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Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms
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abstract
We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points. This settles Smale's tenth problem in the special case of area-preserving diffeomorphisms of closed surfaces.
Forward citations
Cited by 2 Pith papers
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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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