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

与二次映射相关的嵌铺与 Lyubich-Minsky 叶状结构，I：捏半共轭

Tessellation and Lyubich-Minsky laminations associated with quadratic maps, I: Pinching semiconjugacies

本文引入双曲和抛物二次映射填充 Julia 集的嵌铺，利用瓦片组织内部动力学，并基于嵌铺与线性化坐标退化研究构造了双曲到抛物退化的捏半共轭连续族。

A note on hyperbolic leaves and wild laminations of rational functions

We study the affine orbifold laminations that were constructed by Lyubich and Minsky. An important question left open in their construction is whether these laminations are always locally compact. We show that this is not the case. The counterexample we construct has the property that the regular leaf space contains (many) hyperbolic leaves that intersect the Julia set; whether this can happen is itself a question raised by Lyubich and Minsky.

Quatre applications du lemme de Zalcman à la dynamique complexe

We give four applications of Zalcman's lemma to the dynamics of rational maps on the Riemann sphere: a parameter analogue of a proof of the density of repelling cycles in the Julia sets;similarity between the Mandelbrot set and the Julia sets; a construction of the Lyubich-Minsky lamination and its variant; and a unified characterization of conical points by Lyubich-Minsky and those by Martin-Mayer.

On the classification of laminations associated to quadratic polynomials

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 paper, 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.

On density of horospheres in dynamical laminations

In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both domains, starting from the famous Sullivan's no wandering domain theorem in the theory of iterations of rational functions. One of the principal objects used in the study of Kleinian groups is the hyperbolic 3- manifold associated to a Kleinian group, which is the quotient of its…

Global hyperbolicity of renormalization for C^r unimodal mappings

In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of C^r unimodal maps with quadratic critical point. We show that in this space the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r ge 2+ alpha with alpha close to one. As an intermediate step between Lyubich's results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous la…