On algebraic dependencies between Poincaré functions

TL;DR

The paper studies when Poincaré functions $\mathcal{P}_{A,z_0,\lambda}$ attached to non-special rational maps satisfy algebraic relations, refining Ritt-type results on commuting functions. It develops an invariant-curve framework via generalized Lattès maps and semiconjugacies to characterize when a curve $C: f(x,y)=0$ supporting $f(\mathcal{P}_{A_1,z_1,\lambda_1}(z^{d_1}),\mathcal{P}_{A_2,z_2,\lambda_2}(z^{d_2}))=0$ must have genus zero and admit a rational parametrization, tying the relation to a commutative diagram with a common semiconjugate map $B$. The work also extends these ideas to Böttcher functions, proving that algebraic dependencies imply graph-type curves with polynomials commuting with the original maps, and provides a precise dichotomy between trivial and nontrivial dependencies under gcd constraints. Collectively, the results connect algebraic dependencies of dynamical conjugacies to invariant-curve classifications and semiconjugacy structures, sharpening the link between Poincaré/Böttcher theory and Ritt-type phenomena.

Abstract

Let $A$ be a rational function of one complex variable, and $z_0$ its repelling fixed point with the multiplier $λ.$ Then a Poincaré function associated with $z_0$ is a function $\mathcal{P}_{A,z_0,λ}$ meromorphic on $\mathbb C$ such that $\mathcal{P}_{A,z_0,λ}(0)=z_0$, $\mathcal{P}_{A,z_0,λ}'(0)\neq 0,$ and $\mathcal{P}_{A,z_0,λ}(λz)=A\circ \mathcal{P}_{A,z_0,λ}(z).$ In this paper, we investigate the following problem: given Poincaré functions $\mathcal{P}_{A_1,z_1,λ_1}$ and $\mathcal{P}_{A_2,z_2,λ_2}$, find out if there is an algebraic relation $f(\mathcal{P}_{A_1,z_1,λ_1},\mathcal{P}_{A_2,z_2,λ_2})=0$ 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.

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3