On the regular leaf space of the cauliflower

Kawahira, Tomoki

doi:10.2996/kmj/1061901060

Cited by 5 publications

(8 citation statements)

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“…A possible direction is to get M c 0 by a limiting process of finitely many parabolic bifurcations. In fact, if superattracting (or parabolic) f s is given by finitely many parabolic bifurcations (and degenerations) from f 0 , then the topologies of R s , H s and M s are described in detail [9,10].…”

Section: The Original Natural Extension Is Given Bymentioning

confidence: 99%

Topology of the regular part for infinitely renormalizable quadratic polynomials

Cabrera¹,

Kawahira

2010

Fund. Math.

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Section: The Original Natural Extension Is Given Bymentioning

confidence: 99%

Topology of the regular part for infinitely renormalizable quadratic polynomials

Cabrera¹,

Kawahira

2010

Fund. Math.

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“…They are invariant leaves off andĝ with hyperbolic and parabolic dynamics respectively (Figure 12). Essentially the uniformizations of these principal leaves are given by uniformizing the quotient torus and the cylinder of the dynamics downstairs; see [Ka1] and [LM,4.2].…”

Section: On the Other Hand Each λ(β J ) Consists Of Q Leaves Describmentioning

confidence: 99%

Tessellation and Lyubich–Minsky laminations associated with quadratic maps, II: Topological structures of 3-laminations

Kawahira

2009

Conform. Geom. Dyn.

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Abstract. According to an analogy to quasi-Fuchsian groups, we investigate the topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with hyperbolic and parabolic quadratic maps.We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolicto-parabolic degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient 3-laminations of Douady's rabbit, the Cauliflower, and z → z 2 are described.The descriptions employ a new method of tessellation inside the filled Julia set introduced in Part I [Ergodic Theory Dynam. Systems 29 (2009) IntroductionAs an analogue of the hyperbolic 3-orbifolds associated with Kleinian groups, Lyubich and Minsky [LM] introduced hyperbolic orbifold 3-laminations associated with rational maps. For a rational map f :C →C, there exist three kinds of laminations A f , H f , and M f as follows:• The affine lamination A f , a Riemann surface lamination with leaves isomorphic to C or its quotient orbifold.• The H 3 -lamination H f , a hyperbolic 3-lamination with leaves isomorphic to H 3 or its quotient orbifold. (This is a 3D-extension of the affine lamination of f .)As a Kleinian group acts onC and H 3 , there is a natural homeomorphic and leafwise-isomorphic actionf on A f and H f induced from that of f . In particular, the action is properly discontinuous on H f . The third lamination is:• The quotient 3-lamination M f := H f /f , whose leaves are hyperbolic 3-orbifolds.In addition, we have the Fatou-Julia decomposition F f J f = A f of the affine lamination. As a Kleinian group acts on the complement of the limit set properly discontinuously,f acts on F f properly discontinuously. Hence the quotient F f /f forms a Riemann surface lamination and we regard it as the conformal boundary of M f . In this paper we say the quotientLyubich and Minsky applied analogous arguments on rigidity theorems of hyperbolic 3-orbifolds to the hyperbolic 3-laminations, and showed a rigidity result of rational maps that have no recurrent critical points or parabolic points [LM, Theorem 9.1]. However, even for hyperbolic quadratic maps like z 2 − 1 or Douady's rabbit, the precise structure of their laminations have not been investigated. (See the questions in [LM, §10].) Moreover, even for the simplest parabolic quadratic map z 2 + 1/4 ("the Cauliflower") whose 3-lamination has a cuspidal part, its structure had not been precisely known. The aim of this paper is to give a method to describe the topological and combinatorial changes of laminations associated with the motion of parameter c of z 2 + c from one hyperbolic component to another via parabolic parameters. In part...

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“…As interesting examples of polynomials with J non leaf-wise connected, consider: quadratic polynomials with parabolic cycles (see Tomoki Kawahira's paper [14]), or the Feigenbaum quadratic polynomial. In the case of quadratic polynomials with parabolic cycles, leaves in the regular part where the Julia set is disconnected are precisely the periodic leaves corresponding to the parabolic cycle.…”

Section: Leafwise Connectivity Of J Cmentioning

confidence: 99%

“…As for repelling cycles the linearizing coordinate, in this case Fatou's coordinate, gives a uniformization of the periodic leaves on the regular part. A detailed explanation and construction of such uniformization can be found in [14]. It is worth noting that, when f c is parabolic, then for any leaf L the set J c ∩ L consists of finitely many components.…”

Section: Leafwise Connectivity Of J Cmentioning

confidence: 99%

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On the Classification of Laminations Associated to Quadratic Polynomials

Cabrera

2007

J Geom Anal

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Given any rational map f , there is a lamination by Riemann surfaces associated to f . Such laminations were constructed, in general, by Lyubich and Minsky. In this article, we classify laminations associated to quadratic polynomials with periodic critical point. In particular, we prove that the topology of such laminations determines the combinatorics of the parameter. We also describe the topology of laminations associated to other types of quadratic polynomials.

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On the regular leaf space of the cauliflower

Cited by 5 publications

References 1 publication

Topology of the regular part for infinitely renormalizable quadratic polynomials

Topology of the regular part for infinitely renormalizable quadratic polynomials

Tessellation and Lyubich–Minsky laminations associated with quadratic maps, II: Topological structures of 3-laminations

On the Classification of Laminations Associated to Quadratic Polynomials

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