighborhood of A , since the restriction of g to the latterneighborhood is injective. Now take any a P A , and let b be the closestto a point of g ´ p Y q . Then b is ε -close to a , hence b P B . We see that A and B are ε -close, as desired. (cid:3) Equicontinuous motion.
Let A Ă C be any subset and Λ be ametric space with a marked base point τ . A map p τ, z q ÞÑ ι τ p z q fromΛ ˆ A to C is an equicontinuous motion (of A over Λ ) if ι τ “ id A ,the family of maps τ ÞÑ ι τ p z q parameterized by z P A is equicontinu-ous, and ι τ is injective for every τ P Λ. An equicontinuous motion is holomorphic if Λ is a Riemann surface, and each function τ ÞÑ ι τ p z q ,where z P A , is holomorphic. By the λ -lemma of [MSS83], to define aholomorphic motion, it is enough to require that every map ι τ is injec-tive, and ι τ p z q depends holomorphically on τ , for every fixed z . Thenthe family of maps ι τ is automatically equicontinuous. Suppose nowthat F τ : C Ñ C is a family of rational maps such that F τ p A q Ă A .An equicontinuous motion p τ, z q ÞÑ ι τ p z q is equivariant with respect tothe family F τ if ι τ p F τ p z qq “ F τ p ι τ p z qq for all z P A . If the family F τ is clear from the context, then we also say that the holomorphic motion commutes with the dynamics .We first study the equicontinuous motion of the Siegel disk ∆ p P q .The following theorem is an easy consequence of known results. t:sul Theorem 4.3.
Choose an arbitrary base point P P C cλ and an arbi-trary point z P ∆ p P q . There is an equivariant equicontinuous motion ι P of ∆ p P q over C cλ such that ι P “ id . Moreover, ι P p z q has the samepolar coordinates in ∆ p P q as z in ∆ p P q . If P R Z cλ , then this equicon-tinuous motion is holomorphic on C cλ z Z cλ .Proof. Set ι P p z q “ ψ ∆ p P q ˝ ψ ´ p P q . By Corollary 3.3, the function ι P depends continuously on P with respect to the sup-norm. This meansthat ι P is an equicontinuous motion. The equivariance follows from thefact that ψ ∆ p P q conjugates the rotation by θ with P | ∆ p P q .Suppose now that P R Z cλ and P runs through C cλ z Z cλ . The P -orbitof 1 moves holomorphically with P P C cλ z Z cλ ; By [MSS83], this motionextends to an equivariant holomorphic motion of B ∆ p P q , cf. [Che20].By a remark of D. Sullivan [Sul] (see also [Zak16]), there exists anequivariant holomorphic motion ι P : ∆ p P q Ñ ∆ p P q that extends theholomorphic motion of B ∆ p P q and is such that ι P : ∆ p P q Ñ ∆ p P q isa conformal isomorphism taking 0 to 0 and 1 to 1. By the uniquenessof the Riemann map the map ι P is the same as before. In particular, ι P preserves the polar coordinates. (cid:3) Stability of legal arcs.
The equicontinuous motion of ∆ p P q ex-tends to some other dynamically defined subsets. Definition 4.4 (Stability) . Consider P P C cλ and a subset A Ă Y p P q .Since Y p P q is by definition forward invariant, it follows that P n p A q Ă Y p P q for all n ě
0. Set B “ Ť n ě P n p A q . Say that A is stable (or λ -stable ) if there is an equivariant equicontinuous motion t ι BP u of B over an open neighborhood of P in C cλ such that, for every z P B ,the point z x P y “ ι BP p z q has the same multi-angle and the same polarradius as z . Clearly, if such an equicontinuous motion exists, then it isunique. Write A x P y for ι BP p A q etc. If the equicontinuous motion is infact holomorphic, then say that A is holomorphically stable . l:regarc-mov Lemma 4.5.
Take P P C cλ and a point z P Y p P q that has a finitemulti-angle. If z is never mapped to c , or if c P ∆ p P q , then the legalarc I z from to z in K p P q is stable. It is holomorphically stable if P R Z cλ .Proof. Suppose that ~α “ p α , . . . , α k q is the multi-angle of z . If k “ ODELING CORE PARTS OF ZAKERI SLICES I 19 k ą
0. Since ~α is a legal sequence of angles, Π m p ~α q has length onefor some minimal integer m ą
0. The subsequent argument employsinduction on both m and k . Let w be the last point of I z with multi-angle p α , . . . , α k ´ q . Then I z is the concatenation of I w (=the legalarc from 0 to w ), and I r w,z s (=the legal arc from w to z ). By inductionon k , assume that I w is stable. In particular, w x P y is defined for all P close to P , and w x P y has the same multi-angle and polar radius as w .By induction on m , assume that P p I z q , hence also T “ P p I r w,z s q , arestable. Thus T x P y depends continuously on P in the Hausdorff metric.Define I r w,z s x P y as the P -pullback of T x P y containing w x P y . If P isclose to P , then, by Corollary 4.2, the set I r w,z s x P y is close to I r w,z s . Inparticular, I r w,z s x P y connects w x P y with a point z x P y that is close to z .Moreover, I r w,z s x P y contains no critical points of P and maps forwardby P in a homeomorphic fashion. It follows that a suitable inversebranch of P on T x P y defines an equicontinuous motion of I r w,z s .Thus both I w and I r w,z s are stable. It follows that their concatenation I z is also stable, as desired. If P R Z cλ , then the argument goes throughwith “equicontinuous” replaced by “holomorphic”. (cid:3) We now discuss stability of infinite periodic legal arcs. t:stab
Theorem 4.6.
