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PFH spectral invariants and $C^\infty$ closing lemmas
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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 $\delta$ a periodic orbit must appear of period $O(\delta^{-1})$. We also prove a "Weyl law" describing the asymptotic behavior of PFH spectral invariants.
Forward citations
Cited by 3 Pith papers
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Reeb orbits frequently intersecting a symplectic surface
For a nice contact form, some Reeb orbit intersects a given symplectic surface with frequency at least the surface area divided by the contact volume, without any genericity assumption.
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Generic density of periodic orbits of area-preserving maps on punctured surfaces
C-infinity generic area-preserving diffeomorphisms on finitely punctured surfaces have dense periodic orbits, and on punctured spheres the orbits can be chosen equidistributed.
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Conformally symplectic topology from a dynamical viewpoint
Characteristic foliations of contact Hamiltonian manifolds determine convexity of hypersurfaces, with Morse-Smale implying convexity, C0-density of convex hypersurfaces, and C2-robust non-convex examples in dimensions ≥5.
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