Robust Transitivity in Hamiltonian Dynamics

TL;DR

This paper investigates the robust transitivity properties of Hamiltonian systems outside KAM tori, introducing new methods like symplectic blenders and demonstrating the prevalence of ergodic measures in these systems.

Contribution

It introduces $C^r$ open sets of symplectic diffeomorphisms with large robustly transitive sets and develops the concept of symplectic blenders for Hamiltonian dynamics.

Findings

Existence of large robustly transitive sets in Hamiltonian systems.

Closure of these sets contains a priori unstable integrable systems.

Presence of ergodic measures with large support in these systems.

Abstract

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1, 2, ..., \infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibiting "large" robustly transitive sets. We show that the $C^\infty$ closure of such open sets contains a variety of systems, including so-called a priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of the proof is a combination of studying minimal dynamics of symplectic iterated function systems and a new tool in Hamiltonian dynamics which we call symplectic blender.