Combinatorial rigidity for unicritical polynomials

Artur Avila, Jeremy Kahn, Mikhail Lyubich, Weixiao Shen

Published 2005-07-12Version 1

We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the ``Multibrot set'') is locally connected at the corresponding parameter values. It generalizes Yoccoz's Theorem for quadratics to the higher degree case.