Let A be a periodic bubble ray for P P C cλ landing at arepelling periodic point x . Suppose that P i p I x q does not contain c for i ě . Then I x is stable; it is holomorphically stable if P R Z cλ . Note that, under assumptions of Theorem 4.6, the arc I x is the corecurve of A . Proof.
Let m be the minimal P -period of A , then P m p x q “ x . Since x is repelling, there is a small round disk D around x such that D Ť P m p D q , and the map P m : D Ñ P m p D q is a homeomorphism. Supposethat A “ p A n q . Since A lands at x , then A n Ă D for all n ě N forsome N . Also, P m shifts bubbles of A by a certain integer k ě y P I x such that I y “ I x X A Y ¨ ¨ ¨ Y A N ` k . By Lemma 4.5, the legal arc I y is stable. In particular, for P close to P , there is a legal arc I y x P y close to I y and with the same multi-angle.Moreover, I y x P y passes through legal bubbles A x P y , . . . , A N ` k x P y of P and terminates at y x P y . Consider the point z “ P m p y q P I y and the segment I r z,y s of I x from z to y . Then I r z,y s is also stable, thecorresponding segment I r z,y s x P y for P connects z x P y with y x P y . Notealso that I r z,w s Ă D (the point z belongs to the closure of A N ). If P is close to P , then D Ť P m p D q , and P m : D Ñ P m p D q is ahomeomorphism. Abusing notation, write P ´ m for the inverse of thishomeomorphism. Then P ´ m is a well-defined holomorphic map on D depending analytically on P . Since x is repelling, it is stable, so thatthere is a nearby repelling point x x P y for P of the same period. Set I x x P y “ I y x P y Y P ´ m p I r z,y s x P yq Y P ´ m p I r z,y s x P yq Y ¨ ¨ ¨ Y t x x P yu . Every term in the right-hand side moves equicontinuously with P aslong as P stays close to P . The infinite union moves equicontinu-ously since for P ´ m the point x x P y is attracting (the iterates cannotinflate the modulus of continuity). It is also clear that the motion isholomorphic provided that P R Z cλ and P is close to P . (cid:3) Stability of Siegel rays.
Theorem 4.6 parallels a classical resulton stability of periodic external rays landing at repelling points. l:rep
Lemma 4.7 ([DH85a], cf. Lemma B.1 [GM93]) . Let P be a polyno-mial, and z be a repelling periodic point of P . If an external ray R P p θ q with rational argument θ lands at z , then, for every polynomial P sufficiently close to P , the ray R P p θ q lands at a repelling periodicpoint z x P y of P close to z , and z x P y depends holomorphically on P . Consider a periodic bubble ray A for P and its core curve I . ByTheorem 3.14, the bubble ray A lands at a repelling or parabolic point a . Let m be the minimal period of A , then P m p a q “ a . Clearly, I alsolands at a , and it is easy to see that I “ I a is a legal arc from 0 to a . Definition 4.8 (Siegel rays) . Let I and a be as above. By the classicalLanding Theorem for polynomials (see e.g. [Mil06, Theorem 18.11]),one or several periodic external rays for P land at a . Let R be anexternal ray landing at a . Then I Yt a uY R is a simple curve connecting0 with . It is called a Siegel ray . The argument of the Siegel ray I Y t a u Y R is defined as the argument of R .The following are immediate properties of Siegel rays. Every Siegelray originates at 0 and extends to . Every Siegel ray contains preciselyone periodic point a ‰
0; this point a is repelling or parabolic. Twodifferent Siegel rays may have some initial segment in common. Theybranch off either at an iterated preimage of 0 or at a landing point ofsome bubble ray. An external ray for P is either disjoint from a Siegelray or lies in the Siegel ray.Theorem 4.9 below follows from Theorem 4.6 and Lemma 4.7. t:S-stab Theorem 4.9.
