Abstract
In this paper, we treat an open problem related to the number of periodic orbits of Hamiltonian diffeomorphisms on closed symplectic manifolds, the so-called (generic) Conley conjecture. The generic Conley conjecture states that generically Hamiltonian diffeomorphisms have infinitely many simple contractible periodic orbits. We prove the generic Conley conjecture for very wide classes of symplectic manifolds. Our proof is based on applications of the Birkhoff–Moser fixed point theorem and Floer homology theory.
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References
Asaoka, M., Irie, K.: A \(C^{\infty }\) closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geom. Funct. Anal 26, 1245–1254 (2016)
Audin, M., Damian, M.: Morse Theory and Floer Homology. Springer, Berlin (2014)
Conley, C.: Lecture at the University of Wisconsin. April 6 (1984)
Fukaya, K., Ono, K.: Arnold conjecture and Gromov–Witten invariant. Topology 38(5), 933–1048 (1999)
Ginzburg, V.L.: The Conley conjecture. Ann. Math. 172, 1127–1180 (2010)
Ginzburg, V.L., Gürel, B.Z.: On the generic existence of periodic orbits in Hamiltonian dynamics. J. Mod. Dyn. 3, 595–610 (2009)
Ginzburg, V.L., Gürel, B.Z.: Conley conjecture for negative monotone symplectic manifolds. Int. Math. Res. Not. 2012(8), 1748–1767 (2012)
Ginzburg, V.L., Gürel, B.Z.: The Conley conjecture and beyond. Arnold Math. J. 1, 299–337 (2015)
Ginzburg, V.L., Gürel, B.Z.: Conley conjecture revisited. Int. Math. Res. Not. 2019(3), 761–798 (2019)
Ginzburg, V.L., Kerman, E.: Homological resonances for Hamiltonian diffeomorphisms and Reeb flows. Int. Math. Res. Not. 2010(1), 53–68 (2010)
Hein, D.: The Conley conjecture for irrational symplectic manifolds. J. Symplectic Geom. 10(2), 183–202 (2012)
Herman, M.R.: Exemples de flots hamiltoniens dont aucune perturbation en topologie \(C^{\infty }\) n’a d’orbites périodiques sur ouvert de surfaces d’énergies. C. R. Acad. Sci. Paris Sér. I Math. 312(13), 989–994 (1991)
Liu, G., Tian, G.: Floer homology and Arnold conjecture. J. Differ. Geom. 49, 1–74 (1998)
Moser, J.: Proof of a generalized form of a fixed point theorem due to G.D. Birkhoff. Geom. Topol. 597, 464–494 (1977)
Pugh, C.C., Robinson, C.: The \(C^1\) closing lemma, including Hamiltonians. Ergod. Theory Dyn. Syst. 3, 261–313 (1983)
Salamon, D., Zehnder, E.: Morse thoery for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure Appl. Math. 45, 1303–1360 (1992)
Acknowledgements
This work was carried out during my stay as a research fellow at the National Center for Theoretical Sciences. The author thanks NCTS for a great research atmosphere and a lot of support. He also gratefully acknowledges his teacher Kaoru Ono for continuous supports and Victor L. Ginzburg for checking the draft and giving me comments and advice. He is supported by the Research Fellowships of Japan Society for the Promotion of Science for Young Scientists.
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Sugimoto, Y. On the generic Conley conjecture. Arch. Math. 117, 423–432 (2021). https://doi.org/10.1007/s00013-021-01633-w
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DOI: https://doi.org/10.1007/s00013-021-01633-w