RIGIDITY OF HARMONIC MEASURE OF TOTALLY DISCONNECTED FRACTALS

Abstract

Let f:V®U be a generalized polynomial-like map. Suppose that harmonic measure w=w(×,¥) on the Julia set J_f is equal to measure of maximal entropy m for f:J_f. Then the dynamics (f,V,U) is called maximal. We are going to give a necessary condition for the dynamics to be conformally equivalent to a maximal one, that is to be conformally maximal. Namely the purpose of the paper is to prove that if the Julia set is totally disconnected then w»m implies that the system (f,U,V) is conformally maximal. This shows that maximal systems are natural substitutes for polynomials in the class of genereralized polynomial-like mappings.