Meromorphic first integrals of analytic diffeomorphisms

TL;DR

This paper establishes an upper bound on the number of meromorphic first integrals for analytic diffeomorphisms near fixed points, based on eigenvalue resonances, extending Poincaré-type results to discrete systems.

Contribution

It introduces a new bound for meromorphic first integrals of analytic maps near fixed points, linking eigenvalue resonances to integrability constraints.

Findings

Derived an explicit upper bound based on eigenvalue resonances.

Applied the theoretical results to various difference equations.

Extended classical Poincaré results to discrete dynamical systems.

Abstract

We give an upper bound for the number of functionally independent meromorphic first integrals that a discrete dynamical system generated by an analytic map $f$ can have in a neighborhood of one of its fixed points. This bound is obtained in terms of the resonances among the eigenvalues of the differential of $f$ at this point. Our approach is inspired on similar Poincar\'e type results for ordinary differential equations. We also apply our results to several examples, some of them motivated by the study of several difference equations.