Abstract

We prove that the Julia set J(f) of at most finitely renormalizable unicritical polynomial f : z↦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.

Milestones

Received: 1 August 2005

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