97-21   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## On semiconjugation of entire functions

Let $f$ and $h$ be transcendental entire functions and let $g$ be a continuous and open map of the complex plane into itself with $g\circ f=h\circ g$. 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 $J(\cdot)$ denotes the Julia set of a function. We conclude that if $f$ has no wandering domains, then $h$ has no wandering domains. We further show that for given transcendental entire functions $f$ and $h$, there are only countably many entire functions $g$ such that $g\circ f=h\circ g$.

Mathematics Subject Classification (1991): 30D05, 58F23

Bibliographical note: Math. Proc. Cambridge Philos. Soc. 126, 565-574 (1999)