Strong closing lemmas in Hamiltonian dynamics

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.