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PFH spectral invariants and $C^\infty$ closing lemmas

Oliver Edtmair, Michael Hutchings

Abstract

We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove $C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of area-preserving surface diffeomorphisms. In particular, we show that for a $C^\infty$-generic area-preserving diffeomorphism of the torus, the set of periodic points is dense. Our closing lemmas are quantitative, asserting roughly speaking that for a given Hamiltonian isotopy, within time $δ$ a periodic orbit must appear of period $O(δ^{-1})$. We also prove a "Weyl law" describing the asymptotic behavior of PFH spectral invariants.

PFH spectral invariants and $C^\infty$ closing lemmas

Abstract

We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove closing lemmas for certain Hamiltonian isotopy classes of area-preserving surface diffeomorphisms. In particular, we show that for a -generic area-preserving diffeomorphism of the torus, the set of periodic points is dense. Our closing lemmas are quantitative, asserting roughly speaking that for a given Hamiltonian isotopy, within time a periodic orbit must appear of period . We also prove a "Weyl law" describing the asymptotic behavior of PFH spectral invariants.

Paper Structure

This paper contains 18 sections, 23 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Closing lemmas
  3. Quantitative closing lemmas
  4. Background and motivation for the proofs
  5. Acknowledgments.
  6. Periodic Floer homology
  7. PFH generators and holomorphic curves
  8. The chain complex
  9. Examples of PFH
  10. The $U$ map
  11. Filtered PFH
  12. The monotone case
  13. Invariance of PFH under Hamiltonian isotopy
  14. Spectral invariants in PFH
  15. The ball packing lemma
  16. ...and 3 more sections

Key Result

Lemma 1.4

If the Hamiltonian isotopy class $\Phi$ satisfies the $C^\infty$ closing property, then it satisfies the $C^\infty$ generic density property.

Theorems & Definitions (83)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Definition 1.6
  • Example 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 73 more