Combinatorial rigidity for unicritical polynomials

Artur Avila; Jeremy Kahn; Mikhail Lyubich; Weixiao Shen

doi:10.4007/annals.2009.170.783

Combinatorial rigidity for unicritical polynomials

Pages 783-797 from Volume 170 (2009), Issue 2 by Artur Avila, Jeremy Kahn, Mikhail Lyubich, Weixiao Shen

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.

Mathematical Subject Classification

Primary 2000: 37F45

DOI

https://doi.org/10.4007/annals.2009.170.783

2552107

zbMATH

1204.37047

Milestones

Received: 1 August 2005