Local connectivity of Julia sets for unicritical polynomials

Abstract

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 Principal Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in [KL09] that give control of moduli of annuli under maps of high degree.

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Authors

Jeremy Kahn

Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651
United States

Mikhail Lyubich

Department of Mathematics
University of Toronto
40 St. George Street
Toronto, ON  M5S 2E4
Canada