Ergodic closing lemmas and invariant Lagrangians

TL;DR

$C^\infty$-closing questions in higher-dimensional Hamiltonian dynamics are addressed by exploiting heaviness in Floer-theoretic settings. The authors prove that, under rational and bounded conditions with heavy Lagrangians, $C^\infty$-generic perturbations create sequences of periodic orbits whose invariant measures converge to a measure supported on the heavy Lagrangian, and they obtain quantitative recurrence bounds via an autonomous perturbation $G$ with controlled support. A central contribution is Theorem \ref{['thm:precise1']}, a general, quantitative closing result that yields both existence of periodic points near $L$ and explicit lower bounds on visit frequencies, with further implications for dynamics of pseudo-rotations and invariant sets. The work also clarifies the necessity of boundedness through Herman-type counterexamples and connects to KAM-type phenomena when invariant Lagrangians persist. Overall, the paper advances a Floer-theoretic, measure-theoretic approach to the $C^\u221e$ closing problem in higher dimensions and provides tools to understand the statistical behavior of near-periodic dynamics under smooth perturbations.

Abstract

Motivated by the ergodic closing lemma of Mañé, 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.

Theorems & Definitions (25)

• Corollary 1
• Theorem 1
• Remark 1
• Theorem 2
• Theorem 3
• Corollary 2
• Definition 4
• Remark 2
• Definition 5
• Lemma 1