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- W3091887355 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 $Acirc X=Xcirc B$ 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 $Acirc X=Xcirc B$ holds if and only if there exists a Möbius transformation $mu$ such that $A=mu circ A_jcirc mu ^{-1}$ and $X=mu circ X_jcirc B^{circ k}$ for some $j,$ $1leq j leq r,$ and $kgeq 1$. We also show that the number $r$ and the degrees $deg X_j,$ $1leq j leq r,$ 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." @default.
- W3091887355 created "2020-10-15" @default.
- W3091887355 creator A5034534465 @default.
- W3091887355 date "2020-10-07" @default.
- W3091887355 modified "2023-10-16" @default.
- W3091887355 title "Finiteness theorems for commuting and semiconjugate rational functions" @default.
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- W3091887355 doi "https://doi.org/10.1090/ecgd/354" @default.
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