Real Laminations and the Topological Dynamics of Complex Polynomials

Home , Complex dynamics, External ray

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Advances in Mathematics 184 (2004) 207–267 http://www.elsevier.com/locate/aim

Real laminations and the topological dynamics of complex polynomials

Jan Kiwi1 Facultad de Matema´ticas, Pontiﬁcia Universidad Cato´lica, Casilla 306, Correo 22, Santiago, Chile

Received 25 March 2002; accepted 28 March 2003

Communicated by R.D. Mauldin

Abstract

MSC: 37F20

Keywords: Julia sets; Laminations

1. Introduction

The main purpose of this paper is to study the topological dynamics of polynomials f : C-C with connected Julia sets and without irrationally neutral cycles. Inspired by the work of Levin [18] and by classical results in one real- dimensional dynamics (e.g. see [5]), we consider the dynamical system obtained after collapsing the wandering connected sets contained in the Julia set Jð f Þ of such a polynomial f : The main result of this paper is to give a complete description of the dynamical systems which arise from this collapsing procedure. To be more precise we ﬁx a degree dX2 monic polynomial f : C-C with connected Julia set Jð f Þ and without irrationally neutral cycles. We consider the topological space Xf which is the quotient of Jð f Þ by the equivalence relation (see Remark 5.4) that identiﬁes two distinct points if and only if they lie in a wandering

E-mail address: [email protected]. 1 Supported by FONDECYT Grant #1990436.

208 J. Kiwi / Advances in Mathematics 184 (2004) 207–267 connected set CCJð f Þ (i.e., f nðCÞ-f mðCÞ¼| for all positive integers nam). The aim is to describe Ff : Xf -Xf where Ff is the projection of f : Jð f Þ-Jð f Þ: Our description relies on introducing certain equivalence relations in the circle T ¼ R=Z: These equivalence relations are introduced via prime end impressions. The collection formed by the prime end impressions of Jð f Þ is a collection of compact connected subsets of Jð f Þ that cover Jð f Þ: There is a canonical one-to-one correspondence between arguments in T and prime end impressions. That is, to each tAT there corresponds a prime end impression ImpðtÞ: We deﬁne the lamination of f, denoted by lð f Þ; as the smallest equivalence relation in T that identiﬁes t and s whenever ImpðtÞ-ImpðsÞa|: Since the canonical parameterization of prime end impressions has the property that f ðImpðtÞÞ ¼ ImpðdtÞ; multiplication by d acts on the quotient topological space T=lð f Þ: We show that the action of multiplication by d on T=lð f Þ is topologically conjugate to Ff : Xf -Xf (see Corollaries 5.3 and 5.6). Therefore, to describe the dynamics that arise from collapsing wandering connected sets is equivalent to describe the laminations of polynomials with connected Julia sets and without irrationally neutral cycles. According to Carathe´ odory, every prime end impression is a singleton if and only if the Julia set is locally connected. The main properties of l ¼ lð f Þ when Jð f Þ is locally connected are fairly well understood and are the following (see [6,33]):

(R1) l is closed in T Â T: (R2) Each l-equivalence class A is a finite subset of T: (R3) If A is a l-equivalence class, then d Á A is a l-equivalence class. (R4) If A is a l-equivalence class, then A/d Á A is consecutive preserving (see Definition 4.9). (R5) l-equivalence classes are pairwise unlinked (see Definition 4.9).

Under the assumption that f has no irrationally neutral cycles it also follows that: (NR) If gCT=l is a periodic simple closed curve, then the return map is not a homeomorphism. We say that an equivalence relation l in T such that (R1)–(R5) hold is a Real lamination.AReal lamination for which (NR) holds is called a Real lamination with no rotation curves. Without assuming that Jð f Þ is locally connected, but under the fairly general assumption that f is a polynomial with connected Julia set and without irrationally neutral cycles, we show that still lð f Þ has properties (R1)–(R5) and (NR) listed above. Moreover we prove that these properties characterize laminations of polynomials:

Theorem 1. An equivalence relation l in T is the lamination of a polynomial f with connected Julia set and without irrationally neutral cycles if and only if l is a Real lamination without rotation curves. ARTICLE IN PRESS

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Note that we do not even define the lamination of polynomials with irrationally neutral cycles. In fact, it is not known whether every pair of prime end impressions have non-empty intersection for quadratic polynomials with a non-linearizable irrationally neutral fixed point (i.e., a Cremer fixed point). Laminations were introduced in complex dynamics by Thurston [33]. We believe that Real laminations correspond to Thurston’s invariant laminations so there is no claim to originality in the definition. The difference with Thurston’s viewpoint is that we introduce these objects as equivalence relations and not as their useful representation as geodesic laminations in the Poincare´ disk. For more about laminations in complex dynamics see [6,13,29] and the references therein. The combinatorics and topology of polynomials with exactly one critical point (in particular, quadratic polynomials) has already been studied in great detail (see [7,24,32,33]). Although Theorem 1 is not stated in the literature in the case of polynomials with exactly one critical point, an easier proof should follow from the results contained in the references above. Let us now outline the structure of the paper and at the same time state partial and related results.

1.1. Outline

A quick overview of the organization of the paper is as follows. Sections 2–5 are devoted to study the topology of Julia sets of polynomials without irrationally neutral cycles. As a consequence of this topological study we show that the lamination of a polynomial is a Real lamination with no rotation curves. Most of our topological results also apply to polynomials with possibly disconnected Julia sets, so we work in this more general setting. In the last section of the paper, Section 6, given a Real lamination l with no rotation curves we find a polynomial such that its lamination is l: To find such a polynomial we rely on results by Bielefield et al. [1], Douady [6], Poirier [26] and Thurston [33]. We now discuss in more detail the contents of each section. In Section 2 we discuss how the Julia set Jð f Þ of a polynomial f may be ‘‘decomposed’’ into smaller sets in order to study its topology. We are concerned with the decompositions of Jð f Þ into: ‘‘impressions’’ as introduced by Carathe´ odory, ‘‘fibers’’ as introduced by Schleicher [31], and ‘‘wandering continua’’ as studied by Levin [18]. Some of these decompositions originally apply only to connected Julia sets. Since part of our discussion and results include polynomials with disconnected Julia sets, in Section 2 we generalize these decompositions to the context of disconnected Julia sets. The paper is devoted to the understanding of polynomials without irrationally neutral cycles. For these polynomials, in Section 3, we show that the topology of their Julia sets is rather tame around periodic and preperiodic points.

Theorem 2 (Trivial ﬁbers). Consider a polynomial f : C-C without irrationally neutral cycles. If zAJð f Þ is a periodic or preperiodic point of f, then the ﬁber of z is a singleton, i.e., FiberðzÞ¼fzg: ARTICLE IN PRESS

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In a different but equivalent language, we established the theorem above for polynomials with connected Julia set and all cycles repelling (see the author’s Thesis [14, Section 13]). Levin [18] independently proved the same result for polynomials with one critical point and all cycles repelling. This result was written by Schleicher [31] in the language of ﬁbers for polynomials with one critical point and all cycles repelling. The theorem above has a pair of rather immediate corollaries regarding local connectivity and impressions (cf. [14, Theorem 1(a)]).

Corollary 1.1. Consider a polynomial f : C-C without irrationally neutral cycles. Let zAJð f Þ be a periodic or preperiodic point of f. Then the connected component J of Jð f Þ that contains z is (openly) locally connected at z:

Corollary 1.2. Consider a polynomial f : C-C without irrationally neutral cycles. If an impression Imp contains a periodic or a preperiodic point, then Imp is a singleton.

The main idea of the proof of Theorem 2 is to construct an appropriate ‘‘puzzle’’. Section 3 contains the details of this construction, the proof of the theorem and the proof of its corollaries. It is important to understand the relationship among the decompositions of the Julia set into ﬁbers, into wandering connected sets, and into impressions. The key for passing from the decomposition of Jð f Þ into impressions to the other decompositions is the following ﬁniteness result.

Theorem 3 (Finiteness Theorem). Let f : C-C be a polynomial without irrationally neutral cycles. If fzg is not a preperiodic or periodic component of Jð f Þ; then FiberðzÞ contains at least one and at most ﬁnitely many impressions. Moreover, every impression that intersects FiberðzÞ is contained in FiberðzÞ:

Section 4 contains the proof of the previous theorem. For polynomials with all cycles repelling, Theorem 3 is proved in [14], which is ultimately a consequence of Theorem 2 together with the fact that at most ﬁnitely many rays may land at a given point of a connected Julia set. Thurston [33] established this fact for quadratic polynomials. We generalized Thurston’s result to higher degree polynomials (see [14,16]). Levin [18] gave a new proof of Thurston’s result for locally connected Julia sets with one critical point. Levin’s techniques are further developed in [2] leading to new related results that apply to arbitrary degree polynomials. In Section 5, we collect some consequences of Theorems 2 and 3. On one hand, we show that the lamination of a polynomial f with connected Julia set and without irrationally neutral cycles has the desired properties (R1)–(R5) and (NR). On the other, we study the relationship between abstract dynamical systems that may be constructed just from the data contained in the lamination of a polynomial f and the dynamics of f : ARTICLE IN PRESS

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In Section 6, given a Real lamination l without rotation curves we find a polynomial f without irrationally neutral cycles and with connected Julia set such that the lamination of f is l: We find such a polynomial as a limit of post-critically finite polynomials that have laminations which, in certain sense, approach l: To find these post-critically finite polynomials we use results by Poirier [26] (cf. [1]). To study the combinatorics of the limit polynomial f we use the techniques developed in [14] which were also applied in [16] to characterize rational laminations of complex polynomials.

2. Rays, impressions and ﬁbers

In this section, we discuss the decompositions of Julia sets into impressions, as well as into ﬁbers. Section 2.1 is devoted to impressions, while Section 2.2 to the basic properties of ﬁbers. We assume certain familiarity with iteration of complex polynomials. The reader may refer to [4,21,22] for background material on complex dynamics.

2.1. Rays and impressions

Throughout this section, we consider a monic polynomial f : C-C of degree dX2: To study the dynamics of f ; we use the Green function gf and the Bo¨ ttcher map ff (e.g. see [22, Section 18]). The Green function

gf : C -RX0 log f n z / þj ð Þj z lim dn measures the escape rate of points to N: It vanishes at the filled Julia set Kð f Þ; and is positive and harmonic in the basin of infinity C\Kð f Þ: Also, gf ð f ðzÞÞ ¼ dgf ðzÞ: The gradient vector field rgf ; which is well defined in the basin of infinity, vanishes only at precritical points of f : Thus if Kð f Þ is connected, or equivalently if no critical point of f is in the basin of N; then rgf ðzÞa0 for all zAC\Kð f Þ: / d The Bo¨ ttcher map ff conjugates f with z z in a neighborhood of N: The / germ of ff at N is unique up to conjugacy by z zz; where z is a ðd À 1Þth root of unity. Since f is monic, we can normalize ff to be asymptotic to the identity. That is,

f ðzÞ f -1 z - as z N: Also, gf ðzÞ¼log jff ðzÞj near inﬁnity. ARTICLE IN PRESS

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2.1.1. Rays When the Julia set Jð f Þ is connected, the Bo¨ ttcher ff map extends to a conformal isomorphism between the basin of infinity C\Kð f Þ and the complement of the closed \ 2pit t unit disk C D: The preimage of ð1; NÞe under ff ; denoted R ; is called the external ray at argument tAT ¼ R=Z: When the Julia set is disconnected (cf. [20]), the Bo¨ ttcher map ff no longer extends to the whole basin of infinity under iterations of f : Nevertheless, ff extends to the basin of infinity Oðrgf Þ under the flow of the gradient vector field rgf : Now \ ff is a conformal isomorphism from Oðrgf Þ onto a starlike domain Uf CC D 2pit 2pit around N: For tAT; let ðr0; NÞe be the maximal portion of ð1; NÞe contained Ãt in Uf : The external radius R at argument t is

Ãt À1 2pit R ¼ ff ððr0; NÞe Þ:

Ãt In the case r041; the external radius R terminates at some point z where Ãt t rgf ðzÞ¼0: Otherwise, r0 ¼ 1andR is, in fact, the smooth external ray R at argument t. To understand for which arguments we have well-deﬁned smooth external rays, we need to describe the domain Uf : Denote by y1; y; yl the arguments of the external radii that terminate at critical points of f : Also, let r1; y; rl be the corresponding values of gf at these critical points. It follows that zAC\D is not contained in Uf if and only if, under some iterate of z/zd ; the point z maps onto erþ2piyi ; with 0orpri for some i ¼ 1; y; l: Therefore, external rays are well deﬁned for arguments in T\Sf ; where

n Sf ¼ftAT : d tAfy1; y; ylg for some nX0g:

If the Julia set is connected, we simply agree that Sf ¼ | and Uf ¼ C\D: Following Goldberg and Milnor [10], for each tASf ; we introduce the left hand limit ray RtÀ and the right hand limit ray Rtþ: Since the already defined external rays s R ; for seSf ; are naturally parameterized via gf by the interval ð0; NÞ; we can define Rtþ as the pointwise limit of Rs as s-t and s4t: Similarly, RtÀ is defined as the limit of Rs as s-t and sot: External rays of polynomials with connected Julia set are parameterized by the circle T ¼ R=Z: When the Julia set is disconnected, the next definition shows that external rays together with limit rays are naturally parameterized by a Cantor set (see [10, Appendix A]).

Deﬁnition 2.1. Let Sf be the topological space obtained by replacing every element þ À À þ tASf CT by two distinct points t and t ; declaring t ot ; and endowing it with the local order topology.

Remark 2.2. When Jð f Þ is connected, we agreed that Sf ¼ |: Therefore Sf ¼ T: ARTICLE IN PRESS

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In what follows, we sometimes refer to the elements of Sf as arguments. Also, unless explicitly stated, we will not make a distinction between limit rays and smooth rays. That is, for each argument tASf ; there corresponds a ray which is denoted by Rt: The action on T of the multiplication by d map extends naturally to an action t dt t/dt on Sf : Rays map onto rays according to the rule f ðR Þ¼R ; for all tASf : Moreover, the map c : Sf Âð0; NÞ-C\Kð f Þ that takes ðt; rÞ onto the unique t point z ¼ cðt; rÞ in R such that gf ðzÞ¼r is continuous, onto and f ðcðt; rÞÞ ¼ cðdt; drÞ:

2.1.2. Impressions To understand the dynamics of f over Jð f Þ; we study the behavior of the map c : Sf Âð0; NÞ-C\Kð f Þ as the second coordinate of the domain approaches 0.

t Deﬁnition 2.3. Consider an argument tASf : We call AccðtÞ¼R -Jð f Þ the accumulation set of t: We say that zAJð f Þ belongs to the impression of t; denoted t ImpðtÞ; if and only if there exists a sequence fznAR n g converging to z; with ftngCSf converging to t:

Note that the external ray Rt lands if and only if AccðtÞ is a single point. Also, c extends continuously to the point ðt; 0Þ if and only if ImpðtÞ is a single point. Up to the author’s knowledge there is no known example where AccðtÞaImpðtÞ:

Lemma 2.4. Consider an argument tASf : Then the following statements hold:

(1) ImpðtÞ is a closed and connected subset of Jð f Þ: Also, AccðtÞ is a closed and connected subset of Jð f Þ: (2) f ðImpðtÞÞ ¼ ImpðdtÞ and f ðAccðtÞÞ ¼ AccðdtÞ: (3) AccðtÞCImpðtÞ:

Proof. The proofs of statements (2) and (3) are rather immediate. For statement (1), þ consider tASf : We suppose that t ¼ t is a one-sided isolated point. The general case follows along the same lines. To show that ImpðtÞ is connected, we construct a decreasing sequence fVng of \ À1 À1 connected open subsets of Uf CC D such that the closure of c ðff ðVnÞÞ is

fðs; rÞASf Âð0; NÞ : tpsptn; 0oro1=ng; where ft gCS is a sequence converging to t: Once fV g is constructed, it follows n Tf n À1 that ImpðtÞ¼ ff ðVnÞ is a closed and connected set (see Fig. 1). To obtain Vn; let YCT be the set consisting of the arguments of the external radii that terminate at critical points. Denote by R the maximum over all the equipotential levels which k contain a critical point. Consider an integer k such that d =n4R: Let tnAT be such ARTICLE IN PRESS

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Fig. 1. Proof of Lemma 2.4.

C Àk - that the interval ðt; tnÞ T is disjoint from md ðYÞ; where md : T T denotes the multiplication by d map. Now let Vn be the connected open set consisting of the R element zAUf such that jzjoe and arg zAðt; tnÞ: It is not difﬁcult to check that this sequence of domains fVng has the required properties. To show that AccðtÞ is connected, just note that AccðtÞ may be rewritten as a nested intersection of closed and connected sets Cn; where Cn is the closure of the t portion of R below the equipotential gf ¼ 1=n: &

2.2. Fibers

Fibers were introduced by Schleicher [31] (cf. Yoccoz’s structure of [18]). Here we generalize the notion of ﬁbers to disconnected Julia sets. We introduce a deﬁnition which only relies on the topology of the Julia set and not on its embedding in C:

Definition 2.5. Let Jfinð f Þ denote the set formed by all periodic and preperiodic points in Jð f Þ which are not in the grand orbit of a Cremer point. Given zAJð f Þ; we say that x belongs to the fiber of z; denoted FiberðzÞ; if x and z lie in the same connected component of Jð f Þ\Z; for all finite subsets Z of Jfinð f Þ such that xeZ and zeZ:

Remark. The definition above introduces fibers as subsets of the Julia set. In [31], fibers are introduced as subsets of the filled Julia set. For connected Julia sets, it follows from Corollary 2.18 that each fiber, in the sense of the preceding definition, is the intersection of a fiber, in the sense of Schleicher, with the Julia set. Fibers are closed subsets of Jð f Þ:

Lemma 2.6. For all zAJð f Þ; we have that FiberðzÞ is a closed subset of Jð f Þ:

Proof. To see that FiberðzÞ is closed, consider a point xAJð f Þ such that xeFiberðzÞ: Then there exists a ﬁnite set ZCJfinð f Þ which separates x from z: Since Z also separates a neighborhood of x from z; it follows that FiberðzÞ is closed. & ARTICLE IN PRESS

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Distinct fibers are not necessarily disjoint. More precisely, if z1AFiberðz2Þ and z2AFiberðz3Þ; then it may occur that z1eFiberðz3Þ: Note that the intersection of two distinct fibers must be contained in Jfinð f Þ: In Section 3.5, under the extra assumption that f has no irrationally neutral cycle we will show that the fibers of f are pairwise equal or disjoint. To continue studying fibers, we need to show that, given a connected component J of Jð f Þ and a finite subset Z of Jfinð f Þ; the number of connected components of J\Z is finite. We will translate the problem of counting connected components of J\Z to the one of counting accesses to Z: Recall that a continuum is a compact connected set with at least two elements.

Deﬁnition 2.7. Let KCC be a full continuum. We say that zA@K is accessible from C\K if there exists a continuous curve g : ð0; 1Þ-C\K such that gðsÞ-z as s-1: Such a curve g is called an access to z: If Z is a subset of @K and g an access to an element of Z; then sometimes we simply say that g is an access to Z:

Accessibility is closely related to the boundary behavior of conformal maps (e.g. see [27]). In fact, for K as in the previous deﬁnition, let j : C\D-C\K be a conformal isomorphism. We say that j has radial limit z at tAT if jðre2pitÞ-z as r-1: It is well known that j has radial limit z at t if and only if j has non-tangential limit z at tAT (e.g., [27, Corollary 2.17]). Clearly if z is a radial limit of j; then z is accessible from C\K: The following well known result implies the converse.

