FINITENESS THEOREMS FOR COMMUTING AND SEMICONJUGATE RATIONAL FUNCTIONS

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 ±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 A₁, A₂, …, A_r and X₁, X₂, …, X_r such that the equality A ◦ X = X ◦ B holds if and only if there exists a Möbius transformation μ such that A = μ ◦ A_j ◦ μ^–1 and X = μ ◦ X_j ◦ B^◦k for some j, 1 ≤ j ≤ r, and k ≥ 1. We also show that the number r and the degrees deg X_j, 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.