Local connectivity of Julia sets for unicritical polynomials

2005/05/10 by Jeremy Kahn, Kahn, Jeremy, Mikhail Lyubich +1 · 4 citations
Mathematics · #37F45 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37F45

paper · pdf · doi:10.48550/arxiv.math/0505194

LaTeX, 14 pages, 1 figure

arxiv created 2005/05/10 · arxiv updated 2009/12/01

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 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.

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