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Finiteness theorems for commuting and semiconjugate rational functions


Author: Fedor Pakovich
Journal: Conform. Geom. Dyn. 24 (2020), 202-229
MSC (2010): Primary 30D05, 37P05
DOI: https://doi.org/10.1090/ecgd/354
Published electronically: October 7, 2020
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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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Additional Information

Fedor Pakovich
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Beer Sheva, 8410501 Israel
Email: pakovich@math.bgu.ac.il

DOI: https://doi.org/10.1090/ecgd/354
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
Article copyright: © Copyright 2020 by the author