[Paper Review] PFH spectral invariants and $C^\infty$ closing lemmas

Oliver Edtmair, Michael Hutchings|arXiv (Cornell University)|Oct 6, 2021

Geometric and Algebraic Topology26 references4 citations

TL;DR

This paper develops spectral invariants in periodic Floer homology (PFH) for area-preserving surface diffeomorphisms and uses them to establish quantitative $C^inity$ closing lemmas. It proves that for rational Hamiltonian isotopy classes satisfying the $U$-cycle property, periodic orbits must appear within time $\delta$ of period $O(\delta^{-1})$, and shows the set of periodic points is dense for $C^\infty$-generic such maps on $S^2$ or $T^2$. A Weyl law for PFH spectral invariants is also established.

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.

Motivation & Objective

• To develop a theory of spectral invariants in periodic Floer homology (PFH) for area-preserving surface diffeomorphisms.
• To establish quantitative $C^\infty$ closing lemmas for Hamiltonian isotopy classes of such maps.
• To prove that for $C^\infty$-generic area-preserving diffeomorphisms in rational Hamiltonian isotopy classes satisfying the $U$-cycle property, the set of periodic points is dense in the surface.
• To derive a Weyl law describing the asymptotic growth of PFH spectral invariants.

Proposed method

• The authors define PFH spectral invariants associated to periodic orbits of area-preserving diffeomorphisms via the mapping torus construction.
• They use the mapping torus $Y_\phi$ to relate periodic orbits of $φ$ to closed orbits of the vector field $∂_t$, with a fiberwise symplectic form $ω$ and closed 2-form $ω_φ$.
• The theory relies on the cohomology class $[ω_\phi] \in H^2(Y_\phi; \mathbb{R})$, and defines rationality when this class is a real multiple of an integral class.
• They introduce the $U$-cycle property, ensuring certain homology classes are $U$-cyclic of bounded order, which is essential for spectral invariants to grow linearly with period.
• A ball-packing argument in symplectic topology is used to compare spectral invariants with symplectic volumes, leveraging ECH spectral invariants $c_k^{\text{ECH}}(X)$.
• The proof of the Weyl law uses a limiting argument over sequences of orbits with increasing degree, showing spectral invariants scale linearly with the action difference, with rate $O(d^{-1/2})$.

Experimental results

Research questions

• RQ1 Under what conditions on a Hamiltonian isotopy class does the $C^\infty$ generic density property hold for periodic points?
• RQ2 Can spectral invariants in PFH be used to prove quantitative closing lemmas for area-preserving surface diffeomorphisms?
• RQ3 What is the asymptotic behavior of PFH spectral invariants as the period of orbits tends to infinity?
• RQ4 How does the Calabi invariant relate to PFH spectral invariants in the limit of large periodic orbits?
• RQ5 What role does the $U$-cycle property play in controlling the growth of spectral invariants?

Key findings

• For any Hamiltonian isotopy class $\Phi$ that is rational and satisfies the $U$-cycle property, the $C^\infty$ closing property holds: within any open set $\mathcal{U}$, a $C^\infty$-small perturbation creates a periodic orbit intersecting $\mathcal{U}$.
• The paper proves that for $C^\infty$-generic area-preserving diffeomorphisms in rational isotopy classes on $S^2$ or $T^2$, the set of periodic points is dense in the surface.
• A Weyl law is established: for sequences of orbits $\gamma_i$ with $d(\gamma_i) \to \infty$, the normalized spectral invariant difference converges to $A^{-1} \int_{Y_\phi} (H_2 - H_1) \omega_\phi \wedge dt$, where $A = \int_\Sigma \omega$.
• The rate of convergence in the Weyl law is shown to be $O(d^{-1/2})$ when ball packings are chosen carefully.
• The result implies that the Calabi invariant of a Hamiltonian diffeomorphism on a disk is recovered as the limit of normalized PFH spectral invariants.
• The theory confirms that spectral invariants in PFH can detect the Calabi invariant, extending previous results and supporting the simplicity conjecture.