Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface

TL;DR

This paper demonstrates that analytic, symplectic diffeomorphisms on any closed surface can exhibit both chaotic and elliptic behaviors simultaneously, resolving a longstanding open problem.

Contribution

It proves the coexistence of chaotic and elliptic dynamics in analytic, symplectic surface diffeomorphisms, addressing a problem posed by F. Przytycki in 1982.

Findings

Existence of positive metric entropy in these systems

Presence of integrable KAM islands alongside chaos

Resolution of Przytycki's 1982 problem

Abstract

We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (intregrable KAM island) among analytic, symplectic diffeomorphism of any closed surface. In particilar this solves a problem by F. Przytycki (1982).