Ergodic closing lemmas and invariant Lagrangians

TL;DR

This paper extends ergodic closing lemmas to higher-dimensional Hamiltonian systems, showing that under certain conditions, periodic orbits with controllable statistical properties can be generated by small perturbations.

Contribution

It introduces a $C^ abla$ closing lemma for Hamiltonian systems, linking invariant Lagrangian submanifolds to the existence of periodic orbits with specific statistical behaviors.

Findings

Existence of periodic orbits converging to invariant measures on the zero section.

Under generic conditions, small perturbations produce orbits with desired statistical properties.

Application of Floer theory to control the statistical behavior of orbits.

Abstract

Motivated by the ergodic closing lemma of Ma\~n\'e, we investigate the $C^\infty$ closing lemma in higher-dimensional Hamiltonian systems, with a focus on the statistical behavior of periodic orbits generated by $C^\infty$-small perturbations. We demonstrate that, under certain Floer-theoretic conditions, invariant or recurrent Lagrangian submanifolds can give rise to periodic orbits whose statistical properties are controllable. For instance, we show that for Hamiltonian systems preserving the zero section in $T^*\mathbb{T}^n$, $C^\infty$ generically, there exist periodic orbits converging to an invariant measure supported on the zero section.