View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS 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 We characterize the laminations associated to complex polynomials with connected Julia sets and without irrationally neutral cycles. r 2003 Elsevier Science (USA). All rights reserved. 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. 0001-8708/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0001-8708(03)00144-0 ARTICLE IN PRESS 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 J. Kiwi / Advances in Mathematics 184 (2004) 207–267 209 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´ od- ory, ‘‘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 210 J. Kiwi / Advances in Mathematics 184 (2004) 207–267 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 decomposi- tions 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.