Combinatorial rigidity for some infinitely renormalizable unicritical polynomials

Abstract

Here we prove thatinfinitely renormalizableunicritical polynomialsPc:z↦zd+cP_c:z \mapsto z^d+c, withc∈Cc\in \mathbb {C}, satisfyinga priori boundsand a certain “combinatorial” condition arecombinatorially rigid. This implies the local connectivity of the connectedness loci (the Mandelbrot set whend=2d=2) at the corresponding parameters.