Lemma 2.8. Consider a full continuum KCC and a conformal isomorphism j : C\D-C\K:

(1) If g : ð0; 1Þ-C\K is an access to zA@K; then jÀ13gðsÞ has a well-deﬁned limit in @D as s-1: (2) If j has limit z along a curve in C\D ending at e2pit; then j has radial limit z at tAT: Moreover, j has non-tangential limit z at tAT:

For the proofs of assertions (1) and (2), see Proposition 2.14 of [27] and Corollary 2.17 of [27], respectively. A point zA@K may be accessible through several accesses. To distinguish accesses up to trivial deformations, we need the following notion.

Deﬁnition 2.9. Let KCC be a full continuum. Let j : C\D-C\K be a conformal isomorphism. Consider two accesses g and a to a point zA@K: We say that g and a are homotopic if

lim jÀ13gðsÞ¼lim jÀ13aðsÞ: s-1 s-1 ARTICLE IN PRESS

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Remark 2.10. The deﬁnition above is independent of the choice of j and coincides with the usual notions of homotopic accesses.

The next result shows that the number of connected components of J\Z is bounded above by the number of non-homotopic accesses to Z:

Lemma 2.11. Let KCC be a full continuum, and let Z be a ﬁnite subset of J ¼ @K: Suppose that J\Z has at least nX2 connected components. Then there are at least n non-homotopic accesses to Z from C\K:

The proof is analogous to that of Theorem 6.6 in McMullen’s book [21].

Proof. Let A0; y; AnÀ1 be open and closed subsets of J\Z: Since accessible points are dense in J; for each i ¼ 0; y; n À 1; we may choose a point xiAAi accessible from C\K: Let j : C\D-C\K be a conformal isomorphism. As discussed above, xi is the radial limit of j at some tiAR=Z: Reindexing, if necessary, we may suppose that t0; y; tnÀ1 are in cyclic order. - \ \ Consider a path gi : ½0; 1 C connecting xi to xiþ1 through C K (i.e., giðtÞASC K for 0oto1), subscripts mod n: We may choose g1; y; gnÀ1 so that their union gi is a Jordan curve g which bounds a domain U with KCU: In fact, by Lemma 2.8, we \ let gi ¼ jðbiÞ where bi are curves in C D as in Fig. 2. Note that gi,K separates C into two connected domains. Denote by Vi the bounded one. According to the Tietze extension theorem, there exists a continuous function À1 f : C\Z-½0; 1 such that f ði=nÞ¼Ai; for 0pipn À 1: We may assume that f is smooth in C\K: For each i ¼ 0; y; n À 1; choose a regular value ri so that i=norioði þ 1Þ=n: Since Ai cannot be joined to Aiþ1 through an arc contained in À1 À1 Vi\f ðriÞ; there must exist a component C of f ðriÞ-Vi with points arbitrarily close to gi and points arbitrarily close to J: Thus C is homeomorphic to the open interval ð0; 1Þ: Let ai : ð0; 1Þ-C be a homeomorphism such that aiðsÞ approaches J when s-1: Our continuous map f does not take the value ri in J\Z: It follows that every accumulation point of C in J is contained in Z: The set of such accumulation

Fig. 2. Illustration of the proof of Lemma 2.11 where Z ¼fz1; z2g is such that J\Z has four (4) connected components. ARTICLE IN PRESS

J. Kiwi / Advances in Mathematics 184 (2004) 207–267 217 points is connected, therefore aiðsÞ has a well deﬁned limit zAZ as s-1: Hence ai is an access to zAZ that corresponds to some argument ti; where tiotiotiþ1: To complete the proof, note that a0; y; anÀ1 are n accesses to Z which are pairwise non- homotopic. &

For connected Julia sets, it follows by Lemma 2.8 that counting non-homotopic accesses is equivalent to counting external rays, while for disconnected Julia sets, we have the following result of Levin and Przytycki [19, Lemma 2.1].

Proposition 2.12 (Levin–Przytycki). Let K be a periodic or preperiodic connected component of Kð f Þ which is not a single point. Consider an access a to a point z0 in t J ¼ @K: Then there exists an external ray R that is an access to z0 which is homotopic to the access a:

We are now ready to count connected components of J\Z: Recall that Jfinð f Þ is the subset of Jð f Þ consisting of all periodic and preperiodic points which are not in the grand orbit of a Cremer point.

Proposition 2.13. Let J be a connected component of Jð f Þ; and let Z be a ﬁnite subset of Jfinð f Þ: Then J\Z has ﬁnitely many connected components. Moreover, each connected component of J\Z is an open subset of J\Z:

Proof. Without loss of generality, we may suppose that J is not a single point and that ZCJ: Theorem 1 of [19] guarantees that at most finitely many external rays land at a point zAZ: Hence the total number of external rays landing at Z is finite. By Proposition 2.12, there are at most finitely many accesses to Z from C\K; where K is the component of Kð f Þ that contains J: According to Lemma 2.11, the number of accesses to Z is an upper bound for the number of connected components of J\Z: Thus J\Z has finitely many connected components and each component is an open subset of J: &

Now we have the required ﬁniteness to show that ﬁbers are connected and invariant under dynamics. However, we also need the following fact.

Lemma 2.14. Consider two points x and z in Jð f Þ: The following statements are equivalent:

(1) xAFiberðzÞ: (2) For all ﬁnite sets ZCJfinð f Þ; we have that fx; zgCX for some connected component X of Jð f Þ\Z:

Note that in statement (2) we do not require Z to be disjoint from the points z and x: ARTICLE IN PRESS

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Proof. Suppose that xAFiberðzÞ: Denote by J the connected component of Jð f Þ that contains x and z: Consider a finite subset Z of Jfinð f Þ: We must find a connected component of J\Z that contains both points z and x in its closure. The relevant case is when at least one of the elements of fz; xg; say z; lies in Z: Since z and x are in the same fiber, Z1 ¼ Z\fz; xg cannot separate z from x: Therefore, the connected component Y1 of J\Z1 that contains x also contains z: Let Y2 be the connected component of Y1\fzg that contains x: It follows by Proposition 2.13 that zAY 2: If xeZ; then Y2 is a connected component of J\Z that contains both z and x in its closure. If xAZ; then every connected component of Y2\fxg contains x in its closure and at least one connected component Y3 contains z in its closure. Therefore, Y3 is a connected component of J\Z that contains both z and x in its closure. To complete the proof, suppose that xeFiberðzÞ: Then there exists a finite set ZCJfinð f Þ that separates z and x: Denote by X and Y the connected components of J\Z which contain z and x; respectively. Since X-Y is contained in Z; there is no connected component of J\Z containing both points in its closure. &

Corollary 2.15. For all zAJð f Þ; we have that FiberðzÞ is connected and f ðFiberðzÞÞ ¼ Fiberð f ðzÞÞ:

Proof. Note that, for any finite subset Z of Jfinð f Þ and any connected component J of Jð f Þ; each connected component X of f À1ðJÞ\f À1ðZÞ maps onto a connected component of J\Z: The surjectivity is a joint consequence of X being open in f À1ðJÞ and f being an open map. First, we show that f ðFiberðzÞÞCFiberð f ðzÞÞ: Let J be the connected component of Jð f Þ which contains z and consider a finite set ZCJfinð f Þ: Without loss of generality, we may assume that ZCf ðJÞ: Now if xAFiberðzÞ; then both x and z lie in the closure of X; where X is a connected component of J\f À1ðZÞ: Therefore f ðxÞ and f ðzÞ lie in the closure of f ðXÞ; which is a connected component of f ðJÞ\Z: By the previous lemma, f ðxÞAFiberð f ðzÞÞ: Next, for the converse, suppose that ZCJ is a finite subset of Jfinð f Þ: If xAFiberð f ðzÞÞ; then x and f ðzÞ lie in the closure of a connected component X of f ðJÞ\f ðZÞ: Thus there exists a component Y of J\f À1ð f ðZÞÞ such that zAY and which maps onto X: Hence there exists a preimage of x in the closure of Y: Since this occurs for all Z; there exists a preimage of x in the fiber of z; by the previous lemma. To show that FiberðzÞ is connected let Z1SCZ2C? be an increasing sequence of finite subsets of Jfinð f Þ such that Jfinð f Þ¼ Zi: For each iX1; consider the set Yi formed by the union of the closure Xj of the connected components Xj of Jð f Þ\Zi such that zAXj: It follows that YTi is closed and connected, for all i: By the previous lemma we have that FiberðzÞ¼ Yi; which is a connected set. &

In the next corollary, we conclude that trivial ﬁbers imply local connectivity.

Corollary 2.16. Let z be a point in Jð f Þ such that FiberðzÞ¼fzg: Then the connected component J of Jð f Þ that contains z is openly locally connected at z: ARTICLE IN PRESS

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Proof. ConsiderS an increasing collection Z1CZ2Cy of ﬁnite subsets of Jfinð f Þ such that Zi ¼ Jfinð f Þ\fzg: Denote by Xi the connected component of Jð f Þ\Zi that contains z: According to Proposition 2.13, the set Xi is open in J: Since zA-XiCFiberðzÞ¼fzg; the collection fXig is a basis of connected open neighborhoods of z in J: &

Now our aim is to show that, under certain hypotheses, impressions are contained in fibers (see Proposition 2.19). Toward this purpose, our next result shows that each fiber is ‘‘surrounded’’ by a Jordan curve that intersects the Julia set only at periodic and preperiodic points. Hence it establishes a basic relationship between fibers and the embedding of Jð f Þ in C:

Lemma 2.17. Consider a periodic or preperiodic connected component J of Jð f Þ and a finite subset Z of J such that the number of connected components of J\Z is finite. Let X be a connected component of J\Z and choose an open neighborhood N of X: Then there exists a Jordan curve gCN such that J-gCZ and X ¼ U-J; where U is the bounded component of C\g: Moreover, given r40; the curve g is homotopic, rel J, to a curve g* whose intersection with the basin of infinity consists of finitely many arcs a1; y; ak; where each arc ai is either a piece of a ray or a piece of the equipotential gf ¼ r:

Proof. Since J\Z has ﬁnitely many connected components, we may shrink N (if necessary) so that every connected component Y of J\Z; which is not X; contains points in C\N (see Fig. 3). Consider the set A ¼ðC\NÞ,ðJ\ðZ,XÞÞ:

Note that A and X are closed and connected subsets of the normal topological space C\Z: Apply the Tietze extension theorem to obtain a continuous map h : C\Z-½0; 1 such that hÀ1ð0Þ¼A and hÀ1ð1Þ¼X: We may assume that h is smooth in C\J: Let r be a regular value. Then hÀ1ðrÞ is a one-dimensional manifold that separates A from

Fig. 3. Illustration of the proof of Lemma 2.17. ARTICLE IN PRESS

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X in C\Z: Since hÀ1ðrÞ is closed in C\Z; the closure of hÀ1ðrÞ in C; denoted G; is such that G-JCZ: One connected component V of C\G contains X: Moreover, bounded connected components of C\G cannot contain points that belong to a connected component Y of J\Z distinct from X; since YCA: Let F be the smallest full connected set containing V: The interior of F is a Jordan domain U with piecewise smooth boundary gCG such that U-J ¼ X: The rest of the lemma follows from Proposition 2.12. &

Schleicher introduced fibers for connected Julia sets as subsets of the filled Julia set Kð f Þ: The following corollary shows that a fiber in our sense is a fiber in the sense of Schleicher intersected with Jð f Þ: More precisely, two points z and x in Kð f Þ are not in the same fiber in the sense of Schleicher if (1) or (2) of the corollary below hold for some s and t rational (see [31, Definition 2.3]).

Corollary 2.18. Let f be a polynomial with connected Julia set Jð f Þ: If z and x are not in the same ﬁber, then there exist rational external rays Rs and Rt such that one of the following hold:

(1) Rs and Rt land at the same point z and the graph formed by Rs; Rt and their common landing point separates z from x: (2) Rs and Rt land at the same Fatou component U and there exists an open arc bCU joining the landpoints of Rs and Rt such that the graph formed by Rs; Rt; and b separate z from x:

Proof. Let Z be a subset of Jfinð f Þ separating z and x: Let X be the connected component of Jð f Þ\Z that contains x: Let g be the curve from the previous lemma. The Jordan curve g is homotopic rel Kð f Þ in the Riemann sphere C,fNg to a curve g0 which is the union of several Jordan curves that meet at N and such that g0 intersects the basin of inﬁnity in a collection of external rays. One of these Jordan s curves G must separate z and x: It follows that G contains exactly two rays, say Rf t and Rf that land at Jfinð f Þ: Hence, s and t are rational and have the desired properties. &

Under certain conditions, impressions are contained in ﬁbers (cf. [31, Lemma 2.5]).

Proposition 2.19. If zAAccðtÞ; then ImpðtÞCFiberðzÞ:

Proof. Choose a point x which is not in FiberðzÞ: By deﬁnition, there exists Z such that z and x are in distinct connected components of Jð f Þ\Z: Denote by J the connected component of Jð f Þ which contains z: Call X the connected component of J\Z that contains z: Let g* be the Jordan curve given by the previous lemma. It follows that the bounded component of C\g* contains the portion of all the external rays with arguments in a neighborhood of tASf which is enclosed by an equipotential gf ¼ r: Therefore xeImpðtÞ: & ARTICLE IN PRESS

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3. Trivial ﬁbers

In this section, we establish that the ﬁbers of periodic and preperiodic points are trivial, in the absence of irrationally neutral cycles. That is, we prove Theorem 2. The proof relies on the construction of a puzzle.

3.1. Construction of a puzzle

The key for proving Theorem 2 is to construct a puzzle. The proof of this theorem easily reduces to the case in which the polynomial f : C-C under consideration has connected Julia set and every periodic Fatou component has period one. Thus throughout this subsection, we let f : C-C be a degree d monic polynomial with connected Julia set such that every periodic Fatou component of f is, in fact, ﬁxed. Our aim is to construct a puzzle associated with such a polynomial f :

Remark 3.1. For more about ‘‘puzzles’’, the reader may consult the papers where the original constructions by Branner–Hubbard and Yoccoz are explained (e.g., [3,12,23]). Some variations of the original construction are contained in [9,14,18].

The construction of a puzzle boils down to finding an appropriate collection of graphs fGlglX1: For each lX1; the corresponding graph Gl is obtained by putting together one graph per each fixed Fatou component. In 3.1.2, we associate to the J basin of infinity a graph Gl which is contained in the closure of the basin of infinity ( for a similar construction, see Chapter 3 of [14]; also compare with ‘‘Yoccoz’s Structure’’ of [18]). In 3.1.3 and 3.1.4, we associate to each fixed bounded Fatou U component U a graph Gl whose intersection with the Fatou set is contained in the union of the basin of infinity and the bounded Fatou components V such that f lðVÞ¼U: We deal separately with the case in which U is the immediate basin of attraction of a fixed point and that in which U is the immediate basin of a parabolic fixed point. (Our construction of graphs associated to bounded Fatou components is related to and influenced by one due to Faught [9].)

3.1.1. Graphs and puzzles It is convenient to agree on certain terminology before beginning the construction of the collection of graphs mentioned above. A graph in C cuts the complex plane into several regions. We are interested in those regions that contain points of the Julia set.

Deﬁnition 3.2. Let GCC be a graph. A connected component of C\G that contains at least one point of Jð f Þ is called a G-puzzle piece. The support of the G-puzzle is the closed subset of C consisting of the union of the closure of all G-puzzle pieces. ARTICLE IN PRESS

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3.1.2. The graphs associated to the basin of infinity J For a given lX1; in order to construct Gl ; first we consider the set WlCJð f Þ consisting of all the periodic points w of period not greater than l with at least two periodic external rays landing at w: Then we let Zl be the set formed by the points l J zAJð f Þ such that f ðzÞAWl: Finally, we define Gl to be the graph formed by the union of the following subsets of C:

(J1) Zl; t t (J2) all the external rays R such that the landing point of R belongs to Zl; Àl (J3) the equipotential fz : gf ðzÞ¼d g:

Remark 3.3. Essentially, it is only here that our construction differs from the J a J original ones. Speciﬁcally f ðGlþ1Þ Gl ; whereas the equality usually holds for the standard constructions.

In the next statement, which we state without proof and for future reference, we J record some properties of fGl g which are a straightforward consequence of the construction.

J J Lemma 3.4. Denote by Xl the support of the Gl -puzzle (see Deﬁnition 3.2). Then, for lX1; the following hold:

À1 (1) f ð f ðGlÞÞ ¼ Gl: J J C J J C J (2) f ðGl Þ-intðXl Þ Gl and f ðGl Þ-Jð f Þ Gl : (3) J J C J J C J Gl -intðXlþ1Þ Glþ1 and Gl -Jð f Þ Glþ1:

3.1.3. The graphs associated to an attracting ﬁxed point Let U be the immediate basin of an attracting ﬁxed point, say z0: The associated U C graph G1 will consist of a Jordan curve b U which winds once around z0; arcs connecting b to some points in @U; the external rays which land at these points and U U an equipotential (see Fig. 4). Once G1 is constructed, the graphs Gl are obtained U from G1 by successive pull-backs. To obtain b; consider a positive number R such that all the elements of the critical set, as well as the elements of the post-critical set, which lie in U; are at hyperbolic distance strictly less than R from z0: In other words,

n rU ð f ðoÞ; z0ÞoR;

n for all critical points o and integers nX0 such that f ðoÞAU; where rU denotes the hyperbolic metric of U (constant curvature À1). Denote by D0 the open hyperbolic disk of radius R centered at z0: Note that D ¼ f À1ðD0Þ-U is a topological disk. In fact, by the Maximum Principle, each connected component of f À1ðDÞ is a topological disk. Since the degree and the critical points of f : f À1ðD0Þ-U-D0 and f : U-U coincide, we have ARTICLE IN PRESS

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U Fig. 4. Illustration of the construction of the graph G1 for the immediate basin U of an attracting ﬁxed U point z0: The Julia set is sketched with thick lines and the graph G1 with thin lines.

that f À1ðD0Þ-U has only one connected component, by the Riemann–Hurwitz formula. We let b be the Jordan curve which is the boundary of the topological disk D ¼ f À1ðD0Þ-U:

Remark 3.5. The domain D0 is compactly contained in D: In addition, D0CDCf À1ðDÞCf À2ðDÞ?:

Moreover, by the Maximum Principle, each connected component of f ÀlðDÞ is a topological disk. Furthermore, if a bounded Fatou component V is such that f lðVÞ¼U; then it contains exactly one connected component of f ÀlðDÞ; by the Riemann–Hurwitz formula. Also, if f lðVÞaU; then V is disjoint from f ÀlðDÞ:

U To continue with the construction of G1 ; we apply a result from complex dynamics (e.g., [4, Theorem VII.2.1]) which establishes the existence of an arc g : ½0; 1Þ-U such that f ðgÞ*g ending at a boundary fixed point w1A@U (i.e., limt-1 gðtÞ¼w1). We may assume that gð0Þ is in the boundary of the hyperbolic disk D0 and that gCU\D0: The preimage f À1ðgÞ of g consists of d arcs (recall that d is the degree of f ). There are exactly kX2 of these d arcs which are contained in U; where k - is the degree of f : U U: We denote these k arcs by g1; y; gk so that g1Cg: Each arc \ gi is a subset of the annulus U D and connects its inner boundary b with a point wiA@U; where wi is an element in the preimage of the fixed point w1 (i.e., f ðwiÞ¼w1). Note that by the Snail Lemma [22, Section 16.2], w1 is either parabolic with multiplier 1 or repelling. Therefore, some periodic external rays land at w1: The external rays which land at w1 are necessarily fixed, since the curve g ends at w1 and ARTICLE IN PRESS

224 J. Kiwi / Advances in Mathematics 184 (2004) 207–267 f ðgÞ*g: Consequently, for i ¼ 2; y; k; the prefixed point wi is the landing point of some prefixed external rays. U We introduce the graph G1 as the union of the following subsets of C: (A1) the Jordan curve b; (A2) the closure of the arcs g1; y; gk; (A3) all the external rays which land at one of the points w1; y; wk; (A4) the equipotential gf ¼ 1: U To conclude the construction of Gl ; when U is an immediate basin of an U À1 U attracting fixed point, we inductively define Glþ1 as f ðGl Þ:

3.1.4. The graphs associated to a parabolic basin U The graph G1 associated to an immediate basin U of a parabolic ﬁxed point z0 is obtained by putting together a Jordan curve b contained in U; external rays which land at some points in @U and an equipotential (see Fig. 5). À1 More precisely, ﬁrst we construct a curve b such that b-@UCf ðz0Þ: Then U t we let G1 be the graph consisting of the Jordan curve b; all the external rays R such that the landing point of Rt belongs to b-@U; and the equipotential U U U gf ¼ 1: Once we have G1 ; the graph Gl is obtained from G1 by successive pull- backs. To construct b we need an attracting petal which is convex with respect to the hyperbolic metric in U: Recall that a simply connected domain P; where f is univalent, is called an attracting petal for f in the direction v at z0 if: f ðPÞCP; the sequence of iterates f f kg restricted to P converges uniformly to the constant function with range z0; and an orbit under f is absorbed by P if and only if it converges to z0 in the direction v (see [22, p. 105]).

