Finiteness theorems for commuting and semiconjugate rational functions

Abstract

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 $j,$ $1\le j\le r,$ and $k\ge 1$. We also show that the number $r$ and the degrees $°X_j,$ $1\le j\le r,$ 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.

Final version

References in corpus (1)

• On generalized Lattès maps

Cited by in corpus (4)

• Commuting rational functions revisited
• Invariant curves for endomorphisms of ${\mathbb{P}}^1\times {\mathbb{P}}^1$
• Semiconjugate rational functions: a dynamical approach
• Algebraic curves $A^{\circ l}(x)-U(y)=0$ and arithmetic of orbits of rational functions