TY - JOUR
T1 - FINITENESS THEOREMS FOR COMMUTING AND SEMICONJUGATE RATIONAL FUNCTIONS
AU - PAKOVICH, FEDOR
N1 - Publisher Copyright:
© 2020 © 2020 by the author
PY - 2020/7/1
Y1 - 2020/7/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85095585366&partnerID=8YFLogxK
U2 - 10.1090/ECGD/354
DO - 10.1090/ECGD/354
M3 - Article
AN - SCOPUS:85095585366
SN - 1088-4173
VL - 24
SP - 202
EP - 229
JO - Conformal Geometry and Dynamics
JF - Conformal Geometry and Dynamics
IS - 10
ER -