On algebraic dependencies between Poincaré functions

Let A be a rational function of one complex variable of degree at least two, and its repelling fixed point with the multiplier A Poincaré function associated with is a function meromorphic on such that, and In this paper, we study the following problem: given Poincaré functions and, find out if there is an algebraic relation between them and, if such a relation exists, describe the corresponding algebraic curve We provide a solution, which can be viewed as a refinement of the classical theorem of Ritt about commuting rational functions. We also reprove and extend previous results concerning algebraic dependencies between Böttcher functions.