On the Combinatorial Rigidity for Polynomials with Attracting Cycles

Abstract

We show that every polynomial of degree $d\ge 2$ in the connectedness locus with an attracting cycle which attracts at least two critical points and no indifferent cycles is not combinatorially rigid. In particular, we prove that a hyperbolic polynomial with connected Julia set is combinatorially rigid if and only if it is of the ``disjoint type''.