Strong closing lemmas in Hamiltonian dynamics

TL;DR

This survey reviews strong closing lemmas in Hamiltonian dynamics, emphasizing proofs via spectral invariants across low and high dimensions, and explores related results for minimal hypersurfaces.

Contribution

It provides a comprehensive overview of strong closing lemmas proved using spectral invariants in various dimensions and contexts within Hamiltonian dynamics.

Findings

Strong closing lemmas hold in low-dimensional Hamiltonian systems.

Spectral invariants are key tools in proving these lemmas.

Extensions to high-dimensional systems and minimal hypersurfaces are discussed.

Abstract

This survey focuses on strong closing lemmas in Hamiltonian dynamics that are proved using spectral invariants (also known as action selectors) in symplectic geometry. We review strong closing lemmas in low-dimensional Hamiltonian dynamics (Reeb flows on contact three-manifolds and area-preserving maps on symplectic surfaces) and outline the key ideas behind their proofs. We also discuss results concerning strong closing lemmas in high-dimensional Hamiltonian dynamics, as well as analogous results for minimal hypersurfaces.