Combinatorial rigidity for unicritical polynomials


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.