On Fibers and Renormalization of Julia Sets and Multibrot Sets

Abstract

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 Julia sets is preserved under renormalization and other surgery procedures, such as the Branner-Douady homeomorphisms. We conclude with various applications about quadratic polynomials and its parameter space: we identify embedded paths within the Mandelbrot set, and we show that Petersen's theorem about quadratic Julia sets with Siegel disks of bounded type generalizes from period one to arbitrary periods so that they all have trivial fibers and are thus locally connected.