Let Σ be a Siegel ray for P P C cλ . Suppose that the non-zero periodic point in Σ is repelling. Then, for all P P C cλ sufficiently ODELING CORE PARTS OF ZAKERI SLICES I 21 close to P , there is a Siegel ray Σ x P y close to Σ in the spherical metricand having the same argument. Moreover, the periodic point in Σ x P y depends holomorphically on P provided that P R Z cλ . The only reason we require that P R Z cλ in Theorem 4.9 is thatholomorphic functions are defined on Riemann surfaces, and C cλ z Z cλ rather than C cλ has a natural structure of a Riemann surface.4.4. Siegel wedges.
Let Σ and Σ be two Siegel rays for P . By def-inition, they originate at 0 and extend all the way to infinity. Let b be the point where Σ and Σ branch off. Assume that b is an iteratedpreimage of 0 rather than a periodic repelling or parabolic point. Con-sider a wedge W bounded by segments of Σ and Σ from b to infinity.Notice that there are two such wedges; either wedge is called a Siegelwedge (bounded by Siegel rays Σ and Σ ) . We also say that b is the rootpoint of W . Set I b “ Σ X Σ ; of the two wedges bounded by Σ and Σ one contains I b and the other one is disjoint from I b . If W is a Siegelwedge bounded by Σ and Σ and disjoint from I b , call the multi-angleof b the multi-angle of W . Otherwise (i.e. if W contains I b zt b u ), we setthe multi-angle of W to be pq (the empty sequence). If Σ X Σ “ t u ,we set the multi-angle of W to be pq too. The following property ofSiegel wedges is immediate from the definitions. p:SW-ma Proposition 4.10.
Let W be a Siegel wedge of multi-angle ~α . Thenthe multi-angles of all points in W X Y p P q contain ~α as an initialsegment. Fix a Siegel wedge W . Recall that B W X K p P q Ă Y p P q ; the map η P : Y p P q Ñ K p Q q takes z P Y p P q to a unique point w “ η P p z q withthe same multi-angle and polar radius. Then η P pB W X K p P qq is theunion of the core curves of two periodic bubble rays for Q . These corecurves land at some repelling periodic points, say, x and y of Q (theseare endpoints of K p Q q as K p Q q has no periodic cutpoints). Thereare unique external rays landing at x and y . The union Γ Q of theseexternal rays and η P pB W X K p P qq bounds a unique Siegel wedge W Q of Q that contains points of η P p W X Y p P qq . The wedge W Q is saidto correspond to the Siegel wedge W of P . Observe that since theendpoints of B W X K p P q may be cutpoints of K p P q , there may beseveral Siegel wedges of P corresponding to the same Siegel wedge of Q . 5. The dynamical map η P s:dynmap We now define a P -invariant continuum X p P q Ą Y p P q and extendthe map η P : Y p P q Ñ K p Q q to X p P q . If Y p P q contains no parabolic points of P , then we set X p P q “ Y p P q . Suppose now that there is aparabolic periodic cycle in Y p P q ; let a be a point in this cycle. By theFatou–Shishikura inequality, the cycle of a is the only parabolic cycleof P . In this case, let X p P q be the union of Y p P q and the closures ofall parabolic domains attached to the cycle of a . Clearly, X p P q is aforward invariant continuum. ss:XP The structure of X p P q . Consider possible intersections of X p P q with bubbles of P . l:XPbub Lemma 5.1.
Let A be a bubble of P . Suppose that a point z P A X X p P q is different from the root point r p A q of A . Then A is a legalbubble, and the entire bubble chain to z consists of legal bubbles.Proof. Since z ‰ r p A q , then r p A q P Y P (otherwise points like z wouldnot exist) and the legal arc I z Ă K P from 0 to z is non-disjoint from A ; hence A X X p P q ‰ ∅ . Since A is open, A X Y p P q ‰ ∅ , that is, A is legal. Also, I z intersects every bubble in the bubble chain to z , itfollows that all bubbles in this chain are legal. (cid:3) The following is an immediate corollary of Lemma 5.1. c:XPbub
Corollary 5.2.
