Dynamics of Rational Symplectic Mappings and Difference Galois Theory

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 nonintegrability of a rational symplectic map in the noncommutative sense (Mishchenko–Fomenko). As a particular case, we obtain a complete discrete analogue of Morales–Ramis Theorems for nonintegrability in the sense of Liouville.