On algebraic dependencies between Poincaré functions

Let A be a rational function of one complex variable of degree at least two, and z₀ its repelling fixed point with the multiplier λ. A Poincaré function associated with z₀ is a function P_A,z0,λ meromorphic on C such that P_A,z0,λ(0) = z₀, P_Á,z0,λ(0) ≠ 0, and P_A,z0,λ(λz) = A◦ P_A,z0,λ(z). In this paper, we study the following problem: given Poincaré functions P_A1,z1,λ1 and P_A2,z2,λ2, find out if there is an algebraic relation f(P_A1,z1,λ1, P_A2,z2,λ2) = 0 between them and, if such a relation exists, describe the corresponding algebraic curve f(x, y) = 0. 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.