Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials

A priori bounds for some infinitely renormalizable quadratics: III. Molecules

In this paper we prove {\it a priori bounds} for infinitely renormalizable quadratic polynomials satisfying a ``molecule condition''. Roughly speaking, this condition ensures that the renormalization combinatorics stay away from the satellite types. These {\it a priori bounds} imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.

Combinatorial rigidity for unicritical polynomials

We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the ``Multibrot set'') is locally connected at the corresponding parameter values. It generalizes Yoccoz's Theorem for quadratics to the higher degree case.

某些无穷可重整化二次多项式的先验界：I. 有界本原组合

A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics

本文证明了存在无穷本原重整化序列且周期比有界的无穷可重整化二次多项式的先验界，并由此推出Mandelbrot集在对应点处局部连通。

Geometry of quadratic polynomials: moduli, rigidity and local connectivity

A while ago MLC (the conjecture that the Mandelbrot set is locally connected) was proven for quasi-hyperbolic points by Douady and Hubbard, and for boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC for all at most finitely renormalizable parameter values. One of our goals is to prove MLC for some infinitely renormalizable parameter values. Loosely speaking, we need all renormalizations to have bounded combinatorial rotation number (assumption C1) and sufficiently high combinatorial type (assumption C2). For real quadratic polynomials of bounded combinatorial type t…

Renormalization in the Henon family, I: universality but non-rigidity

In this paper geometric properties of infinitely renormalizable real Hénon-like maps $F$ in $\R^2$ are studied. It is shown that the appropriately defined renormalizations $R^n F$ converge exponentially to the one-dimensional renormalization fixed point. The convergence to one-dimensional systems is at a super-exponential rate controlled by the average Jacobian and a universal function $a(x)$. It is also shown that the attracting Cantor set of such a map has Hausdorff dimension less than 1, but contrary to the one-dimensional intuition, it is not rigid, does not lie on a smooth curve, and gene…

Local connectivity of Julia sets for unicritical polynomials

We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in math.DS/0505191 that give control of moduli of annuli under maps of high degree.

Local connectivity of the Mandelbrot set at some satellite parameters of bounded type

We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family.

有界类型塞格尔多项式的刚性

Rigidity of bounded-type Siegel polynomials

本文研究了具有有界类型塞格尔盘的多项式刚性，证明了其朱利亚集的局部连通性和拟共形刚性，并验证了组合刚性猜想。

Local connectivity of Julia sets: expository lectures

The following notes provide an introduction to recent work of Branner, Hubbard and Yoccoz on the geometry of polynomial Julia sets. They are an expanded version of lectures given in Stony Brook in Spring 1992. I am indebted to help from the audience. Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of quadratic Julia sets. It presents only the "easy" part of his work, in the sense that it considers only non-renormalizable polynomials, and makes no effort to describe the much more difficult arguments which are needed to deal with local connectivity in parameter space. …

Rigidity and non local connectivity of Julia sets of some quadratic polynomials

For an infinitely renormalizable quadratic map $f_c: z\mapsto z^2+c$ with the sequence of renormalization periods ${k_m}$ and rotation numbers ${t_m=p_m/q_m}, we prove that if $\limsup k_m^{-1}\log |p_m|>0$, then the Mandelbrot set is locally connected at $c$. We prove also that if $\limsup |t_{m+1}|^{1/q_m}<1$ and $q_m\to \infty$, then the Julia set of $f_c$ is not locally connected and the Mandelbrot set is locally connected at $c$ provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed…

Quasisymmetric rigidity in one-dimensional dynamics

In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary assumptions are satisfied). The aim of this paper is to conclude this programme in a natural class of $C^3$ mappings. Examples of such rigidity were established previously, but not, for example, for real polynomials with non-real critical points. Our results are also new for analytic mappings. The main new ingredients of the proof in the real analytic case are (i…

Feigenbaum点上的MLC

MLC at Feigenbaum points

该研究证明了Feigenbaum点上Mandelbrot集的局部连通性，并揭示了相关动力学和参数组合的标度普遍性。

A priori bounds for GIETs, affine shadows and rigidity of foliations in genus 2

We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman's theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that, if a smooth, orientable foliation with non-degenerate (Morse) singularities on a closed surface of genus two is minimal, then, under a full measure condition for the rotation number, it is differentiably conjugate to a linear foliation. The corresponding result at the level of Poincaré sections is that, for a full measure set of interval exchange transformati…

抛物有界原始重正化的MLC

MLC for parabolically bounded primitive renormalization

针对重正化类型为原始但可接近Mandelbrot集尖点的一类无穷重正化参数，证明了先验界和MLC（Mandelbrot集局部连通性），通过发展薄厚分解、值演算、游荡者定理和波引理等工具控制重正化退化。

中心双曲分支的边界 I：动力学性质

Boundary of the central hyperbolic component I: dynamical properties

本文研究中心双曲分支边界上多项式映射的动力学，证明了 Julia 集的局部连通性和刚性定理，方法基于 Fatou 树构造与拼图技术，并推广到更广的映射类。

Satellite renormalization of quadratic polynomials

We prove the uniform hyperbolicity of the near-parabolic renormalization operators acting on an infinite-dimensional space of holomorphic transformations. This implies the universality of the scaling laws, conjectured by physicists in the 70's, for a combinatorial class of bifurcations. Through near-parabolic renormalizations the polynomial-like renormalizations of satellite type are successfully studied here for the first time, and new techniques are introduced to analyze the fine-scale dynamical features of maps with such infinite renormalization structures. In particular, we confirm the rig…