Combinatorial rigidity for unicritical polynomials

Abstract

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. This implies that the connectedness locus (the “Multibrot set”) is locally connected at the corresponding parameter values and generalizes Yoccoz’s Theorem for quadratics to the higher degree case.

Authors

Artur Avila

CNRS UMR 7599
Laboratoire de Probabilités et Modèles
Aléatoires
Université Pierre et Marie Curie
Boîte courrier 188
75252 Paris Cedex 05
France

Jeremy Kahn

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

Mikhail Lyubich

Department of Mathematics
University of Toronto
Room 6290, 40 St. George Street
Toronto, ON M5S 2E4
Canada

Weixiao Shen

Department of Mathematics
National University of Singapore
2, Science Drive 2
Singapore 117543