Finiteness theorems for commuting and semiconjugate rational functions

F. Pakovich

Published 2016-04-16Version 1

In this paper we study the functional equation $A\circ X=X\circ B,$ where $A$, $B$, and $X$ are rational functions of degree at least two. More precisely, we assume that $B$ is fixed and study solutions of the above equation in $A$ and $X$. Roughly speaking, our main result states that, up to some natural transformations, the set of such solutions is finite, unless $B$ is a Latt\`es function or is conjugated to $z^{\pm d}$ or $\pm T_d$. In more details, we show that there exist rational functions $A_1, A_2,\dots, A_r$ and $X_1, X_2,\dots, X_r$ such that the equality $A\circ X=X\circ B$ holds if and only if $A=\mu \circ A_j\circ \mu^{-1}$ for some $j,$ $1\leq j \leq r,$ and M\"obius transformation $\mu$, and $X=\mu \circ X_j\circ B^{\circ k}$ for some $k\geq 1$. We also show that the degrees of $X_j,$ $1\leq j \leq r,$ and $r$ can be bounded from above by constants depending on degree of $B$ only. As an application we obtain "effective" analogues of classical results about commuting rational functions.