U Fig. 5. Illustration of the construction of G1 for a basin U of a parabolic ﬁxed point z0: The Julia set is U represented with thick lines and the graph G1 with thin lines. ARTICLE IN PRESS

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Lemma 3.6. Let U be a ﬁxed Fatou component which is the immediate basin of a parabolic ﬁxed point z0A@U: Then there exists an attracting petal PCU which is convex with respect to the hyperbolic metric and such that @P\fz0g is a smooth arc. Moreover, this petal P can be chosen as a petal in the Leau–Fatou Flower Theorem (see [22, Theorem 10.5]).

Proof. Consider a conformal isomorphism f : D-U and let B : D-D be the finite Blashcke product BðzÞ¼fÀ13f 3fðzÞ: The Julia set of B is the unit circle @D: Moreover, each of the domains D and C\D is a fixed immediate basin of a parabolic fixed point in @D: We may choose f so that 1A@D is the parabolic fixed point of B: It follows that, as z-1;

BðzÞ¼1 þðz À 1Þþaðz À 1Þ3 þ Oððz À 1Þ4Þ; where a is a negative real number, since B has exactly two attracting petals at z ¼ 1 with ‘‘horizontal’’ attracting directions. In the coordinate

À1 w ¼ ; 2aðz À 1Þ2

B becomes g : C\ðÀN; 0-C which is of the form gðwÞ¼w þ 1 þ oð1Þ; where oð1Þ-0asjwj-N: Therefore, we may choose C40 large enough so that gðfz : Re zXCgÞCfz : Re z4Cg: Let P˜ ¼ hÀ1ðfz : Re z4CgÞ: The domain P˜ is hyperbolic-convex. Indeed, consider / À1 the conformal isomorphism c : z zÀ1 between D and the right half-plane. The image of P˜ under this map is the domain bounded by the hyperbola x2 À y2 ¼À2aC; which is a hyperbolic-convex domain in the right half-plane. Thus P˜ is hyperbolic- convex. To complete the construction of the petal, let P ¼ fðP˜Þ: &

U We now continue with the construction of G1 : Let rU be the hyperbolic metric in n U: Consider R so that rU ðz;@PÞoR; for all zAU such that z ¼ f ðoÞ for some nX0 and some critical point o: Let D0CU be the domain consisting of all the points in U that are at hyperbolic distance strictly less than R from P: Since P is convex, D0 is a Jordan domain. Our choice of R implies that there exists a unique connected component D of f À1ðD0Þ which is contained in U: This domain is also bounded by a 0 Jordan curve, which we denote by b: Since @D -@U ¼fz0g; we have that À1 b-@UCf ðz0Þ:

Remark 3.7. In comparison with the construction for the basin of an attracting ﬁxed 0 0 point, note that now D -@D ¼fz0g; thus D is not compactly contained in D: But still

D0CDCf À1ðDÞCf À2ðDÞ?: ARTICLE IN PRESS

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Moreover, each connected component of f ÀlðDÞ is a topological disk, for all lX1: Furthermore, if a bounded Fatou component V is such that f lðVÞ¼U; then it contains exactly one connected component of f ÀlðDÞ: For otherwise, V would be disjoint from f ÀlðDÞ:

U Let G1 be the graph formed by: (P1) the Jordan curve b; (P2) all the external rays which land at points that belong to b-@U; (P3) the equipotential gf ¼ 1: U Finally, to complete the construction of the graphs Gl associated to an immediate U À1 U parabolic basin U; inductively deﬁne Glþ1 ¼ f ðGl Þ: For both a parabolic basin U and an attracting basin U; it follows that the support U U \ Àl Xl of the Gl -puzzle is the set fz : gf ðzÞp1g f ðDÞ: For future reference and without proof, we next record other immediate consequences of the constructions above.

U Lemma 3.8. Let fGl g be the collection of graphs associated to a ﬁxed bounded Fatou U J component U (attracting or parabolic). Denote by Xl the support of the Gl -puzzle (see Deﬁnition 3.2). Then, for l41; the following hold:

À1 U U (1) f ð f ðGl ÞÞ ¼ Gl : U U C U U C U (2) f ðGl Þ-intðXl Þ Gl and f ðGl Þ-Jð f Þ Gl : (3) U U C J U C J GlÀ1-intðXlÀ1Þ f ðGl Þ and GlÀ1-Jð f Þ f ðGl Þ:

3.2. The puzzle and its properties

In this subsection, let f : C-C be a monic degree d polynomial with connected Julia set Jð f Þ and such that all its periodic Fatou components are ﬁxed. We let fGlg be the collection of graphs associated to such a polynomial f as constructed in the previous subsection. Here our aim is to study the basic properties of the puzzle determined by this collection. To study the basic properties of Gl-puzzle pieces, we introduce the auxiliary collection of graphs given by 0 fGl ¼ f ðGlÞglX2:

Also, for a given zAC; we let PlðzÞ denote the Gl-puzzle piece that contains z; 0 whenever such a piece is well deﬁned. Similarly if well deﬁned, then Pl ðzÞ will denote 0 the Gl -puzzle piece that contains z:

Lemma 3.9. For all integers lX2; the following statements hold:

0 (1) If P is a GlÀ1-puzzle piece or a Gl -puzzle piece, then P is a closed topological disk. Moreover, P-Jð f Þ is connected. ARTICLE IN PRESS

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0 (2) If P is a Gl-puzzle piece, then f ðPÞ is a Gl -puzzle piece and f : P-f ðPÞ is a proper holomorphic map. 0 0 (3) PlðzÞCPl ðzÞCPlÀ1ðzÞ: Also, Gl-Jð f ÞCGl -Jð f ÞCGlÀ1-Jð f Þ: (4) If zAJð f Þ\Gl; then the fiber of z is contained in PlðzÞ: If zAJð f Þ-Gl; then the 1 n fiber of z is contained in P ,?,P ; where P1; y; Pn is the complete list of all Gl-puzzle pieces P such that zA@P: (5) Let Pl be a Gl-puzzle piece such that a fixed point z belongs to Pl: If z is parabolic, then zA@Pl and there exists a repelling petal L of z such that L-Pl is a closed topological disk which contains a relative neighborhood of z in Pl: 0 (6) If P is a Gl-puzzle piece or Gl -puzzle piece and z0 is an attracting fixed point of f, then z0eP:

Proof. For assertion (2), since Lemmas 3.4 and 3.8(1) guarantee that each of the À1 À1 0 members of Gl is the preimage of its image, also f ð f ðGlÞÞ ¼ f ðGl Þ¼Gl; for all lX2: Thus f maps complementary components of Gl onto complementary 0 components of Gl as a proper map. 0 To prove (3), let Xl be the support of the Gl-puzzle. Note that the support Xl of 0 the Gl -puzzle coincides with XlÀ1: It follows from Lemmas 3.4 and 3.8(2) that 0 0 Gl -intðXlÞCGl: Thus PlðzÞCPl ðzÞ: From Lemmas 3.4 and 3.8(3) it follows 0 0 that GlÀ1-intðXlÀ1ÞCGl : Hence Pl ðzÞCPlÀ1ðzÞ: Similarly, we have that 0 Gl-Jð f ÞCGl -Jð f ÞCGlÀ1-Jð f Þ: For statement (1), complementary components are simply connected, since the graphs involved are connected. By construction, the boundary of a connected component of C\G1 is a Jordan curve. It follows by (3) that every puzzle piece is a Jordan domain. The intersection of a Gl-puzzle piece P with Jð f Þ is connected. For otherwise, there would exist an access to Gl-Jð f Þ that is non-homotopic to any access contained in Gl; by Lemma 2.17. But Gl includes at least one access per homotopy class of accesses to Gl-Jð f Þ; which is a contradiction. Assertion (4) follows immediately from the definition of fibers together with Lemma 2.14 and the fact that Jð f Þ-Gl is a finite set of eventually parabolic or eventually repelling points. To establish (5), consider a parabolic fixed point z: By construction, we have that zAG1: Therefore if z lies in the closure of a puzzle piece Pl; then zA@Pl: Moreover, since two immediate basins of z (if they exist) are separated by a pair of fixed rays landing at z; the puzzle piece Pl has non-trivial intersection with at most one immediate basin of z: If the puzzle piece Pl is disjoint from the immediate basin(s) of z; then a neighborhood of z in @Pl consists of two pieces of fixed external rays that land at z through the same repelling direction v: A repelling petal L in this direction is the required petal. If the puzzle piece Pl has non-trivial intersection with the immediate basin U of z; then we let N be a neighborhood of z such that N\fzg is covered by a collection of petals and one of these petals is the convex attracting petal P of Lemma 3.6. It follows that N-@Pl consists of the union of two arcs. One of the arcs lies in a fixed ARTICLE IN PRESS

228 J. Kiwi / Advances in Mathematics 184 (2004) 207–267 external ray which lands at z through a repelling direction, say v: The other arc lies outside P; between the repelling direction v and the attracting direction corresponding to P: It follows that the repelling petal L associated to the direction v is the desired petal. To complete the proof, just note that (6) follows immediately from the construction. &

3.3. Weakly polynomial-like maps

In the proof of Theorem 2, we count the number of ﬁxed points contained in a given puzzle piece. In order to count, we apply some results due to Goldberg and Milnor [10] concerning weakly polynomial-like maps (cf. polynomial-like maps of Douady and Hubbard [8]).

Deﬁnition 3.10. Consider a Jordan domain DCC and a continuous map g : D-C such that gð@DÞ-D ¼ |: We say that g is a degree d weakly polynomial-like map if the induced map in the integer homology

gÃ : H2ðD;@DÞDZ-H2ðC; C\fz0gÞDZ is multiplication by d: Here z0 can be any base point in D:

Remark 3.11. Consider a weakly polynomial-like map g : D-C such that g : D-gðDÞ is a degree k branched covering. Then the degree of g as a weakly polynomial-like map is also k:

Lemma 3.7 of [10] reads as follows:

Lemma 3.12 (Goldberg and Milnor). If g : D-C is a degree d weakly polynomial- like map with isolated fixed points, then each fixed point zi can be assigned a Lefchetz fixed point index ið f ; ziÞ which is a local invariant and such that the sum of the Lefchetz indices is equal to d.

According to Goldberg and Milnor, the Lefchetz index of an interior fixed point z0 can be defined as the local degree of z-gðzÞÀz at z0 which coincides with the multiplicity of z0 when g is holomorphic. In our next result, we count the number of fixed points of a polynomial f : C-C contained in the closure of a puzzle piece, where f is a degree d monic polynomial with connected Julia set such that every periodic Fatou component of f is fixed. In the statement of the proposition, fGlg will denote the puzzle constructed in Section 3.1 for such a polynomial.

Proposition 3.13. Let Ul be a Gl-puzzle piece. If f ðUlÞ*Ul; then U l contains exactly k distinct ﬁxed points, where k is the degree of f : Ul-f ðUlÞ: ARTICLE IN PRESS

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Proof. By Remark 3.11 and Lemma 3.9(2), we have that f : U l-f ðU lÞ is a degree k weakly polynomial-like map. Since f is holomorphic, it has isolated ﬁxed points. We claim that all its ﬁxed points have Lefchetz index +1. In fact, by Lemma

3.9(5), if w is an interior fixed point of f : Ul-C; then w is a simple fixed point. Thus ið f ; wÞ¼þ1: Now if w is a parabolic boundary fixed point, then according to Lemma 3.9(5), there exists a repelling petal L such that L-U l is a closed topological disk which contains a relative neighborhood of w in Ul: Since L is a repelling petal, f ð@LÞ-L ¼ |: Thus f : L-C is a degree one weakly polynomial-like map. Hence its unique fixed point w has Lefchetz index +1. Similarly, if w is a repelling boundary fixed point, then ið f ; wÞ¼þ1: Since all the fixed points of f : U l-f ðU lÞ have Lefchetz index þ1; the preceding lemma implies that f has exactly k fixed points in U l: &

3.4. Proof of Theorem 2

Suppose that f : C-C is a monic degree d polynomial without irrationally neutral cycles. We must show that the ﬁbers of preperiodic and periodic points are trivial. We begin by dealing with the case when Jð f Þ is connected.

3.4.1. Proof of Theorem 2 for connected Jð f Þ Consider a periodic point zAJð f Þ: By passing to an iterate, we may assume that the bounded Fatou components are fixed, that the point z is fixed, and that all the external rays which land at z are also fixed. Consider the collection of graphs fGlg as constructed in Section 3.1. To show that the fiber of the fixed point z is trivial, we let fUlglX1 be a nested collection of Gl- puzzle pieces such that, for all lX1; the fixed point z lies in U l: If z is a repelling point with exactly one external ray landing at it, then this collection is unique, since in this case zeGl; for all lX1: If z is a parabolic fixed point or the landing point of more than one external ray, then zAG1; and therefore z is in the closure of finitely many G1-puzzle pieces. Consequently once U1 is chosen, the rest of the nested collection fUlg is determined. Moreover, every piece that contains z in its boundary participates in one nested collection. By Lemma 3.9(4), to conclude that the fiber of z is trivial it suffices to establish that \ X ¼ U l ¼fzg:

From the fact that z is a fixed point which is landed by fixed ray(s), we conclude that f ðUlÞ*Ul: Since fUlg is nested, the degree of f : Ul-f ðUlÞ eventually stabilizes, say for all lXl0; the degree is k: By Proposition 3.13, the polynomial f has exactly k distinct fixed points in U l; for all lXl0: Therefore f has exactly k fixed points in X: By Lemma 3.9(6), none of these fixed points is attracting. Therefore, these k fixed points are either repelling or parabolic, since f is a polynomial without irrationally neutral cycles. ARTICLE IN PRESS

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Each one of these k fixed points is the landing point of periodic external rays, say of period p: We claim that p ¼ 1: Suppose that there exists a fixed point wAUl with a period p41 ray landing at it. Then wA@Ul and Ul is contained in a sector bounded by rays landing at w: Also, f ðUlÞ must be contained in that sector. But locally, around w; this sector maps outside Ul; thus f ðUlÞ is disjoint from Ul; which is a contradiction. Therefore p ¼ 1: We suppose that X is not a singleton and after some work we obtain a contradiction. Consider a conformal isomorphism h : C\X-C\D: The image of Ul\X under h is a neighborhood of @D\F; where F is a finite set of points. More precisely, if wAF; then there are two arcs in C\D which end at w such that their preimage under h is contained in the boundary of Ul and end at a common point À1 in X: Let g : hðUl\XÞ-C\D be defined by gðzÞ¼h3f 3h ðzÞ: By the Schwartz reflexion principle, g : hðUl\XÞ-C\D extends to @D\F and, by continuity, to an orientation preserving selfcovering G : @D-@D: Counting preimages of a point where hÀ1 has radial limit, G also has degree k: Since each one of the k fixed points of X is accessible from C\X by a fixed external ray, G has at least k fixed points.

Claim. Every ﬁxed point of G is repelling.

Proof of the Claim. We proceed by contradiction and suppose that there exists a one- or two-sided attracting ﬁxed point wA@D of G: Let aChðUl\XÞ be an arc with one endpoint at w and the other in @D so that the open set VCC\D bounded by a is contained in the basin of attraction of w and GðVÞCV: It follows that f f ng is a normal family in hÀ1ðVÞ: Thus hÀ1ðVÞ is contained in a Fatou component U: Since hÀ1ðVÞ contains a cross cut of @U and the dynamics of f over U is conformally - conjugate to a Blashcke product B : DS D with Julia set @D; it follows that at most n À1 one point of U is not contained in nX1 f ðh ðVÞÞ; which is a contradiction. Hence we have established the claim.

Since a degree k orientation preserving selfcovering G : @D-@D with all fixed points repelling has exactly k À 1 fixed points, we obtain a contradiction with the previous count of k fixed points. Thus X is a singleton. Therefore, the fiber of the fixed point z is trivial. To complete the proof, consider a preperiodic point x such that f kðxÞ¼z: Since the fiber of x is a closed and connected subset of Jð f Þ whose image under the finite- to-one map f k is the fiber of z; it follows that the fiber of x is trivial. &

3.4.2. Proof of Theorem 2 for disconnected Jð f Þ We now suppose that Jð f Þ is disconnected. Let z be a periodic point in this disconnected Julia set Jð f Þ: By passing to an iterate, we may assume that z is ﬁxed. If the connected component C of Kð f Þ that contains z is not a singleton, then f restricted to an appropriate neighborhood of C is a polynomial-like map with ﬁlled ARTICLE IN PRESS

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Julia set C (e.g., see [15, Lemma 4.2]). Therefore, the dynamics of f : C-C is topologically conjugate to that of a complex polynomial (of lower degree) g : KðgÞ-KðgÞ with connected Julia set. Since we proved that the fiber of a fixed point in a connected Julia set is trivial and a fiber is mapped onto a fiber under a topological conjugacy, it follows that the fiber of z is trivial. Triviality of the fibers of preperiodic points follows as well. &

3.5. Corollaries

We collect some consequences of Theorem 2 that include the proofs of Corollaries 1.1 and 1.2. Recall that Corollary 2.16 asserts that if the ﬁber FiberðzÞ is trivial, then the connected component of Jð f Þ that contains z is locally connected about z: Thus by the theorem, we immediately obtain Corollary 1.1, which states that a connected component J of Jð f Þ is openly locally connected about its periodic and preperiodic points, provided that f is a polynomial without irrationally neutral cycles. Corollary 1.2, which states that each impression that contains a periodic or preperiodic point is trivial, is proved below together with the following more general result (cf. [31, Lemma 2.5]).

