Proof of the Branner-Hubbard conjucture on Cantor Julia sets

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 Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of quadratic Julia sets.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. …

Cantor Julia sets with Hausdorff dimension two

We prove the existence of Cantor Julia sets with Hausdorff dimension two. In particular, such examples can be found in cubic polynomials. The proof is based on the characterization of the parameter spaces and dynamical planes of cubic polynomials by Branner-Hubbard, and the parabolic bifurcation theory developed by Shishikura in 1990s.

On Fibers and Renormalization of Julia Sets and Multibrot Sets

We continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in IMS preprints 1998/12 and 1998/13a. The question of local connectivity of these sets is discussed in terms of fibers and becomes the question of triviality of fibers. In this paper, the focus is on the behavior of fibers under renormalization and other surgery procedures. We show that triviality of fibers of Mandelbrot and Multibrot sets is preserved under tuning maps and other (partial) homeomorphisms. Similarly, we show for unicritical polynomials that triviality of fibers of …

具有不连通Julia集的C^2双曲多项式自同构的结构

Structure of hyperbolic polynomial automorphisms of C^2 with disconnected Julia sets

本文在温和耗散条件下，对具有不连通Julia集的C^2双曲多项式自同构，用有限多个准螺线管给出Julia集分量的拓扑描述，建立了精细谱分解与二维Branner-Hubbard理论。