Dynamics of rational symplectic mappings and difference Galois theory

TL;DR

This paper explores the connection between the integrability of rational symplectic maps and difference Galois theory, providing conditions to determine non-integrability and extending Morales-Ramis theorems to discrete systems.

Contribution

It introduces a Galoisian criterion for non-integrability of rational symplectic maps and develops a discrete analogue of Morales-Ramis theorems.

Findings

Galoisian condition ensures non-integrability in non-commutative sense

Complete discrete Morales-Ramis theorem for Liouville non-integrability

New criteria for analyzing integrability of symplectic maps

Abstract

In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville.