Closing lemma and KAM normal form

TL;DR

This paper introduces a novel KAM normal form approach to the closing lemma problem, enabling the proof of dense periodic orbits in high regularity, high dimensional, and Riemannian settings, confirming several longstanding conjectures.

Contribution

It develops a constructive, effective method based on KAM normal form that extends closing lemma results to complex settings and confirms multiple conjectures.

Findings

Periodic orbits are asymptotically dense for typical perturbations in nearly integrable systems.

Smooth perturbations of geodesic flows on flat tori have dense periodic orbits, partially solving an open problem.

Hamiltonian and contact perturbations of ellipsoid geodesic flows have dense orbits, supporting conjectures in the field.

Abstract

In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing $C^1$ perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincar\'e and confirm several old and new conjectures with weak formulations. First, for Poincar\'e's original setting of nearly integrable systems, we prove that for typical perturbations, periodic orbits are asymptotically dense as the size of perturbation tends to zero. Second, we prove that typical smooth perturbation of the geodesic flow on the flat torus has asymptotically dense periodic orbits, which partially solves an open problem since Pugh-Robinson's $C^1$-closing lemma. Third, we prove that for typical Hamiltonian or contact perturbation of the geodesic flows of the ellipsoid has asymptotically dense orbit on the energy level, which enhances the recent researches on strong closing lemma, and also confirms partially a conjecture of Fish-Hofer in this setting. We also discuss the relation of our models to the recent researches on many-body localization in physics.