Finiteness theorems for commuting and semiconjugate rational functions

Let $B$ be a fixed rational function of one complex variable of degree at least two. In this paper, we study solutions of the functional equation $A\circ X=X\circ B$ in rational functions $A$ and $X$. Our main result states that, unless $B$ is a Lattès map or is conjugate to $z^{\pm d}$ or $\pm T_d$, the set of solutions is finite, up to some natural transformations. In more detail, we show that there exist finitely many 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 there exists a Möbius transformation $\mu$ such that $A=\mu \circ A_j\circ \mu ^{-1}$ and $X=\mu \circ X_j\circ B^{\circ k}$ for some $j,$ $1\leq j \leq r,$ and $k\geq 1$. We also show that the number $r$ and the degrees $\deg X_j,$ $1\leq j \leq r,$ can be bounded from above in terms of the degree of $B$ only. As an application, we prove an effective version of the classical theorem of Ritt about commuting rational functions.