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Strong closing lemmas in Hamiltonian dynamics

Kei Irie

TL;DR

The article surveys how spectral invariants (action selectors) enable strong closing lemmas in Hamiltonian dynamics, especially in low dimensions via Reeb flows and area-preserving maps, and discusses prospects and obstacles in higher dimensions. It explains the local-sensitivity mechanism for spectral invariants, derives Weyl-law-type growth, and uses these tools to prove that generic perturbations yield dense periodic orbits and equidistribution in 3D settings. It also draws parallel results in minimal hypersurfaces through volume spectra, highlighting deep connections between dynamical closing problems and variational/min-max frameworks. Open questions remain for high dimensions, where slow-growth fails and only special systems currently admit strong closing results. Overall, the work synthesizes spectral-invariant methods as a powerful lens for obtaining robust density and equidistribution results in Hamiltonian and geometric contexts.

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.

Strong closing lemmas in Hamiltonian dynamics

TL;DR

The article surveys how spectral invariants (action selectors) enable strong closing lemmas in Hamiltonian dynamics, especially in low dimensions via Reeb flows and area-preserving maps, and discusses prospects and obstacles in higher dimensions. It explains the local-sensitivity mechanism for spectral invariants, derives Weyl-law-type growth, and uses these tools to prove that generic perturbations yield dense periodic orbits and equidistribution in 3D settings. It also draws parallel results in minimal hypersurfaces through volume spectra, highlighting deep connections between dynamical closing problems and variational/min-max frameworks. Open questions remain for high dimensions, where slow-growth fails and only special systems currently admit strong closing results. Overall, the work synthesizes spectral-invariant methods as a powerful lens for obtaining robust density and equidistribution results in Hamiltonian and geometric contexts.

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.

Paper Structure

This paper contains 26 sections, 28 theorems, 22 equations.

Table of Contents

  1. Introduction.
  2. $C^1$-closing lemmas
  3. $C^r$-closing lemmas for $r \ge 2$
  4. Strong closing lemmas for low-dimensional Hamiltonian dynamics
  5. Strong closing lemmas for high-dimensional Hamiltonian dynamics
  6. Minimal hypersurfaces
  7. Hamiltonian $C^1$-closing lemma and Herman's example
  8. Strong closing lemmas in low-dimensional Hamiltonian dynamics
  9. Contact manifolds and Reeb flows
  10. Reeb flows on contact $3$-manifolds
  11. Closed geodesics on surfaces
  12. Area-preserving maps of surfaces
  13. Spectral invariants and strong closing lemmas
  14. Reeb flows on contact manifolds
  15. Rational symplectic diffeomorphisms
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $H \in \mathcal{H}_{C^2}(M)$ and $p \in M$. For any neighborhood $\mathcal{U}$ of $H$ in $\mathcal{H}_{C^2}(M)$, there exists $H' \in \mathcal{U}$ such that $p$ is a periodic point of $X_{H'}$.

Theorems & Definitions (51)

  • Theorem 2.1: Pugh_Robinson
  • Corollary 2.2
  • Proof 1: Proof (sketch)
  • Remark 2.3
  • Theorem 2.4: Herman
  • Example 3.1: Boundary of symplectic ellipsoid
  • Example 3.2: Geodesic flow
  • Theorem 3.3
  • Definition 3.4
  • Theorem 3.5: Strong closing lemma for $3$-dimensional Reeb flows Irie_JMD
  • ...and 41 more