On semiconjugate rational functions

Abstract

We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of such functions, then either $B$ can be obtained from $A$ by a certain iterative process, or $A$ and $B$ can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.