Rigidity and non local connectivity of Julia sets of some quadratic polynomials
نویسنده
چکیده
For an infinitely renormalizable quadratic map fc : z 7→ z 2+c with the sequence of renormalization periods {nm} and the rotation numbers {tm = pm/qm}, we prove that if lim supn−1 m log |pm| > 0, then the Mandelbrot set is locally connected at c. We prove also that if lim sup |tm+1| 1/qm < 1 and qm → ∞, then the Julia set of fc is not locally connected provided c is the limit of corresponding cascade of successive bifurcations. This quantifies a construction of A. Douady and J. Hubbard, and strengthens a condition proposed by J. Milnor.
منابع مشابه
On the Borderline of Real and Complex Dynamics
1.1. Overview. We will describe recent developments in several intimately related problems of complex and real one-dimensional dynamics: rigidity of polynomials and local connectivity of the Mandelbrot set, measure of Julia sets, and attractors of quasi-quadratic maps. A combinatorial basis for this study is provided by the Yoccoz puzzle. The main problem is to understand the geometry of the pu...
متن کاملTopological, Geometric and Complex Analytic Properties of Julia Sets
In this paper, we discuss several aspects of Julia sets, as well as those of the Mandelbrot set. We are interested in topological properties such as connectivity and local connectivity, geometric properties such as Hausdorff dimension and Lebesgue measure, and complex analytic properties such as holomorphic removability. As one can easily see from the pictures of numerical experiments, there is...
متن کامل§2. Polynomials for which All But One of the Critical Orbits Escape
Introduction 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 Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of quadratic Julia sets.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 " par...
متن کاملOn Fibers and Local Connectivity of Mandelbrot and Multibrot Sets
We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show “shrinking of puzzle pieces” without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set (see Definition 3.2) and show that fibers of certain points are “trivial”, i.e., they consist of single points. ...
متن کاملOn 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 [S3] and [S4]. 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 f...
متن کامل