Combinatorial rigidity for some infinitely renormalizable unicritical polynomials

Abstract

Here we prove that infinitely renormalizable unicritical polynomials Pc: z → zd + c, with c ∈ C, satisfying a priori bounds and a certain “combinatorial” condition are combinatorially rigid. This implies the local connectivity of the connectedness loci (the Mandelbrot set when d = 2) at the corresponding parameters. © 2010 American Mathematical Society.