When do two rational functions have the same Julia set?

Abstract

It is proved that non-exceptional rational functions f and g on the Riemann sphere have the same measure of maximal entropy iff there exist iterates F of f and G of g and natural numbers M, N such that (*) (G-1oG)oGM=(F-1oF)o FN. If one assumes only that f, g have the same Julia set and no singular or parabolic domains of normality for the iterates, one also proves (*).