Let A be a bubble of P . Either A is legal, or A has nopoints of X p P q except possibly r p A q in which case r p A q is eventuallymapped to c , non-strictly before it is mapped to and strictly before A is mapped to ∆ p P q . Suppose now that A is legal but A is not a subset of Y p P q . Thenthere is a point z P A that is eventually mapped to c , say P n p z q “ c .Since P n : A Ñ P n p A q is a homeomorphism mapping points of Y p P q topoints of Y p P q and vice versa, it is enough to consider the case c P A .Since A is open, it follows that A X Y p P q ‰ ∅ . Recall that the multi-angle of A is the multi-angle of its root point r p A q . Since it takes tworadii to pass through a bubble, we may assume that ~α “ p α , . . . , α k q is the multi-angle of A . Then there are two cases: (1) the multi-angleof any point in A X Y p P q looks like p ~α.α k q , or (2) some points in A have multi-angles p ~α.α k q while others have multi-angles p ~α.α k α k ` q .Consider these cases separately.(1) In this case, c also has multi-angle ~α. p α k q . Since all points A X Y p P q have polar angle α k with respect to A , the set A X X p P q isthe legal arc from the root point of A to c . It is also clear that η P isdefined and continuous on this legal arc. The image of A X X p P q is alegal arc in the closure of the bubble of Q corresponding to A .(2) Choose a point of A X Y p P q with multi-angle ~α. p α k , α k ` q ;evidently, α k ` ‰ α k . Set B “ P p A q , then B Ă Y p P q . Let us describe ODELING CORE PARTS OF ZAKERI SLICES I 23 the P -image of A X Y p P q as a subset of B . Let R be the internal rayof B containing the critical value P p c q . Then P p A X Y p P qq includesthe center of B and all internal rays of B but R . On R , a segment T from P p c q to the boundary of B is not in P p A X Y p R qq ; other pointsof R are in. Call T the special segment .The pullback of T is an arc T Ă A that contains c and two arcsconnecting c and two preimages of the point T X B B ; clearly, under themap P the arc T folds two-to-one onto T . The arc T divides A intwo disjoint pullbacks of B z T , and A X Y p P q is one of them. The set A X X p P q contains a unique P -preimage of the center of B and (initialsegments of) rays of all arguments emanating from this point; all raysbut one extend to B A , and one exceptional ray crashes into c and thensplits into two branches. Here by rays we mean integral curves of theradial vector field B{B ρ in A . The set Bp A X Y p P qq consists of c and two“ separatrices ”, i.e., two rays from c to the boundary of A ; i.e., it equals T . Clearly, the map η P extends to these separatrices. The image η P p A X X p P qq coincides with the entire bubble of Q corresponding to A . In this Q -bubble one radial segment (from η P p c q to the boundaryof the bubble) is covered twice. Otherwise, the map is one-to-one.Theorem 5.3 summarizes all that. t:XPinbub Theorem 5.3.
Let A be a legal bubble of P , and A Q be the correspond-ing bubble of Q . Then η P : Y p P q X A Ñ A Q extends to a continuousmap η P : X p P q X A Ñ A Q , and one of the following two cases holds: (1) The set X p P q X A is a terminal segment of the internal ray of A landing at the root point of A . The map η P : X p P q X A Ñ A Q is one-to-one, and η P p X p P q X A q is a terminal segment of theinternal ray of A Q landing at the root point of A Q . (2) The set X p P qX A is a “sector” bounded by two “separatrices” in A . It is mapped under η P onto A Q , the boundary of the sectormapping two-to-one, and otherwise the map being one-to-one. Here, a terminal segment of an internal ray of A means a segmentfrom some point in the ray to B A .5.2. A separation property.
Suppose that W is a Siegel wedge of P . Consider the corresponding wedge W Q in the dynamical plane of Q . Such wedge is called P -adapted (this notion depends on the choiceof P ). Say that a P -adapted wedge W Q separates a point x from apoint x if x P W Q and x R W Q . This relation is symmetric: the wedge C z W Q is also P -adapted, and it separates x from x .Note that, since K p Q q is locally connected, any two points of K p Q q can be connected by a legal arc. The set η P p Y p P qq Ă K p Q q is legal convex , that is, any two points of this set can be connected by a legalarc lying entirely in this set. l:regconv Lemma 5.4.
The set η P p Y p P qq Ă K p Q q is legal convex.Proof. This follows from a more general observation: the closure of alegal convex set is a legal convex set. Indeed, if x n , x n P K p Q q are twosequences converging to x , x , respectively, then the legal arc from x n to x n converges to the legal arc from x to x . (cid:3) Separation of points within the closure of the same bubble of Q isgiven by the following lemma. l:sep-adapt Lemma 5.5.