Corollary 3.14. Consider a polynomial f : C-C without irrationally neutral cycles. (1) If zAImpðtÞ; then ImpðtÞCFiberðzÞ: (2) If zAImpðtÞ is periodic or preperiodic, then ImpðtÞ¼fzg:

Proof. Since (2) follows from (1) together with Theorem 2, we just have to prove (1). Suppose that zAImpðtÞ: We first deal with the case in which AccðtÞ contains a periodic or preperiodic point x: By Proposition 2.19, ImpðtÞ is contained in the fiber of this periodic or preperiodic point x: Theorem 2 implies that this fiber is trivial. Therefore ImpðtÞ is a singleton. It follows that fzg¼ImpðtÞ¼ FiberðzÞ: Next, in the case that AccðtÞ is disjoint from periodic and preperiodic points, consider a finite subset Z of points with finite forward orbit such that zeZ: Denote by X the connected component of Jð f Þ\Z containing z: The Jordan curve g* given by Lemma 2.17, which consists of pieces of rays and equipotentials, encloses X and intersects Jð f Þ only at points in Z: Under our assumption, no piece of the ray Rt with argument t participates in g*: It follows that the bounded component of C\g* contains a neighborhood of the portion of Rt which is below certain equipotential. Therefore ImpðtÞCX: By the definition of fibers, we conclude that ImpðtÞCFiberðtÞ: &

Based on the fact that the ﬁbers of periodic and preperiodic points are trivial, we show that the ﬁbers of arbitrary points are equal or disjoint (cf. [31, Lemma 2.7]). ARTICLE IN PRESS

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This will allow us to quotient C by the ‘‘ﬁber equivalence relation’’. The next proposition will also be useful for studying the properties of the quotient topological space, as well as the map induced by f on it. In order to state the proposition, we need the notion of lim sup of a collection of sets. For a collection fCng of subsets of C; we say zAlim sup Cn if every neighborhood of z meets at least one set of the collection fCng:

Proposition 3.15. Let f : C-C be a polynomial without irrationally neutral cycles. Then the following hold:

(1) Fibers are either equal or disjoint. (2) If zAJð f Þ; then FiberðzÞ is a compact, connected and full subset of C: (3) If zn-z; then lim sup FiberðznÞCFiberðzÞ:

Proof. Concerning (1), consider a point zAJð f Þ and a preperiodic or periodic point xAJfinð f Þ: Suppose that zax: Since FiberðxÞ¼fxg; there exists a finite set ZCJfinð f Þ such that the connected component X of Jð f Þ\Z that contains z does not contain x: Therefore xeFiberðzÞ: Hence the fibers of z and x are disjoint. Now assume that x is not in Jfinð f Þ: If FiberðzÞaFiberðxÞ; then there exists ZCJfinð f Þ such that FiberðzÞ is contained in the closure of X and FiberðxÞ is contained in the closure of Y; where X and Y are connected components of Jð f Þ\Z: Since X-YCZ and the fibers of points in Z are equal or disjoint with the rest of the fibers, it follows that FiberðxÞ and FiberðzÞ are disjoint. To prove (2), we claim that if z has infinite orbit, then FiberðzÞ is full. Otherwise, a bounded Fatou component U would have boundary contained in FiberðzÞ: By Carlesson and Gamelin [4, Theorem VII.2.1] @U contains at least one preperiodic point (cf. [28] where it is shown that preperiodic points are dense in @U). Therefore, FiberðzÞ would contain a preperiodic point. But this is impossible, since fibers are equal or disjoint. It remains to establish assertion (3). Since L ¼ lim sup FiberðznÞ is connected, it follows that L is contained in the unique connected component J of Jð f Þ such that zAJ: Given a finite set ZCJfinð f Þ; the connected component X of J\Z that contains z is surrounded by a Jordan curve g* such that g*-Jð f ÞCZ: Moreover XCV; where V is the bounded component of C\g*: Therefore FiberðznÞCV; for all n sufficiently large. Hence if xAJ is separated from z by Z; then xelim sup FiberðznÞ: &

4. Finiteness Theorem

The aim of this section is to prove that the fiber of a point in the Julia set of a polynomial without irrationally neutral cycles is the union of finitely many impressions (i.e., Theorem 3). With this purpose in mind, we study the quotient obtained from C after collapsing each fiber to a point. ARTICLE IN PRESS

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4.1. Collapsing ﬁbers

We ﬁrst introduce the ﬁber equivalence relation.

Deﬁnition 4.1. Let f : C-C be a polynomial without irrationally neutral cycles. We say that z; wAC are ﬁber-equivalent, denoted zEw; if either z ¼ w; or z; wAJð f Þ and FiberðzÞ¼FiberðwÞ:

The equivalence relation above is well defined, since Proposition 3.15 asserts that fibers are equal or disjoint. Our next result studies the quotient of C by E: Recall that Jfinð f Þ (see Definition 2.5) denotes the set consisting of all periodic and preperiodic points in Jð f Þ which are not in the grand orbit of a Cremer point.

Proposition 4.2. Consider a polynomial f : C-C without irrationally neutral cycles. Let E be the ﬁber equivalence relation of f. Then:

(1) The quotient of C by E is homeomorphic to C: (2) q : Jfinð f Þ-qðJfinð f ÞÞ is a homeomorphism, where q : C-CDC=E denotes the quotient map. (3) Q : C-C is a degree d branched covering, where Q : C-C is the unique map such that QðqðzÞÞ ¼ qð f ðzÞÞ:

Proof. To show that the quotient C=E is homeomorphic to C; we apply a theorem due to Moore [25]. This result guarantees that the quotient of a two-dimensional sphere S2 under an equivalence relation l is homeomorphic to a sphere if l is a closed equivalence relation whose equivalence classes are connected sets with non- empty connected complement. From assertion (3) of Proposition 3.15 it follows that E is a closed equivalence relation, and from assertion (2) that E classes are connected with connected complement. Therefore C=E is homeomorphic to C: We proceed to show that Q : C-C is a degree d branched covering. If zeqðJð f ÞÞ; then Q is a local homeomorphism about z: Thus Q is a degree d branched covering in C\qðJð f ÞÞ: Now if z ¼ qðFiberðzÞÞAqðJð f ÞÞ and no critical value of f lies in FiberðzÞ; then for an arbitrarily small neighborhood N of FiberðzÞ; there exists a Jordan curve gCN such that FiberðzÞ is contained in the domain bounded by g: À1 Also, g-Jð f ÞCJfinð f Þ; by Lemma 2.17. Moreover, N can be chosen so that f ðNÞ consists of d domains which are pairwise disjoint. Hence f À1ðgÞ consists of d pairwise disjoint Jordan curves g1; y; gd : Since gi-Jð f ÞCJfinð f Þ; the map q maps each one of these Jordan curves homeomorphically onto its image. Thus the domain Di bounded by gi has image qðDiÞ; which is also a Jordan domain. Similarly, the domain D bounded by g maps onto the Jordan domain qðDÞ: Therefore, the preimage of qðDÞ under Q consists of the Jordan domains qðD1Þ; y; qðDd Þ each of which is mapped in a one-to-one fashion onto qðDÞ under Q: It follows that Q : C-C is a degree d branched covering. ARTICLE IN PRESS

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By Theorem 2, q : Jfinð f Þ-qðJfinð f ÞÞ is one-to-one. To see that this map is a homeomorphism, we just have to check that its inverse is continuous. For otherwise, there would exist a sequence fzng in Jfinð f Þ not converging to zAJfinð f Þ such that qðznÞ-qðzÞ: After passing to a subsequence we may suppose that zn-xAJð f Þ; with xaz: We would have qðxÞ¼qðzÞ; since q is continuous. Therefore x would belong to qÀ1ðqðzÞÞ ¼ FiberðzÞ; which is equal to fzg by Theorem 2. Hence, x would be equal to z; which is a contradiction. &

We say that the branched covering Q : C-C of the proposition above is the map obtained after collapsing the ﬁbers of f. For this branched covering Q; it is reasonable to deﬁne the sets corresponding to rays, to the Fatou set, to the Julia set, etc.

Definition 4.3. Consider a polynomial f : C-C without irrationally neutral cycles. Let Q : C-C be the map obtained after collapsing fibers of f : We say that: qðJð f ÞÞ is the topological Julia set JtopðQÞ of Q; C\JtopðQÞ is the topological Fatou set F topðQÞ of Q; qðKð f ÞÞ is the topological filled Julia set KtopðQÞ of Q; C\KtopðQÞ is the basin of infinity of Q; t t A Rtop ¼ qðR Þ is the topological ray with argument t Sf ; qðfz=gf ðzÞ¼rgÞ is a topological equipotential of Q:

Note that f : Dr ¼ fz=jzjorg-C is a degree d weakly polynomial-like map, for r large enough (see Section 3.3). Therefore Q : qðDrÞ-C is also a degree d weakly polynomial-like map for r large enough. According to Goldberg and Milnor, the sum of the Lefchetz indices of the fixed points of Q is d (see Lemma 3.12). The quotient map q induces an isomorphism between the local homologies around fixed points. In other words, it is not difficult to check that if z0 is a fixed point of f ; then qÃ : H2ðU; U\fz0gÞ-H2ðV; V\fhðz0ÞgÞ is an isomorphism where U is a small Jordan domain around z0 such that qðVÞ is a Jordan domain around qðz0Þ (see Lemma 2.17). After counting periodic points of Q; in our next result we conclude that periodic fibers are in one-to-one correspondence with periodic points.

Lemma 4.4. Let f : C-C be a polynomial without irrationally neutral cycles. If f nðFiberðzÞÞ ¼ FiberðzÞ; then FiberðzÞ contains a periodic point, so FiberðzÞ¼fzg and f nðzÞ¼z:

Proof. Let Zf be the set consisting of all the periodic points, under f ; of period n n dividing n: For r large enough, f : Dr-C is a degree d weakly polynomial-like map. By Lemma 3.12, X iðx; f nÞ¼dn:

xAZn ARTICLE IN PRESS

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As discussed above, iðx; f nÞ¼iðqðxÞ; QnÞ: Therefore, all the points of period dividing n under Q are the image, under q; of a periodic point in Zf : Hence, if n z ¼ qðFiberðzÞÞ and f ðFiberðzÞÞ ¼ FiberðzÞ; then zAqðZf Þ: Since the ﬁbers of points in Zf are trivial, it follows that zAZf is periodic of period dividing n under f : &

As an immediate consequence we obtain the following.

Corollary 4.5. Let f : C-C be a polynomial without irrationally neutral cycles. If z is a point with inﬁnite forward orbit, then FiberðzÞ is a wandering closed connected set.

In relation to wandering connected sets we have the following.

Corollary 4.6. Let f : C-C be a polynomial without irrationally neutral cycles. If a connected subset C of Jð f Þ contains no periodic nor preperiodic points, then its closure C is a wandering set.

Proof. Suppose CCJð f Þ is a connected set which contains no periodic nor preperiodic points and choose zAC: It follows from the definition of fibers that CCFiberðzÞ (see Definition 2.5). Note that the preceding corollary states that the fiber of a point with infinite orbit must be a wandering closed connected set. Thus CCFiberðzÞ is a wandering closed and connected set and the result follows. &

Since Q is obtained after collapsing the ﬁbers of f ; the topological Julia set of Q is effectively parameterized by Sf :

t Proposition 4.7. For all tASf ; the topological ray Rtop lands at qðImpðtÞÞ: The t A landing point of Rtop; call it wðtÞ; depends continuously on t Sf : Moreover, every point in JtopðQÞ is the landing point of at least one topological ray. That is,

top w : Sf -J ðQÞ is a continuous surjection such that wðdtÞ¼QðwðtÞÞ:

t Proof. Note that R -Jð f ÞCAccðtÞCImpðtÞ: An impression is either contained in a ﬁber or disjoint (see Corollary 3.14). Thus ImpðtÞCFiberðzÞ; for some z: It follows t t that qðR -Jð f ÞÞCqðFiberðzÞÞ ¼ fzg: Hence Rtop lands at z: We show that w is continuous. Note that if tn converge to t; then lim sup ImpðtnÞCImpðtÞ: It follows from the continuity of q : C-C=E that wðtnÞ¼ qðImpðtnÞÞ converge to wðtÞ¼qðImpðtÞÞ: The map w is surjective, since every point of Jð f Þ lies in ImpðtÞ; for some tASf : Moreover, from f ðImpðtÞÞ ¼ ImpðdtÞ it follows that wðdtÞ¼QðwðtÞÞ: & ARTICLE IN PRESS

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Corollary 4.8. Let lðQÞ be the equivalence relation in Sf that identiﬁes t with s if wðtÞ¼wðsÞ: Then lðQÞ has a closed graph in Sf Â Sf : Moreover, the map

top w# : Sf =lðQÞ -J ðQÞ ½t /wðtÞ is a homeomorphism such that w#ð½dtÞ ¼ Qðw#ð½tÞ:

We say that lðQÞ is the landing relation of Q:

Proof of Corollary 4.8. The graph of lðQÞ is closed because w is continuous. By the deﬁnition of quotient topology, w# is continuous. Since w# is continuous and injective from a compact onto a Hausdorff space, it follows that w# is a homeomorphism. &

We are now concerned with certain properties of some lðQÞ-classes. We focus on the lðQÞ-classes which are completely ‘‘contained’’ in T\Sf : More precisely, recall that Sf CT denotes the countable set formed by the arguments for which left limit 7 7 A C rays and right limit rays are deﬁned (see Section 2.1). We let Sf ¼ft : t Sf g Sf \ þ À \ \ and identify Sf Sf ,Sf with T Sf : Note that T Sf is the set of arguments for which smooth external rays are well deﬁned. In what follows, we will use interval notation in the circle T with the agreement that ðt; tÞ¼T\ftg:

Deﬁnition 4.9. Let A and B be closed disjoint subsets of T: We say that A and B are unlinked if there exist disjoint intervals IA and IB contained in T such that ACIA and BCIB: We say that A/d Á A is consecutive preserving if for every connected component ðs; tÞ of T\A we have that ðds; dtÞ is a connected component of T\d Á A:

Corollary 4.10. Suppose that A1 and A2 are distinct lðQÞ-classes such that both A1 and A2 are contained in T\Sf : Then:

(1) A1 and A2 are unlinked. (2) d Á A1 is a lðQÞ class. (3) A1/d Á A1 is consecutive preserving. Moreover, if A1/d Á A1 is one-to-one, then it is cyclic order preserving.

top Proof. Let ziAJ ðQÞ be the landing point of the topological rays with arguments in Ai; for i ¼ 1; 2: Concerning (1), suppose that A1 and A2 are not unlinked. Then there exists a pair of topological external rays with arguments in A1 that, together with z1; chop the complex plane into two regions such that both regions contain topological external rays with arguments in A2: Since topological external rays with arguments in T\Sf are equal or disjoint, we have that z1=z2: Therefore A1 ¼ A2: ARTICLE IN PRESS

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t dt Concerning (2), by continuity of Q and the identity QðRtopÞ¼Rtop; it follows that the topological rays with arguments in d Á A1 land at Qðz1Þ: Conversely, since Q is a s t branched covering, we may lift a topological ray Rtop landing at Qðz1Þ to a ray Rtop which lands at z1: This implies that tAA1 and that dt ¼ s: For assertion (3), consider the closed set [ t X1 ¼fz1g, Rtop: tAA1

À1 Note that X1 is a connected component of Q ðQðX1ÞÞ: Let ðt; sÞ be a connected À1 component of T\A1: Then there exists a connected component U of C\Q ðQðX1ÞÞ that contains topological rays with arguments in ðt; t þ eÞ; as well as rays with arguments in ðs À e; sÞ; for e40 small. The connected component U maps onto a connected component of C\QðX1Þ that contains rays with arguments in ðdt; dt þ deÞ as well as rays with arguments in ðds À de; dsÞ: It follows that ðdt; dsÞ is a connected component of T\d Á A1: Thus A1/d Á A1 is consecutive preserving. Now suppose A1/d Á A1 is one-to-one. Let t1; t2; and t3 be elements of A1: Deﬁne t1 t2 t3 X1 ¼fz1g,Rtop,Rtop,Rtop: Again, since X1 is a connected component of À1 Q ðQðX1ÞÞ; along the lines of the previous paragraph, it follows that A1/d Á A1 is cyclic order preserving. &

4.2. Finiteness

In this subsection we prove Theorem 3, which asserts that, for a polynomial f without irrationally neutral cycles, the number of impressions contained in a fiber is finite. Counting the number of impressions contained in a fiber amounts to counting how many topological external rays land at a common point in JtopðQÞ; where Q : C-C is the map obtained after collapsing fibers of f (see Section 4.1). We use the next result (see [16, Theorem A.2], which is a generalization of [17, Theorem 1.1]) to count the number of topological external rays landing at a common point. For d ¼ 2; it is a result due to Thurston [33]. For related results, see [2,18,33].

Theorem 4.11. Consider a ﬁnite set ACT such that the following hold: (i) dn Á A; dm Á A are disjoint and pairwise unlinked, for all nam: (ii) A/dn Á A is a cyclic order preserving bijection, for all n.

Then the cardinality of A is at most d.

Corollary 4.12. Assume that f is a polynomial without irrationally neutral cycles and top let Q be the map obtained after collapsing ﬁbers of f. Consider the map w : Sf -J ðQÞ of Proposition 4.2. If wÀ1ðzÞ has inﬁnite cardinality, then fzg is a connected component of JtopðQÞ: Moreover, z is a periodic or preperiodic point. ARTICLE IN PRESS

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Proof. Consider a point zAqÀ1ðzÞ: First, we suppose that z is periodic under Q and show that wÀ1ðzÞ has finite cardinality or fzg is a connected component of JtopðQÞ: In fact, assume that fzg is not a connected component of JtopðQÞ: Then z is not a connected component of Jð f Þ: Since z is periodic, z is also periodic (see Lemma 4.4). By Levin and Przytycki [20], only finitely many rays may land at the periodic point z: Therefore if tAwÀ1ðzÞ; t t À1 then Rtop lands at z and R is one of the finitely many rays landing at z: Thus w ðzÞ is finite. À1 Next, since multiplication by d is finite-to-one on Sf ; finiteness of w ðzÞ follows for an arbitrary preperiodic point z such that fzg is not a connected component of JtopðQÞ: Finally, suppose that z has infinite forward orbit under Q: Then fdkwÀ1ðzÞg is a pairwise disjoint collection of subsets of Sf : Since Sf consists of finitely many k À1 backward orbits, for k large enough, say kXk0; we have that d w ðzÞ is contained in T\Sf : We may also assume, after taking a larger k0 if necessary, that Q is locally one- k À1 k0 n to-one around Q ðzÞ; for kXk0: Let A ¼ w ðQ ðzÞÞ: It follows that A/d A is one-to-one. By Corollary 4.10, the set ACT satisfies the hypothesis of Theorem 4.11. Thus A is finite and hence wÀ1ðzÞ is also finite. &

Proof of Theorem 3. Note that ImpðtÞCFiberðzÞ if and only if tAwÀ1ðzÞ; where z ¼ qðFiberðzÞÞ: The theorem now follows from the previous corollary. &

5. Laminations

Recall that the impressions of Jð f Þ are parametrized by the topological space Sf where Sf is the circle T when Jð f Þ is connected and a Cantor set otherwise (see Deﬁnitions 2.1 and 2.3). The smallest equivalence relation in Sf that identiﬁes elements of Sf with impressions that have non-empty intersection is called the lamination of f :

Deﬁnition 5.1. Let f : C-C be a polynomial without irrationally neutral cycles. Given t and s in Sf ; we say that t and s are lð f Þ-equivalent if there exists elements t ¼ t1; y; tn ¼ s of Sf such that ImpðtiÞ-Impðtiþ1Þa|: Call lð f Þ the lamination of f.

