On semiconjugation of entire functions

Let f and h be transcendental entire functions and let g be a continuous an d open map of the complex plane into itself with g o f = h o g. We show tha t 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 en tire functions f and h, there are only countably many entire functions g su ch that g o f = h o g.