Let A Q ‰ ∆ p Q q be a bubble of Q , and x P B A Q X η P p Y p P qq be a point different from the root point of A Q . Then, forany other point x P B A Q , there is a P -adapted wedge W Q with rootpoint in the center of A Q such that W Q separates x from x . Any pairof different points in B ∆ p Q q is also separated by a P -adapted wedgeunless c P B ∆ p P q and P k p c q “ for some k ě .Proof. Suppose first that A Q ‰ ∆ p Q q and x ‰ r p A Q q is a point of B A Q X η P p Y p P qq . The legal arc I x in K p Q q from 0 to x lies in η P p Y p P qq ,by Lemma 5.4. By Lemma 5.1 and since x is not the root point of A Q , the bubble A Q corresponds to some legal bubble A of P . Since x P B A Q X η P p Y p P qq is not equal to r p A Q q , then case (2) of Theorem5.3 holds. Hence the η P -image of A X Y p P q is A Q except for at most asubarc of R (where R is an internal ray of A Q ) not reaching the centerof A Q . Points of B A Q that are root points of other bubbles attachedto A Q are dense in B A Q . All these points except possibly one are in η P p Y p P qq . Therefore, there is a pair of such points b , b in B A Q thatseparate x from x . The legal arcs from the center a of A Q to b and b can be extended to periodic Siegel rays. Moreover, there are periodicSiegel rays Γ b and Γ b for P such that the wedge W bounded by Γ b and Γ b corresponds to a wedge W Q whose boundary intersects B A Q atpoints b and b . Thus there is a P -adapted wedge W Q that separates x from x , as desired.Suppose now that x , x P B ∆ p Q q but either c R B ∆ p P q or c P B ∆ p P q is never mapped to 1 under iterates of P . Then there are two iteratedpreimages b , b of 1 separating x and x in B ∆ p Q q . The correspondingpoints of B ∆ p P q are root points of legal bubbles attached to ∆ p P q , andthe same argument as above works. (cid:3) The following is a more general separation property.
ODELING CORE PARTS OF ZAKERI SLICES I 25 p:sep-w
Proposition 5.6.
A pair of distinct points x , x P J p Q q X η P p Y p P qq is separated by a P -adapted wedge, except when both x and x are in B ∆ p Q q , and c P B ∆ p P q is eventually mapped to .Proof. we ay assume that for some bubble A Q the point x belongs to A Q and the legal arc I from x to x intersects A Q . Let a be the center of A Q . Let x be the point of B A Q , where I intersects B A Q ; we necessarilyhave x ‰ x . Lemma 5.5 is applicable to x and x since at least oneof these two points is different from r p A Q q . By Lemma 5.5, there is a P -adapted wedge W Q with root point a separating x from x . Then W Q will also separate x from x since the legal arc from x to x cannotintersect B W Q .If neither x nor x belongs to the boundary of a bubble, then thereare bubble rays A Q and A Q (not necessarily periodic) landing at x and x , respectively (since J p Q q is locally connected, then any bubble raylands). Since x P η P p Y p P qq , it follows that there are infinitely manybubbles in A Q intersecting η P p Y p P qq . Then in fact all bubbles in A Q intersect η P p Y p P qq , by Lemma 5.1. It follows that there is a bubbleray A for P corresponding to A Q (recall that, by definition, a bubbleray for P consists of legal bubbles). Similarly, there is a bubble ray A for P corresponding to A Q . Take a bubble B Q P A Q but not in A Q .By the above, B Q corresponds to some legal bubble B of P . Therefore,there exists a P -adapted wedge W Q separating x (equivalently, thepoint where the legal arc from the center of B Q to x intersects B B Q )from the root point of B Q . Then W Q also separates x from x . (cid:3) Continuous extension of η P . In this section, we complete theproof of the following theorem. t:etaPcont
Theorem 5.7.
The map η P : Y p P q Ñ K p Q q extends to a continuousmonotone map η P : X p P q Ñ K p Q q . The extended map is denoted by the same letter.
Proof of Theorem 5.7, definition of η P and its continuity. We start byproving that η P extends continuously to Y p P q . Take y P B Y p P qz Y p P q ;by Theorem 5.3, it is enough to assume that y is not in a bubble. Weneed to prove that, for all sequences y n P Y p P q converging to y , theimages η P p y n q converge to the same limit. Assume the contrary: y n , y n P Y p P q are two sequences converging to y such thatlim n Ñ8 η P p y n q “ x ‰ x “ lim n Ñ8 η P p y n q . Set x n “ η P p y n q and x n “ η P p y n q . By Proposition 5.6, there is a P -adapted wedge W Q in the dynamicalplane of Q that separates x from x so that x P W Q and x R W Q . Since W Q is open, x n P W Q for all large n . By definition of a P -adaptedwedge, W Q corresponds to some legal wedge W for P . Thus, y n P W for large n , and for large n , all y n are in some compact subset of W . Itfollows that y P W , hence also y n P W for large n . We conclude that x n P W Q for large n , therefore, x P W Q , a contradiction.Suppose now that X p P q ‰ Y p P q , that is, there is a parabolic cyclein Y p P q . Every point z P X p P qz Y p P q belongs to the closure of aparabolic domain at a parabolic point a z . Set η P p z q “ η P p a z q . Theextension thus defined is continuous. (cid:3) As usual, fibers of η P are defined as preimages of points under η P .We now address the issue of connectivity of fibers. l:intW Lemma 5.8.
