Conley conjecture and local Floer homology
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In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit, the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold, the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.
KeywordsPeriodic orbits Hamiltonian diffeomorphisms Conley conjecture Floer homology
Mathematics Subject Classification53D40 37J10 37J45
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The author is deeply grateful to Başak Gürel and Viktor Ginzburg for their numerous valuable remarks and suggestions. The author also thanks the anonymous referee for critical remarks and helpful comments. A part of this work was carried out during Hamiltonian and Reeb dynamics: New methods and Applications workshop at Lorentz Center (Leiden, Netherlands); and supported by UCSC. The author would like to thank these institutions for their support.
- 7.Ginzburg, V.L., Gürel, B.Z.: Conley conjecture revisited. Int. Math. Res. Notices IMRN (2017). https://doi.org/10.1093/imrn/rnx137
- 11.Pears, J.: Degenerate Critical Points and the Conley Index. University of Edinburg, Edinburg (1995)Google Scholar
- 12.Piunikhin, S., Salamon, D., Schwarz, M.: Floer–Donaldson, symplectic, theory and quantum cohomology. In: Contact and Symplectic Geometry (Cambridge, 1994), pp. 171–200, Publ. Newton Inst., vol. 8, Cambridge University Press, Cambridge (1996)Google Scholar
- 13.Salamon, D.A.: Lectures on Floer homology, In: Symplectic Geometry and Topology, pp. 143–229, IAS/Park City Math. Ser., vol. 7. AMS, Providence (1999)Google Scholar