Remark 5.2. Following Thurston, the lamination of a polynomial may be represented as a geodesic lamination in the unit disk (cf. [33]). Although we introduce lð f Þ as an equivalence relation we still call it ‘‘lamination’’ (cf. [21]).

In this section we study laminations and collect some consequences of Theorems 2 and 3. In Section 5.1 we study the basic properties of laminations as well as the relationship between ﬁbers, wandering connected sets and laminations. In particular, we show that the lamination of a polynomial with connected Julia set is a Real ARTICLE IN PRESS

J. Kiwi / Advances in Mathematics 184 (2004) 207–267 239 lamination without rotation curves. In Section 5.2 we summarize how from a Real lamination l one may obtain a branched covering which is called the topological realization of l: In Section 6 we use the topological realizations of Real laminations to ﬁnd a polynomial with a suitable lamination. In Section 5.3 we study the ‘‘semiconjugacies’’ between a polynomial f and dynamical systems constructed from the lamination of f :

5.1. Laminations, ﬁbers and wandering connected sets

As a ﬁrst consequence of Theorem 3 we establish the exact relation between the different decomposition of the Julia set in our next result.

Corollary 5.3. Let f : C-C be a polynomial without irrationally neutral cycles. Consider two distinct points z and w in Jð f Þ: Then the following are equivalent: (1) z and w are contained in the same ﬁber. (2) There exist impressions Imp1; y; Impn such that zAImp1; wAImpn; and a| Impi-Impiþ1 for all i ¼ 1; y; n À 1: (3) There exists a wandering connected set CCJð f Þ that contains z and w.

Proof. Suppose that z and w are two distinct points contained in the same fiber. Since fibers of preperiodic and periodic points are trivial, both z and w have infinite forward orbit. By Theorem 3, their common fiber is the union of finitely many impressions. Hence (1) implies (2). Also, the common fiber is a wandering connected set, by Corollary 4.5. Thus (1) also implies (3). By Corollary 3.14, impressions are contained in fibers, therefore (2) implies (1). To show that (3) implies (1), note that from the definition of fibers, a wandering connected set CCJð f Þ must be contained in a fiber. &

Remark 5.4. In view of the previous corollary, the reﬂexive relation that relates two distinct points z and w only if there exists a wandering connected set CCJð f Þ containing both z and w is an equivalence relation.

We now show that the lamination of f coincides with the landing relation lðQÞ of the map Q obtained after collapsing ﬁbers of f in Section 4.1. Afterwards, we translate the results already proven for lðQÞ to results about lð f Þ:

Lemma 5.5. Let f : C-C be a polynomial without irrationally neutral cycles with lamination lð f Þ: Consider the branched covering Q : C-C obtained after collapsing ﬁbers of f and let lðQÞ be the landing relation of Q. Then lðQÞ¼lð f Þ:

Proof. By deﬁnition, t is lðQÞ equivalent to s if and only if wðtÞ¼wðsÞ: In view of Proposition 4.7, wðtÞ¼wðsÞ if and only if qðImpðtÞÞ ¼ qðImpðsÞÞ: Now, from the deﬁnition of q we conclude that qðImpðtÞÞ ¼ qðImpðsÞÞ if and only if ImpðsÞ ARTICLE IN PRESS

240 J. Kiwi / Advances in Mathematics 184 (2004) 207–267 and ImpðtÞ are contained in the same ﬁber, which is equivalent to tBs; by Corollary 5.3. &

Recall that in the introduction we denoted by Xf the topological space obtained after collapsing wandering connected sets in Jð f Þ and by Ff : Xf -Xf the projection of the map f : Jð f Þ-Jð f Þ: By Remark 5.4, collapsing ﬁbers is the same as collapsing wandering connected top top top sets. That is, Xf ¼ J ðQÞ and Ff ¼ Q : J ðQÞ-J ðQÞ: Thus we may rewrite Corollary 4.8 as:

À1 Corollary 5.6. The map w# : Xf -Sf =lð f Þ is a homeomorphism such that À1 À1 w# ðFf ð½zÞÞ ¼ d Á w# ð½zÞ (i.e., a topological conjugacy).

Now we restrict our attention to polynomials with connected Julia sets. We will show that the lamination of such polynomials is a Real lamination in the sense of the following deﬁnition (compare with Thurston’s invariant laminations [33]).

Deﬁnition 5.7. An equivalence relation l in the circle T is called a Real lamination if the following hold.

(R1) The graph of l is closed in T Â T: (R2) Each l-equivalence class A is a ﬁnite subset of T: (R3) If A is a l-equivalence class, then d Á A is a l-equivalence class. (R4) If A is a l-equivalence class, then A/d Á A is consecutive preserving. (R5) l-equivalence classes are pairwise unlinked.

Moreover, since we are dealing with polynomials without irrationally neutral cycles we will show that: (NR) If gCT=l is a periodic simple closed curve under the action induced by multiplication by d; then the return map is not a homeomorphism. We say that a Real lamination for which (NR) holds is a Real lamination without rotation curves. The reason will become apparent in the next subsection.

Lemma 5.8. If f : C-C is a polynomial with connected Julia set and without irrationally neutral cycles, then the lamination lð f Þ is a Real lamination without rotation curves.

Proof. We have to show that (R1)–(R5) of Definition 5.7 hold for lð f Þ: By Lemma 5.5, the equivalence relation lð f Þ coincides with lðQÞ: Corollary 4.8 says that lðQÞ; and hence lð f Þ; is closed, i.e., (R1). Now Corollary 4.12 implies that lð f Þ-classes are finite, i.e., (R2). To complete the proof that lð f Þ is a Real lamination, note that (R3)–(R5) follow from Corollary 4.10. To show that (NR) holds consider a period p simple closed curve g in T=lð f Þ: Using the notation of Corollary 4.8 it follows that w#ðgÞCJtopðQÞ is periodic under iterations of the map Q : C-C obtained after collapsing fibers of f : The Fatou ARTICLE IN PRESS

J. Kiwi / Advances in Mathematics 184 (2004) 207–267 241 domain UCF topðQÞ bounded by w#ðgÞ is also periodic under Q: Moreover, Qp : U-U is a branched covering of the same degree than Qp : w#ðgÞ-w#ðgÞ: In the Fatou set F topðQÞ the map Q is topologically conjugate to the polynomial f : Since f has no rotation domains, every periodic Fatou component of f has return map of degree strictly greater than one. Therefore, the return map to g is not a homeomorphism. &

5.2. Real laminations and branched coverings

Given a Real lamination l; there exists a branched covering P : C-C such that its dynamics resembles that of a polynomial g with locally connected Julia set JðgÞ; where JðgÞ is homeomorphic to T=l: Such a branched covering P is obtained from lð f Þ via a ‘‘pinched disk’’ construction (cf. [6,33]). More precisely, we identify T with @D and extend l to an equivalence relation lC in C as follows. Two distinct points zAC and wAC are lC-equivalent if and only if both points lie in the Euclidean convex hull of the same l-class AC@DDT: According to a result of Moore, C=lC is homeomorphic to C: So denote the quotient map by p : C-CDC=lC: Now according to Proposition 4.14 of [16], there exists a branched covering P : C-C such that (i) through (iv) below hold. Such branched covering is called a topological realization of l:

(i) The following diagram commutes:

(ii) If U is a bounded component of C\pð@DÞ; then U is a Jordan domain which is eventually periodic under iterations of P: (iii) If U is a period p bounded component of C\pð@DÞ; then Pp :U-U is topologically conjugate to one of next two maps:

e2piyðÁÞ : D -D z /e2piyz;

for some yAR\Q; or

ðÁÞd : D -D z /zd;

for some integer d41:

(iv) There exists a collection of homeomorphisms hU : D-U such that the following hold: ARTICLE IN PRESS

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If U is NOT a period p component such that Pp : U-U is topologically conjugate to an irrational rotation, then for some integer dU X1; the following diagram commutes:

If U is a period p component such that Pp : U-U is topologically conjugate to an irrational rotation z/e2piyz; then the following diagram commutes:

By (iii), we have that l has no rotation curves if and only if P has no rotation domains.

Remark 5.9. The topological realization of a Real lamination is unique modulo topological conjugacy.

For future reference we introduce some natural deﬁnitions associated to a topological realization.

Definition 5.10. Let P : C-C be a topological realization of a Real lamination l: Denote by p : C-CDC=lC the quotient map. We say that: pð@DÞ is the topological Julia set JtopðPÞ of P; C\JtopðPÞ is the topological Fatou set F topðPÞ of P; pðDÞ is the topological filled Julia set KtopðPÞ of P; C\KtopðPÞ is the basin of infinity of P; t 2pit RP ¼fpðre Þ : rAð1; NÞg is the topological ray with argument tAT:

t We now introduce extended topological rays. Consider a ray RP landing at a point z which lies in @U; where U is a bounded topological Fatou component. For some 2pis sAT; the point z ¼ hU ðe Þ; where hU : D-U is the homeomorphism given by the ˆ t above proposition. An extended topological ray R P is the arc obtained as the union t 2pis of RP; fzg and hU ð½0; 1e Þ: Therefore, extended rays connect N to the center of a Fatou component. We omit the word topological and the superscript top when clear from the context. Note that by the deﬁnition of topological realizations, all of the critical points of P (i.e., the branch points) which are contained in the topological Fatou set have ﬁnite ARTICLE IN PRESS

J. Kiwi / Advances in Mathematics 184 (2004) 207–267 243 forward orbit. It follows that a polynomial f with locally connected Julia set is the topological realization of its lamination lð f Þ provided that the Fatou critical points have ﬁnite forward orbits. Also, call the support of lC; denoted supp lC; the union of the convex hulls of l–classes AC@DDT: Note that the support of lC is a closed subset of C that contains @D and that U is a bounded Fatou component if and only if pÀ1ðUÞ is a connected component of D\supp lC: In Section 6 we need the following result.

Lemma 5.11. Let P : C-C be a topological realization of a Real lamination l: Let U and V be two distinct bounded Fatou components of P. Then there exist a pair of s t s external rays RP and RP that land at a common point z such that the arc formed by RP; t RP and fzg separates U from V.

À1 Proof. Pick a point w1AU and a point w2AV: The segment joining p ðw1Þ and À1 p ðw2Þ in D must cross a segment g that connects two l-equivalent arguments sA@DDT and tA@DDT since g must intersect the support of lC: It follows that s and t have the desired property. &

5.3. Semiconjugacies

The action of a polynomial f without irrationally neutral cycles on its Julia set is semiconjugate to the action induced by multiplication by d on the quotient Sf =lð f Þ: In the next result we state the basic properties of the semiconjugacy as well as necessary and sufﬁcient conditions for the semiconjugacy to be a topological conjugacy.

Theorem 5.12. Consider a degree d polynomial f : C-C without irrationally neutral cycles. Let lð f Þ be the lamination of f. Then lð f Þ has a closed graph in Sf Â Sf ; thus Sf =lð f Þ is a Hausdorff topological space. Moreover, consider the map

h : Jð f Þ -Sf =lð f Þ; z /½t; if zAImpðtÞ; where ½t denotes the lð f Þ-class of tASf : Then h is a continuous monotone surjection that semiconjugates the action of f with that induced by multiplication by d. That is, h3f ðzÞ¼d Á hðzÞ: Also, h is a homeomorphism between Jfinð f Þ and its image, where Jfinð f Þ is the set of all zAJð f Þ with ﬁnite forward orbit. Furthermore, the following statements are equivalent:

(1) h : Jð f Þ-Sf =lð f Þ is a homeomorphism. (2) FiberðzÞ¼fzg; for all zAJð f Þ: (3) ImpðtÞ is a singleton, for all tASf : (4) If CCJð f Þ is a wandering connected set, then C is a singleton. ARTICLE IN PRESS

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(5) Every non-trivial connected set contained in Jð f Þ contains a periodic or preperiodic point.

In addition, if any of statements (1)–(5) hold, then every connected component of Jð f Þ is locally connected. Also, under the assumption that Jð f Þ is connected, statements (1)–(5) are equivalent to saying that Jð f Þ is locally connected.

For connected Julia sets of unicritical polynomials, Levin [18] showed that statements (4) and (5) are equivalent to local connectivity (also see [2]). Moreover, it is worth mentioning that Levin also studied the relationship between the absence of non-trivial wandering connected sets and ‘‘backward stability’’.

Proof of Theorem 5.12. The equivalence relation lð f Þ is closed, since it coincides with lðQÞ; which is a closed equivalence relation, according to Corollary À1 top 4.8. Also by Corollary 4.8 the map w# : J ðQÞ-Sf =lð f Þ is a homeomorphism. Note that h ¼ w#À13q: It follows that h is monotone, since q is monotone, and that h is a continuous semiconjugacy. By Proposition 4.2, the map q restricted to Jfinð f Þ is a homeomorphism, and therefore h is a homeomorphism between Jfinð f Þ and its image. To prove that (1)–(5) are equivalent, note that h is a homeomorphism iff q : C-C is injective. By its deﬁnition, the map q is injective iff every ﬁber is trivial. Thus (1) is equivalent to (2). Statement (2) is equivalent to (3) and (4), by Corollary 5.3. Now, from Corollary 4.6 we conclude that (4) is equivalent to (5). Also, (2) implies that every connected component of Jð f Þ is locally connected (Corollary 2.16). To complete the proof of the theorem, note that if Jð f Þ is connected and locally connected, then Carathe´ odory’s theorem implies (2). That is, ImpðtÞ is a single point for all tASf : &

When the Julia set Jð f Þ is connected we compare the dynamics of f with that of the topological realization of its lamination:

Theorem 5.13. Consider a monic degree dX2 polynomial f : C-C with connected Julia set Jð f Þ and without irrationally neutral cycles. Let lð f Þ be its lamination. Consider a topological realization P of the Real lamination lð f Þ: Then there exists a continuous monotone surjection H : C-C such that

P3HðzÞ¼H3f ðzÞ; for all zAJð f Þ,Oð f Þ where Oð f Þ is the basin of inﬁnity. Also, H is a homeomorphism between Jfinð f Þ and its image, where Jfinð f Þ denotes the set of all zAJð f Þ with ﬁnite forward orbit. ARTICLE IN PRESS

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Furthermore, the following statements are equivalent: (1) H : C-C is a homeomorphism. (2) FiberðzÞ¼fzg; for all zAJð f Þ: (3) ImpðtÞ is a singleton, for all tAS: (4) If CCJð f Þ is a wandering connected set, then C is a singleton. (5) Every non-trivial connected set contained in Jð f Þ contains a periodic or preperiodic point. (6) Jð f Þ is locally connected.

Proof. Let f : C\D-C\Kð f Þ be the Bo¨ ttcher map. As usual, denote the ﬁber collapsing map by q : C-CDC=E: Also, call p : C\D-C the map conjugating z/zd with P: We construct a homeomorphism G : C-C where the domain is the dynamical plane of Q and the range is the dynamical plane of P: In JtopðQÞ; we let G ¼ w# : JtopðQÞ-T=lð f Þ¼pð@DÞ: For z in the unbounded component of C\JtopðQÞ; let GðzÞ¼p3fÀ1ðqÀ1ðzÞÞ: It remains to deﬁne G in bounded components of C\JtopðQÞ: We extend G over each bounded component U of C\JtopðQÞ as follows. Since U is a Jordan domain, Gð@UÞ is a Jordan curve that bounds a domain V which is a bounded component of C\pð@DÞ: Hence there exist homeomorphisms from U onto V that extend G : @U-@V: Let G : U-U be an arbitrary choice among these extensions. It follows that G : C-C is a homeomorphism, hence the continuous map H ¼ G3q has the required properties. To complete the proof of the theorem, just remark that the equivalence of statements (1)–(6) follows from Theorem 5.12. &

6. Conformal realization

The aim of this section is to show that for any given Real lamination l without rotation curves there exists at least one polynomial f : C-C without irrationally neutral cycles such that l ¼ lð f Þ: To find the polynomial f we apply ideas similar to those used by the author to characterize rational laminations of complex polynomials (see [16]). Throughout let l be a Real lamination with topological realization P: In Section 6.1, following Bielefield, Fisher, Hubbard and Poirier, we use critical portraits to describe the location of the critical points in the filled Julia set of the topological realization P: A critical portrait Y is a finite collection of finite subsets of T: If a critical portrait Y is compatible with P; we say the Y is a critical marking for P: In Sections 6.2–6.5, we show how via symbolic dynamics one may recover l from a critical marking of P: In Section 6.5, using Poirier characterization of critical markings of post-critically finite polynomials, we show that by perturbing Y we may ðnÞ find a sequence Y of critical markings of post-critically finite polynomials fn: By ARTICLE IN PRESS

246 J. Kiwi / Advances in Mathematics 184 (2004) 207–267 passing to a subsequence, since fn live on a compact subset of parameter space, we may suppose that fn converge to a polynomial f : In Section 6.7, we prove some separation results which will allow us to show, in Section 6.8, that f is a polynomial without irrationally neutral cycles which has Real lamination l:

6.1. Critical portraits and critical markings

The location of the critical points of a topological realization P in the ﬁlled Julia set KtopðPÞ will be described with the aid of critical portraits (cf. [1,14,16,26]).

Deﬁnition 6.1. Consider a pair of collections

F ¼fF1; y; Flg;

J ¼fJ1; y; Jmg of ﬁnite subsets of T: We say that Y ¼ðF; JÞ is a formal critical portrait of degree d if the following hold.

(C0) jFjjX2P and jJkjX2: Also,P jd Á Fjj¼1andjd Á Jkj¼1: (C1) d À 1 ¼ ðjFjjÀ1Þþ ðjJkjÀ1Þ: (C2) F and JÀ are unlinked. That is, for e40 small enough, we have that

F1; y; Fl; J1 À e; y; Jm À e are disjoint and pairwise unlinked. (C3) F (resp. J) is hierarchic. That is, for any two elements y; y0AT that participate i j 0 in F (resp. J) such that d y and d y lie in Fk (resp. Jk), for some i; j40; we have that diy ¼ d jy0: (C4) Given an element y that participates in F; there exists a periodic argument diy which also participates in F: (C5) None of the arguments that participate in J are periodic.

Following Poirier’s ideas certain formal critical portraits efﬁciently capture the location of the critical points of a topological realization. To make this precise we need the notion of supporting arguments and rays.

Deﬁnition 6.2. Let U be a bounded Fatou component of P and let zA@U: Denote by t1; y; tk the arguments of the external rays which land at z indexed respecting cyclic t1 t2 order. Without loss we assume that the sector bounded by RP and RP contains U: We say that t1 (resp. t2) is the left (resp. right) supporting argument of U at z and that t1 t2 the external ray RP (resp. RP ) left (resp. right) supports U.

