Locally connected models for Julia sets

arXiv:0809.3754v2 [math.DS] 25 Aug 2010 Locally connected models for Julia sets Alexander M. Blokh ∗,1, Clinton P. Curry 2, Lex G. Oversteegen 3 University of Alabama at Birmingham, Department of Mathematics, Birmingham, AL 35294-1170, USA Abstract Let P be a polynomial with a connected Julia set J. We use continuum theory to show that it admits a finest monotone map ϕ onto a locally connected continuum J∼P , i.e. a monotone map ϕ : J →J∼P such that for any other monotone map ψ : J →J′ there exists a monotone map h with ψ = h ◦ϕ. Then we extend ϕ onto the complex plane C (keeping the same notation) and show that ϕ monotonically semiconjugates P|C to a topological polynomial g : C →C. If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi’s fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map ϕ not to collapse all of J into a point. Key words: Complex dynamics, Julia set, core decomposition 1 Introduction 1 A major idea in the theory of dynamical systems is that of modeling an ar- 2 bitrary system by one which can be better understood and treated with the 3 help of existing tools and methods. To an extent, the entire field of symbolic 4 dynamics is so important for the rest of dynamical systems because symbolic 5 dynamical systems serve as an almost universal model. A different example, 6 ∗Corresponding author. Email addresses: ablokh@math.uab.edu (Alexander M. Blokh), clintonc@uab.edu (Clinton P. Curry), overstee@math.uab.edu (Lex G. Oversteegen). 1 This author was partially supported by NSF grant DMS-0456748. 2 This author was partially supported by NSF grant DMS-0353825. 3 This author was partially supported by NSF grant DMS-0405774. Preprint submitted to Elsevier 26 June 2018 coming from one-dimensional dynamics, is due to Milnor and Thurston who 7 showed in [16] that any piecewise-monotone interval map f of positive en- 8 tropy can be modeled by a piecewise-monotone interval map of constant slope 9 h (i.e., f is monotonically semiconjugate to h). For us however the most in- 10 teresting case is that of modeling complex polynomial dynamical systems on 11 their connected Julia sets by so-called topological polynomials on their (topo- 12 logical) locally connected Julia sets. Let us now describe more precisely what 13 we mean. 14 Consider a polynomial map P : C →C; denote by JP the Julia set of P, by KP 15 its filled-in Julia set, and by U∞(P) = C\KP its basin of attraction of infinity. 16 In this paper we always assume that JP is connected. A very-well known fact 17 from complex dynamics (see, e.g., Theorem 9.5 from [15]) shows that there 18 exists a conformal isomorphism Ψ from the complement of the closure of the 19 open unit disk D onto U∞(P) which conjugates zd|C\D and P|U∞(P ). The Ψ- 20 image Rα of the radial line of angle α in C\D is called an (external) ray. By [9] 21 external rays with rational arguments land at repelling (parabolic) periodic 22 points or their preimages (i.e., the rays compactify onto such points). If JP is 23 locally connected, Ψ extends to a continuous function Ψ which semiconjugates 24 zd|C\D and P|U∞(P ). 25 External rays have been extensively used in complex dynamics since the ap- 26 pearance of the papers by Douady and Hubbard [9]. The fundamental idea of 27 using the system of external rays in order to construct special combinatorial 28 structures in the disk (called laminations or geometric laminations) is due 29 to Thurston [25] (see also the paper [8] by Douady). Laminations allow one 30 to relate the dynamics of P and the dynamics of the map zd|S1. Below we 31 describe a few approaches to laminations. 32 Set ψ = Ψ|S1 and define an equivalence relation ∼P on S1 by x ∼P y if and 33 only if ψ(x) = ψ(y). The equivalence ∼P is called the (d-invariant) lamination 34 (generated by P). The quotient space S1/ ∼P= J∼P is homeomorphic to JP 35 and the map f∼P : J∼P →J∼P induced by zd|S1 ≡σ is topologically conjugate 36 to P|JP. The set J∼P is a topological (combinatorial) model of JP and is often 37 called the topological Julia set. On the other hand, the induced map f∼P : 38 J∼→J∼serves as a model for P|JP and is often called a topological polynomial. 39 Moreover, one can extend the conjugacy between P|JP and f∼P : J∼P →J∼P 40 (as the identity outside JP) to the conjugacy on the entire plane. In fact, 41 equivalences ∼similar to ∼P can be defined abstractly, in the absence of any 42 polynomial. Then they are called (d-invariant) laminations and still give rise 43 to similarly constructed topological Julia sets J∼and topological polynomials 44 f∼. 45 In his fundamental paper [13] Kiwi extended this to all polynomials P with 46 no irrational neutral periodic points (called CS-points), including polynomials 47 2 with disconnected Julia sets. In the case of a polynomial P with connec

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