Local connectivity of Julia sets for unicritical polynomials

TL;DR

This paper proves local connectivity of Julia sets for certain unicritical polynomials with all periodic points repelling, extending previous results and introducing new analytic tools for controlling moduli of annuli.

Contribution

It establishes local connectivity for a broad class of unicritical polynomials using novel analytic techniques and a modified puzzle piece approach.

Findings

Julia sets are locally connected for finitely renormalizable unicritical polynomials with all periodic points repelling.

The proof employs a priori bounds derived from new analytic tools controlling annuli moduli.

Extension of Yoccoz's results to higher degrees and more general conditions.

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 Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in math.DS/0505191 that give control of moduli of annuli under maps of high degree.