# Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Finiteness theorems for commuting and semiconjugate rational functionsHTML articles powered by AMS MathViewer

by Fedor Pakovich
Conform. Geom. Dyn. 24 (2020), 202-229

## 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 $j,$ $1\leq j \leq r,$ 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.
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