On semiconjugation of entire functions

Abstract

Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g∘f=h∘g. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g−1(J(h))=J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that g∘f=h∘g.