The same deﬁnition applies to introduce supporting rays and arguments of bounded Fatou component of polynomials g : C-C: ARTICLE IN PRESS

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Deﬁnition 6.3. Let l be a Real lamination of degree d with topological realization - l m P : C C: We say that a formal critical portrait Y ¼ðfFigi¼1; fJgi¼1Þ is a critical marking for P if the following compatibility conditions hold:

(1) For each i ¼ 1; y; m; the external rays with arguments in Ji land at a critical point c of P and the local degree of P at c is jJij: (2) For each i ¼ 1; y; l; the external rays with arguments in Fi left support a bounded Fatou component U that contains a critical point c and the local degree of P at c is jFij: (3) d j s t Suppose that sAJi and that t participates in Y: If, for some jX1; RP and RP land at the same point, then d jt ¼ s: (4) Suppose that sAFi and that U is a bounded Fatou component which contains a j j critical point. If d s is a left supporting argument of U; then d s belongs to Fk; for some k:

Lemma 6.4. Given a Real lamination l with topological realization P there exists at least one and at most ﬁnitely many critical markings for P.

We omit the proof of the lemma. It consists of a construction which is described in [26, Section I.2] for post-critically ﬁnite polynomials and in [16, Section 6.1] for topological realizations.

6.2. Real laminations and critical markings

We would like to recover the Real lamination from the critical marking of the corresponding topological realization. This subsection contains some preliminaries in that direction. l m Throughout, we let Y ¼ðF ¼fFigi¼1; J ¼fJigi¼1Þ be a formal critical portrait of degree d: We now consider the partition of the circle into Y-unlinked classes.

Deﬁnition 6.5. We say that s and t are in the same Y-unlinked class L if and only if s and t lie in the same connected component of T\Fi; for all i ¼ 1; y; l; as well as in \ the same component of T Ji; for all i ¼ 1; y; m:

Note that unlinked classes are often disconnected. For example consider the degree 3 Chebychev polynomial f ðzÞ¼z3 À 3z with preﬁxed critical points 71: The external rays at arguments 71=3 land at À1 and the external rays at arguments 71=6 land at þ1: Since f is post-critically ﬁnite f is a topological realization of its lamination. Therefore, a critical marking for f is Y ¼ðF ¼ |; J ¼ ff1=3; 2=3g; f1=6; 5=6gÞ: The Y-unlinked classes are L1 ¼ð5=6; 1=6Þ; L2 ¼ ð1=3; 2=3Þ and L3 ¼ð1=6; 1=3Þ,ð2=3; 5=6Þ: Note that L3 is disconnected. We refer the reader to [26] for more examples. We collect, without proof, the basic properties of unlinked classes in the next lemma. ARTICLE IN PRESS

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Lemma 6.6. There are exactly d unlinked classes. Moreover, we may write each unlinked class L as the union of ﬁnitely many open intervals:

L ¼ðy0; y1Þ,?,ðy2pÀ2; y2pÀ1Þ with subscripts mod 2p and respecting cyclic order. The total length of these intervals is 1=d: Additionally, dy2iÀ1 ¼ dy2i for all i ¼ 1; y; p: Furthermore, md : L-d Á Lisa cyclic order preserving bijection and d Á L consists of T with the ﬁnite set of points fdy0; y; dy2pÀ2g removed.

If Y is a critical marking, then we will obtain l back from Y via the symbolic dynamics of the multiplication by d map md according to the partition of T into Y- unlinked classes. Throughout we denote by L1; y; Ld the Y-unlinked classes.

À Deﬁnition 6.7. For tAT; we let aYðtÞ¼i if there exists e40 such that ðt À e; tÞCLi: À þ We call aYðtÞ the left address of t: We say that aYðtÞ¼i if there exists e40 such that C þ ðt; t þ eÞ Li: We call aYðtÞ the right address of t: A þ þ þ For t T; we say that sYðtÞ¼ðaYðtÞ; aYðdtÞ; yÞ is the right symbol sequence of t. À À À Similarly, sYðtÞ¼ðaYðtÞ; aYðdtÞ; yÞ is the left symbol sequence of t.

We omit the subscript Y when clear from the context. Now given a critical portrait Y we may produce an equivalence relation in T:

Deﬁnition 6.8. The relation generated by Y; denoted by LTðYÞ; is the smallest equivalence relation in T such that if one of the following holds then s and t are LTðYÞ-equivalent:

À À (1) sYðsÞ¼sYðtÞ À i À i j j C (2) There exists j such that aYðd sÞ¼aYðd tÞ for i ¼ 0; y; j À 1 and fd s; d tg Jk for some k:

Not necessarily LTðYÞ is a Real lamination. Nevertheless, after some work we will show that LTðYÞ¼l provided that Y is a critical marking for a topological realization P of l whenever l has no rotation curves (equivalently P has no rotation domain).

6.3. Sides and cylinders

We continue under the assumption that

l m Y ¼ðF ¼fFigi¼1; J ¼fJigi¼1Þ is a formal critical portrait of degree d and that L1; y; Ld denote the Y-unlinked classes. ARTICLE IN PRESS

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In order to study the dynamics of multiplication by d according to this partition, for each symbol sequence

i ¼ði ; i ; yÞAf1; y; dgN,f0g; % 0 1 we consider the sets

A j A y Li0yin ¼ft T : d t Lij for j ¼ 1; ; ng and \ Xi ¼ Li yi : % 0 n nX0

When Y is a critical marking for l; we will establish a relationship between l and Xi % in terms of the ‘‘sides’’ of Xi: %

! Deﬁnition 6.9. Given a subset XCT; we say that st is a side of X if the interval ðs; tÞ is a connected component of T\X:

! At the end of this subsection we will establish that if st is a side of Xi; then s and t % are l-equivalent. A ﬁrst step in this direction is the basic fact contained in the next lemma.

Lemma 6.10. If Y is a critical marking for a topological realization P and! st is a side s t of a Y-unlinked class, then RP and RP land at the same point or at the boundary of the same bounded Fatou component.

Proof. From the deﬁnition of unlinked classes it follows that there exists a minimal chain of arguments s ¼ t0; y; tn ¼ t which belong to the interval ½s; t and such that, for j ¼ 0; y; n À 1; we have that ftj; tjþ1g is contained in an element YjAJ,F: In view of property (C2) of formal critical portraits, if YjAF; then j þ 1 ¼ n: Also, if YjAJ; then j þ 1 ¼ n or Yjþ1AF: Therefore, n ¼ 1or2: If n ¼ 1; then the conclusion of the lemma follows. s t1 If n ¼ 2; then fs; t1gCY0AJ and ft1; tgCY1AF: Therefore, RP and RP land at a common point which is in the boundary of the bounded Fatou t1 t s t component U left supported by the rays RP and RP: Hence, RP and RP land at @U: &

With the purpose of showing that the previous lemma also holds for the sides of

Li0yin we analyze the basic properties of the sides of cylinders sets Li0yin in the three lemmas below. ARTICLE IN PRESS

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Lemma 6.11. Consider two distinct arguments s and t in Li: ! ! (1) ds ¼ dt if and only if st or ts is a side of Li: À þ ! (2) ds ¼ dt and a ðsÞ¼a ðtÞ¼i if and only if st is a side of Li:

Proof. It is an immediate consequence of Lemma 6.6. &

! Lemma 6.12. Assume that st is a side of Li0i1yin : Then the following hold: ! (1) if ds ¼ dt; then st is a side of L : i0 a ! (2) if ds dt; then ðdsÞðdtÞ is a side of Li1yin : ! Proof. By Lemma 6.11, if ds ¼ dt then st is a side of Li0 : C - Since Li0i1yin Li0 ; it follows that md : Li0i1yin Li1yin is a cyclic order preserving C \ a C \ bijection. Therefore, if ðs; tÞ T Li0i1yin and ds dt; then ðds; dtÞ T Li1yin : Hence, ! ðdsÞðdtÞ is a side of Li1yin : &

A Lemma 6.13. Suppose that s; t Li0yin : Then the following are equivalent: (a) ! st is a side of Li0yin ! (b) k k ðd sÞðd tÞ is a side of Lik ; for some kpn: k k kþ1 kþ1 À k þ k (c) d sad t; d t ¼ d s; and a ðd sÞ¼a ðd tÞ¼ik; for some kpn:

Proof. Statements (b) and (c) are equivalent, by Lemma 6.11(2). We proceed by induction on the length of the word i0yin to show that (a) implies (b). In fact, for n ¼ 0; statement (b) becomes statement (a). Suppose that y ! (a) implies (b) for all words j0 jnÀ1 of length n: If st is a side of Li0yin which is ! not a side of Li0 ; then ðdsÞðdtÞ is a side of Li1yin ; by Lemma 6.12. Now using ! l l the inductive hypothesis, ðd dsÞðd dtÞ is a side of Lilþ1 ; for some lpn À 1: Hence, (a) implies (b). To complete the proof of the lemma we establish that (b) implies (a). In fact, ! k k assume that ðd sÞðd tÞ is a side of Lik ; for some kpn: The proof continues by ! A contradiction. Suppose that st is not a side of Li0yin : Then there exists u ðs; tÞ which k - lies in Li0yin : Since d : Li0yin Likyin is cyclic order preserving, it follows that k A k k k A C k k d u ðd s; d tÞ and d u Likyin Lik : Hence, ðd s; d tÞ is not contained in the ! k k complement of Lik which contradicts our assumption that ðd sÞðd tÞ is a side & of Lik :

Proposition 6.14. Assume that Y is a critical marking for a topological realization P. ! s1 s2 Suppose that s1s2 is a side of Li0i1yin : Then RP and RP land at a common point in JðPÞ or at the boundary of a common bounded Fatou component U where U is contained in an attracting basin. ARTICLE IN PRESS

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! k k Proof. From the preceding lemma, we have that ðd s1Þðd s2Þ is a side of Lik for some k k t1 kpn: To simplify notation let t1 ¼ d s1 and t2 ¼ d s2: From Lemma 6.10, RP and t2 ˆ t1 RP land at a common critical point c in JðPÞ or there exists extended rays R P and ˆ t2 A R P connecting N with a common Fatou critical point c: Denote by Yc F,J the t1 t2 marking of c: We consider the arc G formed by RP ; RP and fcg; in the ﬁrst case, and ˆ t1 ˆ t2 by R P and R P in the latter. The arc G cuts C into two regions. Denote by S the one that contains the rays with arguments in ðt2; t1Þ: Let NCS be a collar neighborhood of G such that all the critical values of Pk are outside N: Consider the connected component N˜ of PÀkðNÞ s1 ˜ ˜ k such that RP C@N: It follows that N is mapped, under P ; homeomorphically, onto * ˜ s1 k N: Therefore, the connected component G of @N that contains RP maps, under P ; * s homeomorphically, onto G: Hence, G contains an external ray RP which maps in k t2 iterates onto RP : To complete the proof it sufﬁces to show that s ¼ s2 or equivalently that À j þ j a ðd s1Þ¼a ðd sÞ for j ¼ 0; y; k À 1: By contradiction suppose that À j þ j a ðd s1Þaa ðd sÞ for some jok: Then there exists fy1; y2gCYi for some j j j j YiAJ,F; such that y1Aðd s; d s1Þ and y2A½d s1; d s: Hence the external ray y1 j * y1 RP lands at P ðGÞ if YiAJ; or RP left supports a Fatou component V with center kÀj j * A d y1 A in P ðGÞ if Yi F: It follows that the external ray RP lands at G if Yi J; and kÀj d y1 RP left supports a Fatou component U with center in G if YiAF: By properties kÀj j (3) and (4) of critical markings we conclude that d y1AYc: But since P ðN˜Þ maps, kÀj kÀj under P homeomorphically onto N; it follows that d y1Aðt2; t1Þ; which is a contradiction because ðt2; t1Þ-Yc ¼ |: &

Recall that for each i i ; i ; y A 1; y; d N,f0g we let % ¼ð0 2 Þ f g \ Xi ¼ Li yi : % 0 n n

Note that Xi consists of all the arguments with left or right symbol sequence i: % % Lemma 6.15. Assume that Y is a critical marking for a topological realization P. s t Suppose that RP and RP land at @U where U is a bounded Fatou component contained s t in an attracting basin. If fs; tgAXi; then R and R land at the same point. % P P Proof. Let us ﬁrst note some basic facts. A bounded Fatou component V which contains a critical point is supported by an element FV of F: The extended rays with arguments in FV joining N to the center of V cut @V into jFV j¼ degðP : V-PðVÞÞ pieces. Moreover, @V is parameterized via hV by T (see Section 5.2). Each piece of @V corresponds to an interval of length 1=degðP : V-PðVÞÞ: Furthermore, if two arguments t and s of external rays that land at @V are in the same Y-unlinked class, then the landing points are in the same piece of @V: ARTICLE IN PRESS

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s To prove the lemma, denote by z be the landing point of RP and by w the landing t D point of RP: The boundary @U is parameterized by T @D via hU : In view of the preceding paragraph, by induction it follows that if s and t lie in Li0yin ; the angular n n distance between hU ðzÞ and hU ðwÞ is less than 1=degðP : U-P ðUÞÞ: Therefore, if s and t lie in Li0yin ; for all n; then hU ðzÞ¼hU ðwÞ: Since hU is a homeomorphism, the lemma follows. &

Corollary 6.16. Assume that Y is a critical marking for a topological realization P of ! s t l: If st is a side of Xi; then R and R land at the same point. That is, s and t are l- % P P equivalent.

! ! Proof. Since st is a side of Xi; there exist sides sntn of Li yi such that sn-s and % 0 n - sn tn tn t: By Lemma 6.14, either sn and tn are l-equivalent, or RP and RP land at @Un where Un is a bounded Fatou component. In the ﬁrst case, it follows that s and t are l-equivalent, since l is closed. In the second, after passing to a subsequence, we may suppose that UnaUm if nam; by the preceding Lemma. Hence, there exists l- sn tn equivalent arguments sn and tn such that the external rays RP and RP separate Un from Unþ1 (see Lemma 5.11). It follows that sn-s (or t) and tn-t (resp. or s). This implies that s and t are l-equivalent, since l is a closed relation. &

6.4. Finiteness of Xi %

Under the extra hypothesis that P has no rotation domain we will establish that Xi % is a ﬁnite set for all iA 1; y; d N,f0g: Then, from the preceding corollary, we will % f g conclude that Xi is contained in a l-class. In the next subsection this fact will be the % key to describe l in terms of symbolic dynamics. As usual we denote by s : f1; y; dgN,f0g-f1; y; dgN,f0g the one-sided shift.

Lemma 6.17. Consider two distinct elements i and joff1; y; dgN,f0g: Then the % % following hold.

(1) Xi and Xj are weakly unlinked. That is, there exists two closed intervals ICT and % % JCT with disjoint interiors such that XiCI and XjCJ: % n % (2) If yAXi-Xj; then d yAYi for some YiAJ,F: % % - (3) The multiplication by d map md : Xi XsðiÞ extends to a degree one monotone circle % % map. Moreover, it is a cyclic order preserving bijection if XiCLi : % 0

a Proof. Let n be the smallest integer such that in jn: For statement (1), since Li0yin and Lj0yjn are unlinked, we have that Li0yin and Lj0yjn are weakly unlinked. C C Therefore, Xi Li0yin and Xj Lj0yjn are weakly unlinked. Statement (2) follows % % A n A after observing that if y Li0yin -Lj0yjn ; then d y Lin -Ljn : Statement (3) is a consequence of Lemma 6.6 and the fact that XiCLi : & % 0 ARTICLE IN PRESS

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Corollary 6.18. Assume that Y is a critical marking for a topological realization P of l: If i has infinite forward orbit under the one-sided shift s; then Xi is finite. Moreover, % % the external rays with arguments in Xi land at a common point in JðPÞ: % If i is periodic then one of the following two statements hold: % (a) Xi is a finite set of arguments with a common period. The external rays with these % arguments land at a common point. (b) Xi is infinite and the landing points of the external rays with arguments in Xi form % % the boundary of a rotation domain of P.

Proof. Suppose that i has infinite orbit under the one sided shift s: Note that % each argument is contained in at most two sets of the form Xi: Since the number of % arguments that participate in Y is finite, we have that only finitely many Xj contain % arguments which participate in Y: Therefore, for n large enough, say nXN; XsnðiÞ is % disjoint from Yi for all YiAF,J: By Lemma 6.17, the collection fXsnðiÞ: nXNg is % k - pairwise disjoint and unlinked. Moreover, d : XsN ðiÞ XsNþkðiÞ is a cyclic order % % preserving bijection. Therefore, by Theorem 4.11, XsN ðiÞ is finite and thus Xi is also % % finite. Now suppose that i is periodic under the one sided shift. Without loss of generality % suppose that i is fixed. That is, i ¼ði0i0i0yÞ for some i0Af1; y; dg: In Xi; consider % % % the equivalence relation B which identifies s with t if there exists a chain of ! ! arguments t0 ¼ s; y; tn ¼ t such that titiþ1 or tiþ1ti is a side of Xi: Note that each B- % class is contained in a l-class and hence is finite. Moreover, B is a closed equivalence relation. Therefore, Xi=B is a compact and Hausdorff space. It is not difficult to % check that Xi=B is connected. If Xi=B is not a singleton, then after removing two % % points from Xi=B we are left with two connected components. It follows that Xi=B % % is a singleton or homeomorphic to the circle S1 (e.g., see [2–5,11]). If Xi=B is a singleton, then Xi is finite and it is contained in a fixed l class. % % If Xi=B is homeomorphic to the circle, then md : Xi-Xi extends to a degree 1 % % % monotone circle map with irrational rotation number. For otherwise, the rotation number would be rational and Xi would have periodic orbits which attract % uncountably many points of Xi under iterations of md which is impossible. It follows % that every orbit of md : Xi-Xi is dense (e.g. see [19]). Therefore md : Xi-Xi projects % % % % to a homeomorphism g : Xi=B-Xi=B which is conjugate to an irrational rotation. % % If two arguments s and t in Xi are l-equivalent, then s and t are B equivalent. % Otherwise, there would be infinitely many distinct pairs of the form fdns; dntg which are linked with fs; tg and therefore the l-class of s would be infinite. It follows that top Xi=B is embedded in J ðPÞ as the boundary of a rotation domain. & %

6.5. From critical markings to Real laminations

Recall that given a formal critical portrait Y; in Deﬁnition 6.8, we introduced the equivalence relation LTðYÞ generated by Y: ARTICLE IN PRESS

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Our aim now is to prove the following result.

Proposition 6.19. Assume that Y is a critical marking for the topological realization P of a Real lamination l: If P has no rotation domain or equivalently l has no rotation curve, then LTðYÞ¼l:

The last ingredient needed for the proof of the proposition is the following lemma.

Lemma 6.20. Assume that Y is a critical marking for a topological realization P. A À À Consider s; t T such that aYðsÞ¼aYðtÞ: Then the following hold: dt ds t s (1) If RP and RP land at a common point, then RP and RP land at a common point. (2) If dt and ds are LTðYÞ-equivalent, then s and t are LTðYÞ-equivalent.