A nonempty intersection of finitely many Siegel wedgesfor P is connected and has a connected intersection with X p P q .Proof. A Siegel wedge disjoint from the legal arc connecting 0 and itsroot point is said to be of type 1; all other Siegel wedges are said to beof type 2. Finitely many Siegel wedges of type 1 intersect over a Siegelwedge of type 1. Finitely many Siegel wedges of type 2 intersect overa set B whose complement is the closure of a finite union of pairwisedisjoint type 1 Siegel wedges. Finally, the intersection of a set B likeabove and a Siegel wedge of type 1 either coincides with the later, or is B minus the closure of a finite union of pairwise disjoint type 1 Siegelwedges contained in B . Let U be a nonempty intersection of finitelymany Siegel wedges for P . The description given above implies that U is connected, and U X Y p P q is connected. In case X p P q “ Y p P q ,the lemma follows because U X X p P q is a superset of U X Y p P q and asubset of U X Y p P q . If there are parabolic points in U X X p P q , thentheir parabolic domains are also in U X X p P q , and the lemma alsofollows. (cid:3) Proof of Theorem 5.7, the monotonicity part.
Take x P η P p X p P qq ; con-sider the fiber η ´ P p x q . If x is in a bubble of Q , then the fiber of x isa singleton. Thus we may assume that x P J p Q q . First suppose that x P B ∆ p Q q and c P B ∆ p P q is eventually mapped to 1. By definition, x P Y p P q , and η P is defined on x . Moreover, η P : B ∆ p P q Ñ B ∆ p Q q isa homeomorphism. Thus, the fiber η ´ P p x q is a singleton contained in B ∆ p P q unless some point y R ∆ p P q is mapped to x . Let us show thatthe latter is impossible. If y is on the boundary of a legal bubble A of ODELING CORE PARTS OF ZAKERI SLICES I 27 P , then A can be chosen so that y ‰ r p A q . Then η P p y q is on the bound-ary of the bubble A Q of Q corresponding to A , and η P p y q ‰ r p A Q q . If y is in the limit set of a bubble ray of P consisting of legal bubbles,then η P p y q is the landing point of the corresponding bubble ray of Q .In both cases it is impossible that η P p y q P B ∆ p Q q .Now assume that x R B ∆ p Q q or that c R B ∆ p P q or that c P B ∆ p P q is never mapped to 1. Let Z x be the intersection of all P -adaptedwedges containing x . By the separation property, Proposition 5.6, wehave Z x X J p Q q “ t x u . Apart from x , the set Z x may include certainexternal rays of Q as well as certain internal rays in bubbles of Q .Thus Z x X K p Q q is the union of t x u and possibly several internal raysin bubbles A Q such that x P B A Q . Moreover, every such bubble mayintersect Z x by at most one internal ray.Consider the preimage Z x “ η ´ P p Z x X K p Q qq . This is the union of η ´ P p x q and possibly several internal rays in legal bubbles A of P . Everysuch bubble may intersect Z x by at most one internal ray. It followsthat connectivity of Z x will imply the connectivity of the fiber η ´ P p x q .The rest of the proof deals with connectivity of Z x .Note that Z x is the intersection of all Siegel wedges of P containing x with X p P q . Moreover, it is enough to intersect countably many Siegelwedges W , . . . , W n , . . . : Z x “ X p P q X č n “ W n “ X p P q X č n “ U n , U n “ W X ¨ ¨ ¨ X W n . The sets X p P q X U n are connected by Lemma 5.8 and form a nestedsequence. Therefore, their intersection is also connected. (cid:3) The parameter maps Φ cλ and Φ λ s:par Consider a map P “ P c P C cλ . Suppose that c P X p P q . Define Φ cλ p P q as η P p c q . Let D cλ denote the domain of the map Φ cλ , that is, the set ofall P P C cλ such that c P X p P q . Observe that Z cλ Ă D cλ .6.1. Immediate renormalization.
Recall the notions of a polynomial-like map and an immediate renormalization. Write U Ť V if U Ă V .Let U Ť V be Jordan disks. The following classical definitions are dueto Douady and Hubbard [DH85]. d:pl Definition 6.1 (Polynomial-like maps [DH85]) . Let f : U Ñ V be aproper holomorphic map. Then f is said to be polynomial-like (PL).By definition, a quadratic-like (QL) map is a PL map of degree two.The filled Julia set K p f q of f is defined as the set of points in U , whoseforward f -orbits stay in U . Similarly to polynomials, the set K p f q is connected if and only if allcritical points of f are in K p f q . The following is a greatly simplifiedand weakened version of a much stronger classical theorem of Douadyand Hubbard [DH85]. t:DH-pl Theorem 6.2 (PL Straightening Theorem [DH85]) . A PL map f : U Ñ V is topologically conjugate to a polynomial of the same degree re-stricted on a Jordan neighborhood of its filled Julia set. The next result allows one to locate some PL Julia sets. t:thmb
Theorem 6.3 (Theorem B [BOPT16]) . Let P : C Ñ C be a polyno-mial, and Y Ă C be a full P -invariant continuum. The following as-sertions are equivalent: (1) the set Y is the filled Julia set of some polynomial-like map P : U ˚ Ñ V ˚ of degree k , (2) Y is a component of the set P ´ p P p Y qq , and, for every attract-ing or parabolic point y of P in Y , the immediate attractingbasin of y or the union of all parabolic domains at y is a subsetof Y . A cubic polynomial P P C cλ is immediately renormalizable if P : U Ñ V is a QL map for some U , V . p:Dc Proposition 6.4.
