FINITENESS THEOREMS FOR COMMUTING AND SEMICONJUGATE RATIONAL FUNCTIONS

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Abstract

Let B bea fixed rational function of one complex variable of degree at least two. In this paper, we study solutions of the functional equation A ◦ X = X ◦ B in rational functions A and X. Our main result states that, unless B is a Lattès map or is conjugate to z±d or ±Td, the set of solutions is finite, up to some natural transformations. In more detail, we show that there exist finitely many rational functions A1, A2, …, Ar and X1, X2, …, Xr such that the equality A ◦ X = X ◦ B holds if and only if there exists a Möbius transformation μ such that A = μ ◦ Aj ◦ μ–1 and X = μ ◦ Xj ◦ B◦k for some j, 1 ≤ j ≤ r, and k ≥ 1. We also show that the number r and the degrees deg Xj, 1 ≤ j ≤ 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.

Original languageEnglish
Pages (from-to)202-229
Number of pages28
JournalConformal Geometry and Dynamics
Volume24
Issue number10
DOIs
StatePublished - 1 Jul 2020

ASJC Scopus subject areas

  • Geometry and Topology

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