Proof. Statement (2) follows immediately from the deﬁnition of LTðYÞ: Therefore, À À dt ds we proceed to prove (1). Suppose that aYðsÞ¼aYðtÞ and that RP and RP land at a common point z1: There are two possibilities: s À If the landing point z0 of RP is not a critical point, then the left address aYðtÞ¼ À t t0 aYðsÞ for all t such that RP lands at z0: Moreover, there exists a ray RP that maps dt 0 onto RP which lands at z0: It follows that t ¼ t since there is only one argument that À maps onto dt and that has left address aYðsÞ: s If the landing point z0 of RP is a critical point, then, for some i; the rays with arguments in JiAJ land at z0: Note that locally, around z0; the sector bounded by two consecutive rays in Ji maps one-to-one onto a slitted neighborhood z1: It 0 0 follows that there exists a ray with argument t such that dt ¼ dt which lands at z0 À 0 À 0 and has left address aYðt Þ¼aYðsÞ: Hence, t ¼ t and the lemma follows. &

Proof of Proposition 6.19. To show that LTðYÞCl consider arguments s and t which are LTðYÞ-equivalent. Without loss of generality, assume that either (1) or (2) of À À Deﬁnition 6.8 holds. If (1) holds, then s and t belong to Xi where i ¼ s ðsÞ¼s ðtÞ: % % Y Y By Corollary 6.18, s and t are l-equivalent. If (2) holds, then apply j times the previous lemma to conclude that s and t are l-equivalent. To show that lCLTðYÞ; let A be a l-class. Note that, for e40 small, A À e is unlinked with Fi; for all i: Moreover, if A is not a critical class, then A is unlinked À A with Ji; for all i: Hence, either A is a critical class or aYðtÞ coincide for all t A: À j * Therefore, either sYðtÞ coincide for all tAA or d A Ji for some i and a minimal j: In the ﬁrst case A is contained in a LTðYÞ-class. In the second, we proceed by induction on the number of critical classes in the forward orbit of A: Since the forward orbit of d jþ1A contains one critical class less than the forward orbit of A; we jþ1 0 suppose that d A is contained in a LTðYÞ-class. Consider t; t AA: Note that À j À À j 0 À 0 0A aYðd tÞ¼aYðsÞ and aYðd t Þ¼aYðs Þ for some s; s Ji: From the previous lemma, j j 0 0 j j 0 d t and s are LTðYÞ-equivalent, as well as d t and s : Hence, d t and d t are 0 LTðYÞ-equivalent. Applying the previous lemma j times we conclude that t and t are also LTðYÞ-equivalent. & ARTICLE IN PRESS

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The next lemma will allow us to apply a result by Poirier.

Lemma 6.21. Let Y be a critical marking for a topological realization P without rotation domains. Then the following statements hold:

(C6) Consider a periodic argument t which participates in F and an argument 0A þ þ 0 0 t Q=Z: If sYðtÞ¼sYðt Þ; then t ¼ t : A 0A À i À 0 (C7) Consider preperiodic arguments t Jl and t Jk: If sYðd tÞ¼sYðt Þ; for some i. i Then d tAJk:

Proof. For (C6), note that if sþ t sþ t0 i; then t and t0 belong to the finite set Yð Þ¼ Yð Þ¼% t t0 Xi of l-equivalent arguments. Therefore, R and R land at the same point. By the % P P definition of critical markings, t is a left supporting argument. Since aþðtÞ¼aþðt0Þ; it follows that t ¼ t0: dit t0 For (C7), by the previous proposition RP and RP land at the same point. By Definition 6.3(4), we have that dit ¼ t0: &

6.6. Approximation

In this subsection, we fix a Real lamination l with no rotation curves. Our aim is to find a polynomial f : C-C such that its lamination lð f Þ coincides with l: We will obtain f as a limit of post-critically finite polynomials fn: These post-critically finite polynomials will be obtained from the following theorem [26, Theorem I.3.9].

Theorem 6.22 (Poirier). Let Y# be a formal critical portrait such that (C6) and (C7) of Lemma 6.21 hold. Suppose that all the arguments that participate in Y# are rational. Then there exists a post-critically ﬁnite polynomial g such that Y# is a critical marking for g.

Let P : C-C be a topological realization of l: Suppose that Y ¼ðF; JÞ is a critical marking for P: Denote, by c1; y; ck the critical points of P which have infinite forward orbit. It follows that these critical points are in the Julia set JðPÞ: Let ; y; denote the elements of associated to these critical points. Note that Jc1 Jck J C \ y Jci R=Z Q=Z; for i ¼ 1; ; k: By changing Y slightly we will obtain critical markings of post-critically finite polynomials. To make this precise we introduce the following definition.

Deﬁnition 6.23. We say that fYðnÞ ¼ðF; JðnÞÞg is a sequence of admissible critical portraits converging to ðnÞ y Y if J is obtained after replacing the elements Jc1 ; ; Jck of by en; y; en; where en ; y; en are sequences of positive real J Jc1 À 1 Jck À k f 1g f kg numbers such that: n- - (a) ei 0asn N; for all i: ARTICLE IN PRESS

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nC (b) Jci À ei Q=Z; for all i and n: (c) YðnÞ is a formal critical portrait such that (C6) and (C7) of Lemma 6.21 hold.

Proposition 6.24. Suppose that Y is a critical marking for the topological realization P. Then there exists a sequence of admissible critical portraits converging to Y:

n n Proof. For each integer nX1; choose positive real numbers e1; y; ek with n n maxfei go1=n such that: all the arguments in Jci À ei are strictly preperiodic, the n eventual period pi of Jci À ei is different from the eventual period of any argument e n ðnÞ ðnÞÀ t Jci À ei which participates in Y ; and F; J are unlinked (see (C2) in Definition 6.1). We have to check that YðnÞ is a formal critical portrait. It follows immediately that (C0)–(C2) of Definition 6.1 hold. A n j To check that (C3) holds note that if y Jci À ei ; then for all 40; the argument d jy does not participate in YðnÞ since, by construction, the eventual period of y is distinct from the eventual period of the arguments that participate in YðnÞ which are n not in Jci À ei : It also follows directly from the construction that (C4) and (C5) of Definition 6.1 hold. Therefore YðnÞ is a formal critical portrait. It remains to check that (C6) and (C7) of Lemma 6.21 holds. For (C6), note that if t is periodic of period p and i is the right or left symbol % sequence of t; then all the elements of Xi are periodic of the same period, since % p m : Xi-Xi extends to a degree one monotone circle map. Hence, it is enough to d % % check (C6) for finitely many t0: Therefore, if (C6) is false for infinitely many n’s, there would exist a periodic argument t which participates in F and an argument t0 (at) such that sþ ðt0Þ¼sþ ðtÞ; for infinitely many n’s. Thus we would have that sþðt0Þ¼ YðnÞ YðnÞ Y þ 0 sYðtÞ: By Lemma 6.21 we conclude that t ¼ t : It follows that (C6) holds for n large enough. ðnÞ A ðnÞ 0A ðnÞ To check that (C7) holds for Y just note that if t Jl and t Jk are such that sÀ dit sÀ t0 ; then the eventual period of t and t0 coincide. Therefore, t and t0 YðnÞ ð Þ¼ YðnÞ ð Þ ðnÞ ðnÞ participate in the original J: That is, Jl ¼ Jl and Jk ¼ Jk : Hence, for n large, À i À 0 i A ðnÞ sYðd tÞ¼sYðt Þ: Thus d t Jk ¼ Jk : &

6.7. A separation result

As mentioned before our candidate to be a polynomial f with a given lamination l is obtained as the limit of post-critically ﬁnite polynomials. A key step to establish that the lamination of f is in fact l is to show that f is a polynomial without neutral cycles. To show that f is a polynomial without neutral cycles we rely on the following separation result. ARTICLE IN PRESS

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Proposition 6.25. Let P be a topological realization of a Real lamination l: Assume that P has no rotation domain. Suppose that z and w are distinct eventually periodic points in JðPÞ: Let N be such that PN ðzÞ and PN ðwÞ are periodic. Then there exist s and t in Q=Z such that dN s and dN t are not periodic arguments and one of the following hold:

t s (1) RP and RP land at the same point and separate z from w. t s m (2) RP and RP are left supporting rays of a bounded Fatou component U and d t as well as dms participate in F for some m. Moreover, the corresponding extended rays joining N with the center of U separate z from w.

To prove the preceding proposition we need a discussion involving several deﬁnitions and lemmas. To motivate the discussion, let us consider the situation given in the statement of the proposition. We denote by AðzÞ and AðwÞ the arguments of the rays landing at z and w and observe that AðzÞ is contained in ðs; tÞ ! where st is a side of AðwÞ: It is possible to separate z from w by a pair of external rays landing at a common point if !st is not ‘‘isolated’’ in the sense of the following deﬁnition:

Deﬁnition 6.26. Let A be a l-class. We say that a side !st of A is isolated if there exists e40 such that ðs; s þ eÞÂðt À e; tÞ-l ¼ |: We say that the class A is isolated if all the sides of A are isolated.

! If the side st of AðwÞ is isolated, then the next lemma will enable us to separate z from w with the help of extended external rays connecting inﬁnity to the center of a Fatou component.

! s t Lemma 6.27. A side st of a l-class A is isolated if and only if RP and RP land at a point z in the boundary of a bounded Fatou component U, s is a left supporting argument for U, and t is a right supporting argument for U.

Proof. Suppose that !st is isolated and take e40 as in the deﬁnition. Let T be the isosceles triangle contained in the Euclidean convex hull of ½s; tC@D that has as base the segment BCD joining s with t and angle e=2 at both of these vertices s and t: Note that T bounds a region R which is contained in the complement of the support of lC (see Section 5.2). It follows that pðRÞ is contained in a bounded Fatou s t component U which is supported by RP and RP: The converse is straightforward as well. &

! By deﬁnition, if the side st of AðwÞ is not isolated, then there are rays, rational or irrational, separating z from w: For the proposition we are interested in rational rays separating z from w: Unfortunately, the rational part of l is not necessarily ‘‘dense’’ in l: That is, according to Lemma 4.17 in [16], the smallest closed equivalence ARTICLE IN PRESS

258 J. Kiwi / Advances in Mathematics 184 (2004) 207–267 relation that contains l-Q=Z Â Q=Z is l if and only if no critical point of P that has inﬁnite orbit is in the boundary of a bounded Fatou component. These type of critical points are closely related to isolated classes:

Lemma 6.28. Let A be an irrational l-class such that jAjX2: Denote by zAJðPÞ the landing point of the rays with arguments in A. The following are equivalent:

(1) A is an isolated class. (2) A has an isolated side. (3) The forward orbit of z is inﬁnite and contains a critical point c which is in the boundary of a bounded Fatou component.

Proof. Clearly (1) implies (2). By the previous lemma (3) implies (2). To show that (2) implies (3) and (1) let !st be an isolated side of A: According to s t the previous lemma RP and RP support a bounded Fatou component U: Without dks loss of generality, we may suppose that U is ﬁxed under P: It follows that RP and dkt RP are also left and right supporting rays of U: Since s and t are irrational, we conclude that the collection fðdks; dktÞg is a pairwise disjoint of non-trivial intervals k k or d 0 s ¼ d 0 t for some k0: The former implies that the lim sup of the length of ðdks; dktÞ is greater than 1=d which is impossible since these intervals are pairwise disjoint. Hence, dk0 s ¼ dk0 t: By the consecutive preserving property (see Deﬁnition 5.7), it follows that dk0 A is a singleton, say fsg: Therefore, A is a precritical class. ! ! That is (3) holds. To conclude (1), note that ss is an isolated side. Moreover, if s0t0 k0 k0 ! ! is a side of A; then s ¼ d s0 ¼ d t0: It follows that s0t0 is isolated. Otherwise, ss would not be isolated. &

Lemma 6.29. Let! st be a side of a non-isolated irrational l-class A. Then there exist sequences fsng and ftng contained in ðs; tÞCT which converge to s and t (respectively) such that, for all n, both sn and tn belong to the same l-class An and one of the following holds:

(1) An is isolated. (2) AnCQ=Z:

Proof. Let l0 ¼ l and inductively deﬁne lm as the equivalence relation obtained from lmÀ1 after removing isolated classes. That is, A is a non-trivial lm-class if and only if A is a non-isolated lmÀ1-class. It is not difﬁcult to check that lm is a Real lamination.

We claim that there exists m0 such that lm0þ1 ¼ lm0 : In fact, we proceed by contradiction and suppose that, for all m; lmalmÀ1: It follows that there exists a critical lmÀ1-class CmÀ1 which is isolated. Therefore, CmÀ1 is not a lm-class. Hence, CmaCl if mal: Since Cm is a critical l-class, for all m; and there are at most d À 1of those, we obtain a contradiction. ARTICLE IN PRESS

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Note that lm0 has no isolated classes. Therefore, is the smallest closed equivalence relation that contains lm0 -ðQ=Z Â Q=ZÞ: It follows that the lemma holds for l ¼ lm0 : By construction, if the lemma holds for lmþ1; then it also holds for lm: Therefore, the lemma follows for l ¼ l0: &

Proof of Proposition 6.25. Recall that AðzÞ and AðwÞ denote the arguments of the ! external rays that land at z and w: Also AðzÞCðs; tÞ where st is a side of AðwÞ: By induction on N we will prove the following claim:

! Claim. Let st be a side of AðwÞ where wAJðPÞ is such that PN ðwÞ is periodic. Then, for arbitrarily small e40 there exist rational arguments sAðs; s þ eÞ and tAðt À e; tÞ such that dN s and dN t are not periodic and Rs and Rt land at a common point or left support a common Fatou component. In the latter case we may choose s and t so that k k d s and d t belong to some F0AF:

Since, for e40 small enough, AðzÞ is contained in T\ðs; s þ eÞ,ðt À e; tÞ; the proposition will follow once the claim is proven. Under the hypothesis of the claim consider any e40 small enough so that ðs; s þ eÞ and ðt À e; tÞ are disjoint and multiplication by d is one-to-one in ðs; s þ eÞ,ðt À e; tÞ: From Lemmas 6.27 and 6.29 there exist rational arguments s0Aðs; s þ eÞ and t0Aðt À e; tÞ such that one of the following holds:

s0 t0 (a) RP and RP land at a common point. s0 t0 k k (b) RP and RP land at @U and d s0 as well as d t0 lie in F0AF; for some k:

s t Denote by G the arc formed by RP; RP and fwg: Also let G0 be the arc s0 t0 formed by RP ; RP and their common landing point in case (a), and by the ˆ s0 ˆ t0 extended rays R P ; R P joining N with the center of U in case (b). Note that the region R bounded by G and G0 maps homeomorphically onto its image under P: For N ¼ 0 we have that s and t are periodic. By passing to an iterate, if necessary, m we may suppose that s and t are ﬁxed under multiplication by d: Let smAðs; s þ e=d Þ m m m and tmAðt À e=d ; tÞ be such that d sm ¼ s0 and d tm ¼ t0: Since P : R- PðRÞ*R,G0 is a homeomorphism, by successively lifting G0 it follows that the rays with arguments sm and tm land at a common point or at the boundary of a common Fatou component. For m large enough, s ¼ sm and t ¼ tm are strictly preperiodic and the claim follows for N ¼ 0: For the inductive step, suppose that PNþ1ðwÞ is periodic. Note that P maps R homeomorphically onto its image PðRÞ which contains the rays with arguments in ðds; ds þ deÞ and ðdt À de; dtÞ: By the inductive hypothesis we may suppose that there exists s0Aðds; ds þ deÞ and t0Aðdt À de; dtÞ as in the claim. It follows that the unique sAðs; s þ eÞ and tAðt À e; tÞ such that ds ¼ s0 and dt ¼ t0 fulﬁll the requirements of the claim. & ARTICLE IN PRESS

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Corollary 6.30. Given two irrational arguments s and t which are not l-equivalent there exist s; tAQ=Z such that one of the two following hold:

s t s t (1) RP and RP land at a common point and separate RP and RP: m m s t (2) there exist m such that d s and d t participate in F; the rays RP and RP left support a bounded Fatou component U and the extended rays joining N with the s t center of U separate RP and RP:

! Proof. Let A be the l-class of s and B be the l-class of t: Then ACðb; b0Þ where bb0 is a side of B: There are two possibilities: ! (a) If bb0 is an isolated side, then we may find s and t such that (2) holds, by Lemma 6.27. !0 (b) If bb is not isolated, then there exists An as in Lemma 6.29. In the case that AnCQ=Z we find s and t such that (1) holds. In the other case, as in (a), we may find s and t such that (2) holds. &

6.8. Realization

In what follows, we let l be a Real lamination with topological realization P: We assume that P has no rotation domain. We also let Y ¼ðF; JÞ be a critical marking for P and YðnÞ be a sequence of admissible critical markings converging to Y (see Definition 6.23). By Porier’s result (Theorem 6.22), there exist post- ðnÞ critically finite polynomials fn with critical markings Y : By compactness of the connectedness loci (e.g. see [3]), without loss of generality we may suppose that fn converge to some polynomial f with connected Julia set. The aim of this subsection is to prove that f has no irrationally neutral cycles, and that the lamination of f is l: 7 In order to simplify notation we let Rt ¼ Rt ; a7 ¼ a ; etc. n fn n YðnÞ Our first result compares the landing pattern of rational rays of fn with that of P:

Proposition 6.31. Let t and s be elements of Q=Z:

t s (1) There exists NX0 such that: Rn and Rn land at the same point for all nXN if and t s only if RP and RP land at the same point. (2) Assume that there exists mX0 such that dmt and dms participate in F: Then there t s exists N such that: Rn and Rn left support the same bounded Fatou component Un t s for all nXN if and only if RP and RP left support the same Fatou component.

Proof. For statement (1), suppose that dlt and dls are periodic of period p: Let Q be the subset of Q=Z formed by all the arguments t such that dlt is periodic of period p: Since YðnÞ is a sequence of admissible critical markings converging to Y in the sense À of Deﬁnition 6.23, there exists N; such that for all t1; t2AQ we have that: an ðt1Þ¼ ARTICLE IN PRESS

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À À À an ðt2Þ for all nXN if and only if aYðt1Þ¼aYðt2Þ: Therefore statement (1) follows from Proposition 6.19. s t To prove (2) suppose that RP and RP left support the bounded Fatou component U: Denote by ws and wt the corresponding landing points. Let g be the closed arc in @U joining ws with wt following the standard orientation of @U: Without loss of generality, we may assume that Pmþ1 maps the interior of g injectively onto its image. ! þ k 0A 0 For kX0; let ik ¼ aYðd sÞ: It follows that there exists t ðs; t such that t s is a side t0 of the Y-cylinder Li0yim : By Proposition 6.14, RP lands in @U and therefore in g: By kþ1 0 kþ1 t0 Lemma 6.13, d t ¼ d s for some kpm: Hence, k ¼ m: Therefore RP either s lands at ws or at wt: The former is impossible since RP is left supporting. We t0 conclude that RP lands at wt: ! For N large enough, it follows from Lemma 6.13 and the deﬁnition of YðnÞ; that t0s ðnÞ is a side of a Y -cylinder for all nXN: Denote by Un the bounded Fatou component t0 s s such that Rn and Rn land at @Un: It follows that Rn left supports @Un (otherwise 0 0 s there would exist s Aðs; t Þ that lands at the same point than Rn but this contradicts s t statement (1) and the fact that RP and RP left support the same component). By t t0 t statement (1), we may assume that Rn and Rn land at a common point wn: If Rn is not left supporting for Un then there exists tAðt; sÞ landing at wn: Again by statement t t (1), we may assume that RP and RP land at the same point, which is impossible since t t RP is left supporting for U: Hence Rn is also left supporting for Un: t s For the converse, suppose that Rn and Rn left support Un: Without loss we may ! assume that, for all nXN; there exists t0Aðs; t such that t0s is a side of a YðnÞ-cylinder t t0 !0 and that the rays Rn and Rn land at a common point. As above, it follows that t s is a t s side of a Y-cylinder and that the rays RP and RP left support the same bounded Fatou component U: &

To continue we will need to recall some standard definitions. Assume that g is a polynomial and suppose that each bounded Fatou component V of g contains a unique point z0 which is in the grand orbit of some critical point. We say that z0 is the center of V: Thus we may consider a conformal isomorphism c : V-D 2piy such that cðz0Þ¼0: The preimage of a radial segment ½0; 1Þe under c is called an internal ray I of V: It is not difficult to see that internal rays of V map onto internal rays of gðVÞ: The internal ray I lands at z if, as we approach the Julia set along I; the limit is z: The union of an internal ray, fzg and an external t ˆ t ray Rn landing at z is called an extended ray R n at argument t joining N with z0: Note that if there exists an extended ray at argument t joining N with z0; then it is unique. Consider a sequence of (extended) external rays fSngCC: The set lim sup Sn consists of points zAC such that every neighborhood of z intersects infinitely many Sn: ARTICLE IN PRESS

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The next lemma from [16] combined with the separation results of Section 6.7 and a result of Goldberg and Milnor will imply that all periodic orbits of f are either repelling or superattracting.

