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On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics

Yoshihiro Sugimoto

TL;DR

The paper proves a non-contractible version of the Hofer-Zehnder conjecture for broad classes of closed symplectic manifolds by combining non-contractible Floer theory with Z_p-equivariant and Tate constructions. It shows that if a Hamiltonian diffeomorphism has finitely many 1-periodic orbits in a nonzero homology class γ and at least one orbit has nontrivial local Floer cohomology, then for all large primes p there exists a simple p-periodic orbit in p·γ, yielding infinitely many simple periodic orbits in multiples of γ. The approach relies on a local-to-global perturbation framework (X_K-modules) to manage degeneracies, together with a Z_p-equivariant pair-of-pants product that links the behavior of HF(φ^p) to HF(φ) via Tate cohomology and spectral sequences. The results generalize prior works by accommodating non-contractible classes and extend to both toroidally monotone and, with additional technical work, weakly monotone manifolds via a robust homological algebra toolkit and geometric moduli-space constructions.

Abstract

In this paper, we treat an open problem related to the number of periodic orbits of Hamiltonian diffeomorphisms on closed symplectic manifolds. Hofer-Zehnder conjecture states that a Hamiltonian diffeomorphisms has infinitely many periodic orbits if it has "homologically unnecessary periodic orbits"". For example, non-contractible periodic orbits are homologically unnecessary periodic orbits because Floer homology of non-contractible periodic orbits is trivial. We prove Hofer-Zehnder conjecture for non-contractible periodic orbits for very wide classes of symplectic manifolds.

On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics

TL;DR

The paper proves a non-contractible version of the Hofer-Zehnder conjecture for broad classes of closed symplectic manifolds by combining non-contractible Floer theory with Z_p-equivariant and Tate constructions. It shows that if a Hamiltonian diffeomorphism has finitely many 1-periodic orbits in a nonzero homology class γ and at least one orbit has nontrivial local Floer cohomology, then for all large primes p there exists a simple p-periodic orbit in p·γ, yielding infinitely many simple periodic orbits in multiples of γ. The approach relies on a local-to-global perturbation framework (X_K-modules) to manage degeneracies, together with a Z_p-equivariant pair-of-pants product that links the behavior of HF(φ^p) to HF(φ) via Tate cohomology and spectral sequences. The results generalize prior works by accommodating non-contractible classes and extend to both toroidally monotone and, with additional technical work, weakly monotone manifolds via a robust homological algebra toolkit and geometric moduli-space constructions.

Abstract

In this paper, we treat an open problem related to the number of periodic orbits of Hamiltonian diffeomorphisms on closed symplectic manifolds. Hofer-Zehnder conjecture states that a Hamiltonian diffeomorphisms has infinitely many periodic orbits if it has "homologically unnecessary periodic orbits"". For example, non-contractible periodic orbits are homologically unnecessary periodic orbits because Floer homology of non-contractible periodic orbits is trivial. We prove Hofer-Zehnder conjecture for non-contractible periodic orbits for very wide classes of symplectic manifolds.

Paper Structure

This paper contains 13 sections, 28 theorems, 355 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Elementary notations
  4. Floer cohomology theory
  5. Local Floer cohomology
  6. $\mathbb{Z}_p$-equivariant Floer theory
  7. Local (${\mathbb{Z}_p}$-equivariant Tate) Floer cohomology and homological perturbation theory
  8. Proof of the main theorem: toroidally monotone case
  9. Weakly monotone case
  10. Preliminary
  11. ${X_K}$-module, ${X_K}$-morphism and ${X_K}$-homotopy
  12. Directed family of $X_K$-modules and $X_K$-morphisms
  13. Geometric constructions

Key Result

Theorem 2

Let ${(M,\omega)}$ be a closed weakly monotone symplectic manifold, and let ${\phi \in \textrm{Ham}(M,\omega)}$ be a Hamiltonian diffeomorphism generated by a periodic Hamiltonian function ${H\in C^{\infty}(S^1\times M)}$. Suppose that the number of ${1}$-periodic orbits of $\phi$ in a non-trivial c

Theorems & Definitions (54)

  • Conjecture 1: Hofer-Zehnder conjecture
  • Theorem 2
  • Definition 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • Remark 7
  • Definition 8
  • Remark 9
  • Remark 10
  • ...and 44 more