On algebraic dependencies between Poincar\'e functions

TL;DR

This paper studies algebraic relations between Poincaré functions associated with rational functions, extending classical results and providing criteria for when such functions are algebraically dependent.

Contribution

It offers a new criterion for algebraic dependencies between Poincaré functions, refining Ritt's theorem and extending results on Böttcher functions.

Findings

Characterization of algebraic relations between Poincaré functions.

Extension of Ritt's classical theorem to Poincaré functions.

Reproof and extension of results on algebraic dependencies of Böttcher 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 Poincar\'e function associated with $z_0$ is a function $\mathcal{P}_{A,z_0,\lambda}$ meromorphic on $\mathbb C$ such that $\mathcal{P}_{A,z_0,\lambda}(0)=z_0$, $\mathcal{P}_{A,z_0,\lambda}'(0)\neq 0,$ 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 Poincar\'e 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 B\"ottcher functions.