Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms
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 n...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.2110.02925 |