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

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

2006-09-01

arxiv.org/abs/math/0609046v2

Mathematics

Dynamical Systems

LaTeX, 29 pages, 2 figures

Scientific paper

A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we prove {\it a priori} bounds. They imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.

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