Lemma 6.32. Consider a sequence fgng of post-critically ﬁnite polynomials converging to g. If tAQ=Z is periodic and zA lim sup Rt J g ; then z is periodic under g. ð gn Þ- ð Þ If tAQ=Z is not periodic and zA lim sup Rt J g ; then z is not periodic under g. ð gn Þ- ð Þ Moreover, assume that there exists an extended ray Rˆ t ; for all n. gn If tAQ=Z is periodic and zA lim sup Rˆ t J g ; then z is periodic under g. ð gn Þ- ð Þ If tAQ=Z is not periodic and zA lim sup Rˆ t J g ; then z is not periodic under g. ð gn Þ- ð Þ

See [16, Lemma 7.6] for the proof.

Lemma 6.33 (Goldberg and Milnor [10, Lemma B.1]). Consider a polynomial g0 and an external ray Rt which lands at a repelling periodic point z : Then Rt lands at the g0 0 g analytic continuation of z0; for all g in a sufﬁciently small neighborhood V of g0: t Moreover, if g -g; then lim sup Rt R : n gn ¼ g

Now we may prove the following result.

Lemma 6.34. Every periodic orbit of f is either repelling or superattracting. Moreover, each bounded Fatou component contains a unique element in a critical grand orbit.

It follows that every critical point of f which belongs to the Fatou set is eventually periodic.

Proof. To rule out parabolic periodic points we proceed by contradiction. Assume p0 that z0 is a parabolic periodic point of f : That is, f ðz0Þ¼1 for some pX1: It follows that there exist distinct points zn and wn periodic under fn of period dividing p such that zn-z0 and wn-z0: Since the attracting periodic points of fn have multiplier 0; zn and wn are repelling periodic points. The number of periodic points of p period dividing p in Jð fnÞ and JðPÞ coincide for all n; since it is d minus the number of attracting periodic points of period dividing p: By Proposition 6.31, for n large enough the classes of l of period dividing p coincide with the classes of lð fnÞ of period dividing p: Hence, by passing to a subsequence if necessary, we may assume that there exist periodic arguments s and t; independent of n; such that the rays Rs fn and Rt land at z and w respectively, for all n: Now say that Rs lands at z and Rt fn n n P P lands at w: Then there exists a graph GPCC formed by (extended) strictly preperiodic rays with arguments s and t separating z from w (Proposition 6.25). Thus the corresponding graph Gn formed by the (extended) strictly preperiodic rays of fn with arguments s and t separates wn from zn (Proposition 6.31). Hence, z0Alim sup Gn which contradicts Lemma 6.32. ARTICLE IN PRESS

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We also proceed by contradiction to show that f is free of irrationally neutral cycles. If z0 is a period p irrationally neutral point of f ; then there exist period p periodic points zn of fn converging to z0: As above, without loss of generality, we may suppose that there exists a periodic argument t such that zn is the landing point of Rt for all n: But some repelling point w of f is the landing point of Rt : Therefore, fn 0 f by Lemma 6.33, the analytic continuation of w ; say w ; is the landing point of Rt : 0 n fn Hence, wn ¼ zn and w0 ¼ z0; which is impossible. To finish the proof we must show that each bounded Fatou component of f contains a unique point that belongs to some critical grand orbit. Since every Fatou component V of f is a connected component of a (super) attracting basin, every compact subset K of V is contained in the Fatou set of fn for n sufficiently large. Therefore, if V contains two distinct points in critical grand orbits, then also a Fatou component of fn contains two distinct points in a critical grand orbit. But since fn is post-critically finite, it follows that V may contain only one point in a critical grand orbit. &

Lemma 6.35. The following statements hold:

(1) t t If tAQ=Z; then lim sup Rn ¼ Rf : (2) Consider internal rays In of fn which join the center cn of a Fatou component with pþl l - - znAJð fnÞ such that for ﬁxed pX1 and lX0 fn ðInÞ¼fn ðInÞ: If cn c and zn z; then lim sup In ¼ I where I is the internal ray of f joining c with z: A s t (3) Consider s; t Q=Z: Suppose that Rf and Rf land at a common point. Then, for n s t large, Rn and Rn land at a common point. (4) Let U be a bounded Fatou component of f. Suppose tAQ=Z and that ICUisan t eventually periodic internal ray such that Rf and I land at the same point. Then, for n large, there exist bounded Fatou components Un of fn and internal rays In such t t that In and Rn have a common landing point and lim sup In ¼ I: Moreover, if Rf is t left supporting for U, then Rn is left supporting for Un:

Proof. To prove (1) note that if t is periodic, then the assertion follows from Lemma t t 6.33. By induction, it suffices to prove that lim sup Rn ¼ Rf provided that dt dt t lim sup Rn ¼ Rf : In fact, since Kð f Þ-lim sup Rn is compact, connected and its t image f ðKð f Þ-lim sup RnÞ under the finite to one map f is the singleton dt dt t Kð f Þ-lim sup Rn ¼ Kð f Þ-Rf ; it follows that Kð f Þ-lim sup Rn is a singleton. Since the Bo¨ ttcher map depends continuously on the polynomial, we also have that t t Oð f Þ-lim sup Rn ¼ Rf where Oð f Þ denotes the basin of infinity under f : Hence, t t lim sup Rn ¼ Rf : For statement (2), if In is periodic (i.e., l ¼ 0), then the Goldberg and Milnor [10, Appendix B] proof applies as well. By induction, as in the previous paragraph, the statement follows for preperiodic In: ARTICLE IN PRESS

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s t In order to establish statement (3), suppose that Rf and Rf land at a common s t point and that Rn and Rn do not, for all n arbitrarily large. Let N be an integer so N N t s that d t and d s are periodic. Then, by Proposition 6.31, we have that RP and RP do not land at a common point. Hence, by Proposition 6.25 we may find arguments s and t such that dN s and dN t are periodic and the graph G formed by (extended) rays t s with arguments t and s separate RP from RP: Applying Proposition 6.31 again, we obtain the corresponding graphs Gn in the dynamical planes of fn: It follows that the t s common landing point of Rf and Rf is contained in lim sup Gn which is a t s contradiction to Lemma 6.32. Hence, Rn and Rn land at the same point for all n sufficiently large. Under the hypothesis of statement (4), denote by c be the center of U; by z the t t landing point of Rf and by zn the landing point of Rn: Suppose that pX1; lX0 are integers such that f pþlðIÞ¼I: For e40 sufficiently small, there exists a unique center cn of a Fatou component Un of fn which is e-close to c; since an e- neighborhood of c is contained in the Fatou set of fn for n large and each Fatou component of fn contains a unique element in a critical grand orbit. Note that the - 0 conformal isomorphisms jn : D Un such that jnð0Þ¼cn and jnð0Þ40 converge uniformly on compact sets to a conformal isomorphism j : D-U such that jð0Þ¼c 0 pþl l and j ð0Þ40: Consider the internal ray In such that fn ðInÞ¼fn ðInÞ so that the À1 À1 * % angle between jn ðInÞ and j ðIÞ is minimal. It follows that lim sup In I: 0 0 Denote by zn the landing point of In: We may suppose that for a fixed argument t t0 0 0 a t t0 the ray Rn lands at zn for all n: If zn zn for arbitrarily large n; then RP and RP land at distinct points. Note that dlt and dlt0 are periodic. As above we may find arguments s and t such that dlt and dlt are not periodic and the graph G formed by t t0 the (extended) rays with arguments s and t separates RP from RP: The t corresponding graphs Gn in the dynamical plane of fn separate In from Rn: t * Therefore, lim sup Gn contains z since z belong to lim sup Rn and lim sup In I: This t is a contradiction with Lemma 6.32. We conclude that In and Rn land at the same point zn: t To complete the proof of the Lemma, we must show that if Rn is not left t t supporting, then Rf is not left supporting. In fact, if Rn is not left supporting, then s there exists Rn landing at zn: These two rays cut the plane into two sectors. The sector that spans the interval ðs; tÞ at infinity must contain cn: By statement (1) we t conclude that the same picture holds in the dynamical plane of f : Hence Rf is not left supporting. &

Lemma 6.36. There exists a critical marking Yf ¼ðFf ; Jf Þ for a topological realization of lð f Þ such that F ¼ Ff :

Proof. It is not difﬁcult to show that for any collection F for which (2) and (4) of Deﬁnition 6.3 hold one may construct Jf such that Yf ¼ðF; Jf Þ is a critical marking for the topological realization of lð f Þ (e.g. see [26, Section I.2; ARTICLE IN PRESS

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16, Section 6.1]). Since every Fatou critical point of f is eventually periodic, it follows that the map Q obtained after collapsing fibers of f is a topological realization of lð f Þ: Hence, to check that (2) and (4) of Definition 6.3 hold for F and the topological realization Q of lð f Þ it is enough to show that (2) and (4) of Definition 6.3 hold for F and the polynomial f : ðnÞ Consider FiAF and let ci be the critical point of fn marked by Fi: By passing to ðnÞ- - a subsequence if necessary we may suppose that ci ci as n N for all i: Moreover, ciacj if iaj: Hence, the local degree of f at ci is jFij: By Lemma 6.35(1) A ˆ t ˆ t ˆ t and (2), for all t Fi; we have that lim sup R n ¼ R f where R n is the extended ray ðnÞ ˆ t joining N to ci and R f is the extended ray joining N to ci: It follows that t left supports the Fatou component U centered at ci: Otherwise, there would exist other s t external Rf ray landing at the same point than Rf such that the critical point ci s t would be contained in the sector bounded by Rf and Rf that spans the interval ðs; tÞ at infinity. The same picture would be true for fn which is impossible since t left ðnÞ supports the component centered at ci : We conclude that (2) of Definition 6.3 holds d j t for Fi and f : To show that (4) of the definition holds, suppose that Rf left supports d j t U where U is the component centered at ci: Then Rn left supports Un; the ðnÞ j component centered at ci : Hence d tAFi: Therefore (4) holds and the lemma follows. &

Lemma 6.37. l ¼ lð f Þ:

Proof. A first step to show that these two equivalence relations coincide is to prove they coincide in Q=Z: More precisely, we will establish that lQ ¼ l-Q=Z Â Q=Z and lQð f Þ¼lð f Þ-Q=Z Â Q=Z coincide. We now prove that lQClQð f Þ: By Proposition 6.31 (1), if s and t are lQ- s t equivalent, then Rn and Rn land at a common point, for all n sufficiently large. It s t then follows, from Lemma 6.35(1), that Rf and Rf land at a common point. Hence, s and t are lQð f Þ-equivalent. To prove that lQð f ÞClQ; consider lQð f Þ-equivalent arguments s and t: By s t Lemma 6.35(3), the rays Rn and Rn land at a common point, for n sufficiently large. s t It follows, from Proposition 6.31(1), that RP and RP land at the same point. We conclude that lQ ¼ lQð f Þ: Our next step is to prove that lð f ÞCl: Consider two irrational arguments s and t which are not l-equivalent. By Corollary 6.30, there exist rational arguments s and t such that the (extended) external rays of P with arguments s and t form a graph that s t separates RP from RP: From Proposition 6.31 we know that there exists a s t corresponding graph Gn in the dynamical plane of fn that separates RP from RP: Lemma 6.32 implies that ðlim sup GnÞ-Jð f Þ is formed by periodic and preperiodic points. Therefore, Impf ðtÞ-Impf ðsÞ is either empty or formed by eventually periodic | points. By Lemma 3.14, the latter is impossible. Hence, Impf ðtÞ-Impf ðsÞ¼ : It follows that lð f ÞCl: ARTICLE IN PRESS

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To establish that lClð f Þ; consider two irrational arguments s and t which are not lð f Þ-equivalent. According to Corollary 2.18, there exist rational arguments s and t such that the graph G formed by the rays or extended rays with arguments s and t s t separates Rf from Rf : Moreover, if extended rays are needed we may suppose that dms and dmt participate in F: By Lemma 6.35(3) and (4), the corresponding graphs s t Gn separate Rn from Rn; for n sufﬁciently large. From Proposition 6.31 we conclude s that there exists a graph GP formed by rational (extended) rays separating RP from t RP: Therefore, s and t are not l-equivalent. This implies that lClð f Þ which completes the proof. &

Acknowledgments

Some of the results (Theorems 2 and 3) are sharper versions of unpublished results obtained in my Ph. D. Thesis. I thank J. Milnor for his support and advise during my graduate studies. I thank Juan Rivera-Letelier for his questions, suggestions and recent paper [30], which were a main motivation for this work. I am grateful to Sasha Blokh for helpful conversations. I am also thankful for the hospitality of CIMAT at Guanajuato, Mexico and IMPA, Brazil where part of this research was undertaken.

References

[1] B. Bieleﬁeld, Y. Fisher, J.H. Hubbard, The classiﬁcation of critically preperiodic polynomials as dynamical systems, J. Amer. Math. Soc. 5 (1992) 721–762. [2] A. Blokh, G. Levin, An inequality for laminations, Julia sets and ‘growing trees’, Ergodic Theory Dynamical Systems 22 (2002) 63–97. [3] B. Branner, J.H. Hubbard, The iteration of cubic polynomials, Part I: the global topology of parameter space, Acta Math. 160 (1988) 143–206. [4] L. Carlesson, T.W. Gamelin, Complex Dynamics, Springer, Berlin, 1993. [5] W. de Melo, S. van Strien, One-dimensional Dynamics, Springer, Berlin, 1993. [6] A. Douady, Description of compact sets in C; in: L. Goldberg, A. Phillips (Eds.), Topological Methods in Modern Mathematics, Publish or Perish, Berkeley, CA, 1993, pp. 429–465. [7] A. Douady, J.H. Hubbard, E´ tude dynamique des polynomes complexes, I & II, Publ. Math. Orsay, 1984–85. [8] A. Douady, J.H. Hubbard, On the dynamics of polynomial-like mapping, Ann. Sci. E´ cole Norm. Sup. Paris 4e se´ rie 18 (1985) 287–344. [9] D. Faught, Local connectivity in a family of cubic polynomials, Thesis, Cornell University, 1992. [10] L.R. Goldberg, J. Milnor, Fixed point portraits of polynomial maps, Part II: Fixed Point Portraits, Ann. Sci. E´ cole Norm. Sup. Paris 4e se´ rie 26 (1993) 51–98. [11] J. Hocking, G. Young, Topology, Dover, New York, 1988. [12] J.H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, in: Topological Methods in Modern Mathematics, Publish or Perish, Berkeley, CA, 1993. [13] K. Keller, Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot set, in: Lecture Notes in Mathematics, Vol. 1732, Springer, Berlin, 2000. [14] J. Kiwi, Rational rays and critical portraits of complex polynomials, Thesis, SUNY at Stony Brook, 1997 (available as SUNY-IMS preprint #1997/15). [15] J. Kiwi, Non-accessible critical points of Cremer polynomials, Ergodic Theory Dynamical Systems 20 (2000) 1391–1403. ARTICLE IN PRESS

J. Kiwi / Advances in Mathematics 184 (2004) 207–267 267

[16] J. Kiwi, Rational laminations of complex polynomials, in: M. Lyubich et al. (Eds.), Laminations and Foliations in Dynamics, Geometry and Topology, Contemp. Math. 269 (2001) 111–154. [17] J. Kiwi, Wandering orbit portraits, Trans. Amer. Math. Soc. 354 (4) (2002) 1473–1485. [18] G. Levin, On backward stability of holomorhic dynamical systems, Fund. Math. 158 (1998) 97–107. [19] G. Levin, F. Przytycki, External rays to periodic points, Israel J. Math. 94 (1995) 29–57. [20] G. Levin, M. Sodin, Polynomials with disconnected Julia sets and Green maps, preprint, Hebrew University of Jerusalem, 1990. [21] C.T. McMullen, Complex Dynamics and Renormalization, Princeton University Press, Princeton, NJ, 1994. [22] J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Vieweg, Braunschweig, 1999. [23] J. Milnor, Local connectivity of Julia sets: expository lectures, in: Tan Lei (Ed.), The Mandelbrot set, Theme and Variation, Lecture Notes in Science, Vol. 274, Cambridge University Press, Cambridge, 2000, pp. 67–116. [24] J. Milnor, in: M. Flexor, P. Sentenac, J.-C. Yoccoz (Eds.), Periodic Orbits, External Rays and the Mandelbrot Set: An Expository Account, Aste´ rique, Vol. 261, Socie´ te´ Mathe´ matique de France, 2000, pp. 277–331. [25] R.L. Moore, Concerning Upper Semi-continuous Collection of Continua, Trans. Amer. Math. Soc. 27 (1925) 412–428. [26] A. Poirier, On postcritically ﬁnite polynomials, part I, Stony Brook IMS preprint #1993/5. [27] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992. [28] F. Przytycki, A. Zdunik, Density of periodic sources in the boundary of the basin of attraction of iteration of holomorphic maps: geometric coding trees technique, Fund. Math. 145 (1994) 65–77. [29] M. Rees, A partial description of parameter space of rational maps of degree two: part 1, Acta Math. 168 (1992) 11–87. [30] J. Rivera–Letelier, Rational maps with decay of geometry: rigidity, Thurston’s algorithm and local connectivity, Stony Brook IMS preprint #2000/9. [31] D. Schleicher, On ﬁbers and local connectivity of compact sets, Stony Brook IMS preprint #1998/12. [32] D. Schleicher, in: M. Flexor, P. Sentenac, J.-C. Yoccoz (Eds.), Rational Parameter Rays of the Mandelbrot Set, Aste´ rique, Vol. 261, Socie´ te´ Mathe´ matique de France, 2000, pp. 405–445. [33] W.P. Thurston, On the combinatorics of iterated rational maps, preprint, Princeton, 1985.