Suppose that P P C cλ z D cλ . Then P is immediatelyrenormalizable with X p P q being the corresponding quadratic-like Juliaset.Proof. Since P P C cλ z D cλ , then c R X p P q . The set X p P q is compact;also, it is easy to see that X p P q is a component of P ´ p X p P qq (itsuffices to consider the set Y P ). There are no parabolic periodic pointsof P in X p P q ; otherwise c would be in one of the parabolic domainsadded to X p P q . By Theorems 6.2 and 6.3, there is a Jordan domain U Ą X p P q such that P : U Ñ P p U q is a quadratic-like map whosefilled Julia set coincides with X p P q . (cid:3) Recall that the set P cλ is the subset of C cλ consisting of polynomialsthat can be approximated by sequences P n P C cλ n with | λ n | ă P n in the immediate basin of 0. c:PcDc Corollary 6.5.
The set P cλ is a subset of D cλ .Proof. Take any P P C cλ z D cλ . By Proposition 6.4, there is a quadratic-like map P : U Ñ V with filled Julia set X p P q , and we may choose U and V so that c R U . Take any µ very close to λ but with | µ | ă f P C cµ very close to P . Then f ´ p V q is close to P ´ p V q by ODELING CORE PARTS OF ZAKERI SLICES I 29
Lemma 4.1, and a component U f of f ´ p V q is close to U . It followsthat f : U f Ñ V is a quadratic-like map whose filled Julia set containsan attracting point 0. Therefore, the filled Julia set of f : U f Ñ V isa Jordan disk, on which f is two-to-one. In particular, it is impossiblethat f is in the principal hyperbolic component, and so P R P cλ . (cid:3) Continuity.
It will be established in this section that Φ cλ is con-tinuous.Recall that P is the polynomial with P P C ˚ λ such that c “ cλ at P . l:P1 Lemma 6.6.
Suppose that a sequence P c n P D cλ converges to P (sothat c n Ñ ). If η P cn p c n q converges, then the limit is equal to .Proof. Assume the contrary: c Q,n “ η P cn p c n q converges to a point dif-ferent from 1. Let I n denote the legal arc for P c n from 0 to c n . Passingto a subsequence, assume that I n Ñ I in the Hausdorff metric, where I Ă K p P q is a continuum. For any bubble A of P , the set I X A is inthe union of at most two internal rays of A and o A . Indeed, for every ρ ă
1, the set t z | ρ z ă ρ u Ă K p P q moves equicontinuously with P P C cλ near P . In particular, the intersection of any internal ray with this setmoves equicontinuously. The set I intersects infinitely many bubbles of P ; otherwise there is a contradiction with Lemma 3.12. Now let A bea bubble of P such that P p A q “ ∆ p P q and I X A ‰ ∅ . That is, A is one of the two bubbles attached to ∆ p P q at 1. Choose internal rays R and L of A that separate the internal rays in I X A and are even-tually mapped to the internal ray of ∆ p P q from 0 to 1. Then R and L are segments of core curves R and L , respectively, of some periodicbubble rays of P . Here we use a somewhat relaxed definition of bubblerays, different from Definition 3.10. Namely, we do not require that thebubbles of the considered bubble rays are all legal (in fact, there are nolegal bubbles for P except for ∆ p P q ). However, the landing theoremfor bubble rays, Theorem 3.14, is still valid in our setting. Adding thelanding points of R , L and a pair of external rays landing at thesepoints to R Y L , we obtain a cut Γ that divides the plane into twowedges. This cut Γ separates the initial segment I of I (the segmentconnecting 0 with ce p A q ) from the rest of I . In particular, I z I doesnot come close to 1, a contradiction with I n Ñ I and c n Ñ (cid:3) Theorem 6.7 completes the proof of Theorem 2.3. t:contoutZ
Theorem 6.7.
The map Φ cλ : D cλ Ñ K p Q q is continuous. Proof.
