Abstract
We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period. This theorem generalizes, for instance, a recent result of Hingston establishing the Conley conjecture for tori.
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TITLE = {Multiple periodic points of the {P}oincaré map of {L}agrangian systems on tori},
JOURNAL = {Math. Z.},
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VOLUME = {233},
YEAR = {2000},
NUMBER = {3},
PAGES = {443--470},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {37J45 (58E05 70H12)},
MRNUMBER = {2001a:37093},
MRREVIEWER = {Maria Letizia Bertotti},
DOI = {10.1007/s002090050481},
ZBLNUMBER = {0984.37074},
} -
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G. Lu, "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems," J. Funct. Anal., vol. 256, iss. 9, pp. 2967-3034, 2009.
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JOURNAL = {J. Funct. Anal.},
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VOLUME = {256},
YEAR = {2009},
NUMBER = {9},
PAGES = {2967--3034},
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[Ma] M. Mazzucchelli, The Lagrangian Conley conjecture, 2008.
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author={Mazzucchelli, M.},
TITLE={The Lagrangian Conley conjecture},
NOTE={preprint, to appear in {\it Comment. Math. Helv.}},
YEAR={2008},
ARXIV={0810.2108},
} -
[mdsa-intro] D. McDuff and D. Salamon, Introduction to Symplectic Topology, New York: The Clarendon Press Oxford University Press, 1995.
@book {mdsa-intro, MRKEY = {1373431},
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[mdsa-book] D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, Providence, RI: Amer. Math. Soc., 2004, vol. 52.
@book {mdsa-book, MRKEY = {2045629},
AUTHOR = {McDuff, Dusa and Salamon, Dietmar},
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@incollection {Poz, MRKEY = {1736217},
AUTHOR = {Po{\'z}niak, Marcin},
TITLE = {Floer homology, {N}ovikov rings and clean intersections},
BOOKTITLE = {Northern {C}alifornia {S}ymplectic {G}eometry {S}eminar},
SERIES = {Amer. Math. Soc. Transl. Ser. 2},
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YEAR = {1999},
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ZBLNUMBER = {0948.57025},
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[Sa:london]
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MRNUMBER = {92a:58028},
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@incollection{Sa, MRKEY = {1702940},
AUTHOR={Salamon, Dietmar},
TITLE = {Lectures on Floer homology},
BOOKTITLE={Symplectic Geometry and Topology},
SERIES = {IAS/Park City Math. Series},
VOLUME = {7},
PAGES={145--229},
NOTE = {Lectures from the Graduate Summer School held in Park City, UT, June 29--July 19, 1997 , (Eliashberg, Y. and Traynor, L., eds.)},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
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} -
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YEAR = {1993},
PAGES = {x+235},
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MRCLASS = {58E05 (55N35 57R70)},
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[We77] A. Weinstein, Lectures on Symplectic Manifolds, Providence, R.I.: Amer. Math. Soc., 1977.
@book {We77, MRKEY = {0464312},
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NOTE = {Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8--12, 1976, Regional Conference Series in Mathematics, No. 29},
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MRNUMBER = {1507138},
DOI = {10.2307/2371062},
ZBLNUMBER = {0013.28401},
}