On algebraic dependencies between Poincaré functions

Abstract

Let $A$ be a rational function of one complex variable, and $z_0$ its repelling fixed point with the multiplier $\lambda .$ Then a Poincaré function associated with $z_0$ is a function ${\mathcal{P}}_{A,z_0,\lambda }$ meromorphic on $\mathbb{C}$ such that ${\mathcal{P}}_{A,z_0,\lambda }(0)=z_0$, ${\mathcal{P}}_{A,z_0,\lambda }^{\prime }(0)\ne 0,$ and ${\mathcal{P}}_{A,z_0,\lambda }(\lambda z)=A\circ {\mathcal{P}}_{A,z_0,\lambda }(z).$ In this paper, we investigate the following problem: given Poincaré functions ${\mathcal{P}}_{A_1,z_1,{\lambda }_1}$ and ${\mathcal{P}}_{A_2,z_2,{\lambda }_2}$, find out if there is an algebraic relation $f({\mathcal{P}}_{A_1,z_1,{\lambda }_1},{\mathcal{P}}_{A_2,z_2,{\lambda }_2})=0$ 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.

The final version, to appear in Ergod. Th. Dynam. Sys

References in corpus (2)

• Commuting rational functions revisited
• Invariant curves for endomorphisms of ${\mathbb{P}}^1\times {\mathbb{P}}^1$