Take P P D cλ . Suppose that P n “ P c n P D cλ converge to P “ P c .We show that c Q,n “ Φ cλ p P n q converge to c Q “ Φ cλ p P q . If not, then bychoosing a suitable subsequence, we may assume that c Q,n Ñ c Q ‰ c Q .Now consider several cases.First, suppose that c Q belongs to a bubble A Q of Q . By definition, A Q corresponds to a legal bubble A of P containing c . The sequence c n converges to c . The bubble P p A q is stable, in particular, there isa unique bubble B n of P n close to P p A q , for large n . Moreover, B n contains the critical value P n p c n q . By Lemma 4.1 it follows that acomponent A n of P ´ n p B n q contains the critical point c n and is closeto A . All A n have the same multi-angle, thus they all correspond to A Q . Now, since both c Q and c Q lie in the same bubble A Q , they havedifferent images Q p c Q q ‰ Q p c Q q . On the other hand, η ˜ P p ˜ P p ˜ c qq dependscontinuously on ˜ P “ P ˜ c near P (because of the stability of P p A q ). Itfollows that Q p c Q,n q “ η P n p P n p c n qq Ñ η P p P p c qq “ Q p c q . On the otherhand, Q p c Q,n q Ñ Q p c Q q since c Q,n Ñ c Q . Thus we must have c Q “ c Q in the considered case.Suppose now that c Q P J p Q q and either c R B ∆ p P q or c P B ∆ p P q isnever mapped to 1 under P . By Proposition 5.6, there is an adaptedwedge W Q separating c Q from c Q . Since W Q is open, c Q,n P W Q for allsufficiently large n . In fact, c Q,n even lie in some compact subset C Q of W Q for all large n . Let W be the Siegel wedge of P correspondingto W Q . Since the boundary of W is stable, there are Siegel wedges W n for P n close to W that correspond to the same W Q . It follows from c Q,n P C Q that c n P W n for large n . Then c P W , a contradiction.Finally, suppose that c P B ∆ p P q and P k p c q “
1. If k “
0, then thetheorem follows from Lemma 6.6. If k ą
0, then P is conjugate toanother polynomial ˜ P P Z cλ via a linear map that takes c to 1. Thisconjugacy takes 1 to a critical point ˜ c of ˜ P such that ˜ P k p q “ ˜ c . Since ˜ c is never mapped to 1 under ˜ P , the argument given above is applicableto ˜ P . (cid:3) The unmarked map Φ λ . We now study the unmarked map Φ λ : P λ Ñ ˜ K p Q q . Recall that ˜ K p Q q was defined as a model space obtainedas a quotient of K p Q qz ∆ p Q q . Namely, points of B ∆ p Q q that are ψ Q -images of complex conjugate points in S are identified in ˜ K p Q q . Let π : K p Q qz ∆ p Q q Ñ ˜ K p Q q be the quotient map. The map π ˝ Φ cλ : P cλ Ñ ˜ K p Q q is then well defined and continuous. It suffices to prove that P c and P { c have the same images under π ˝ Φ cλ . Then the map π ˝ Φ cλ descends to a continuous map Φ λ from P λ to ˜ K p Q q , as is claimed inthe following lemma. ODELING CORE PARTS OF ZAKERI SLICES I 31 l:welldef
Lemma 6.8.
The points Φ cλ p P c q and Φ cλ p P { c q have the same π -imagesin ˜ K p Q q .Proof. Note that P c and P { c are affinely conjugate; the difference isonly in how the critical points are marked. Thus the angular differencebetween the two critical points in the boundary of the Siegel disk isthe same up to a sign for P c and P { c . Here, by the angular difference between two points a , b P B ∆ p P q , we mean α ´ β P R { Z , where a “ ψ ∆ p P q p e πiα q and b “ ψ ∆ p P q p e πiβ q . It follows that Φ cλ p P c q and Φ cλ p P { c q have the same angular difference with 1 up to a sign in B ∆ p Q q . Bydefinition, such points are identified in ˜ K p Q q . (cid:3) Lemma 6.8 implies the Main Theorem.
Acknowledgements.
The second named author was partially supportedby NSF grant DMS-1807558. The fourth named author has been sup-ported by the HSE University Basic Research Program.
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ODELING CORE PARTS OF ZAKERI SLICES I 33 (A. Blokh, L. Oversteegen)
Department of Mathematics, University ofAlabama at Birmingham, Birmingham, AL 35294-1170 (A. Shepelevtseva, V. Timorin)
Faculty of Mathematics, HSE University,Russian Federation, 6 Usacheva St., 119048 Moscow (A. Shepelevtseva)
Scuola Normale Superiore, 7 Piazza dei Cavalieri,56126 Pisa, Italy (Vladlen Timorin)
Independent University of Moscow, Bolshoy VlasyevskiyPereulok 11, 119002 Moscow, Russia
Email address , Alexander Blokh: [email protected]
Email address , Lex Oversteegen: [email protected]
Email address , Anastasia Shepelevtseva: [email protected]
Email address , Vladlen Timorin:, Vladlen Timorin: