A priori bounds for some infinitely renormalizable quadratics: II. Decorations

无限可重整 Lorenz 映射的 Lyapunov 指数与随机稳定性

Lyapunov Exponent and Stochastic Stability for Infinitely Renormalizable Lorenz Maps

本文研究无限可重整压缩 Lorenz 映射，证明在有界几何或先验界条件下其临界值满足慢回归，并据此得到零逐点 Lyapunov 指数与随机稳定性。

Topology of the regular part for infinitely renormalizable quadratic polynomials

In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topology is rigid modulo its combinatorics.

Renormalization for Lorenz maps of monotone combinatorial types

Lorenz maps are maps of the unit interval with one critical point of order rho>1, and a discontinuity at that point. They appear as return maps of leafs of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.

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…

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.

The Renormalization Method and Quadratic-Like Maps

The renormalization of a quadratic-like map is studied. The three-dimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the {\sl a priori} complex bounds, the local connectivity of its Julia set is proved by using the three-dimensional Yoccoz puzzle. The generalized version of Sullivan's sector theorem is discussed and is used to prove his result that the Feigenbaum quadratic polynomial has the {\sl a priori} complex bounds and is unbranched. A dense subset on the boundary of the Mandelbrot set is constructed so …

无先验界的映射

Maps with no a priori bounds

该研究针对有理函数卫星情形下的类多项式映射，给出仅依赖相对周期与次数的模上界，否定无界相对周期下的先验界，并将工具拓展至无穷可重整化场景的相关几何量研究。

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。

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…