# Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

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
DOI: https://doi.org/10.1090/ecgd/354
Published electronically: October 7, 2020

## 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.
References
Similar Articles
• Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 30D05, 37P05
• Retrieve articles in all journals with MSC (2010): 30D05, 37P05
Bibliographic Information
• Fedor Pakovich
• Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Beer Sheva, 8410501 Israel
• MR Author ID: 602219
• Email: pakovich@math.bgu.ac.il
• Received by editor(s): April 23, 2019
• Received by editor(s) in revised form: July 11, 2020
• Published electronically: October 7, 2020
• Additional Notes: This research was partially supported by the ISF, Grants No. 1432/18