On the Hofer-Zehnder conjecture

Abstract

We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold, then it must have an infinite number of contractible periodic points. This constitutes a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994.

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      issn = {1073-7928},
      mrclass = {53D35 (53D40 53D45)},
      mrnumber = {1979584},
      mrreviewer = {Ignasi Mundet-Riera},
      doi = {10.1155/S1073792803210011},
      url = {https://doi.org/10.1155/S1073792803210011},
      zblnumber = {1047.53055},
      }
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    @INCOLLECTION{EntovPolterovich-semisimple,
      author = {Entov, Michael and Polterovich, Leonid},
      title = {Symplectic quasi-states and semi-simplicity of quantum homology},
      booktitle = {Toric Topology},
      series = {Contemp. Math.},
      volume = {460},
      pages = {47--70},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2008},
      mrclass = {53D45 (53D40)},
      mrnumber = {2428348},
      mrreviewer = {Martin Pinsonnault},
      doi = {10.1090/conm/460/09010},
      url = {https://doi.org/10.1090/conm/460/09010},
      zblnumber = {1146.53066},
      }
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    @ARTICLE{EntovPolterovich-rigid,
      author = {Entov, Michael and Polterovich, Leonid},
      title = {Rigid subsets of symplectic manifolds},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {145},
      year = {2009},
      number = {3},
      pages = {773--826},
      issn = {0010-437X},
      mrclass = {53D40 (53D12 53D35)},
      mrnumber = {2507748},
      mrreviewer = {Martin Pinsonnault},
      doi = {10.1112/S0010437X0900400X},
      url = {https://doi.org/10.1112/S0010437X0900400X},
      zblnumber = {1230.53080},
      }
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    @ARTICLE{Floer1,
      author = {Floer, Andreas},
      title = {Proof of the {A}rnol\cprime d conjecture for surfaces and generalizations to certain {K}ähler manifolds},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {53},
      year = {1986},
      number = {1},
      pages = {1--32},
      issn = {0012-7094},
      mrclass = {58F22 (53C55 53C57 58F05)},
      mrnumber = {0835793},
      mrreviewer = {A. Morimoto},
      doi = {10.1215/S0012-7094-86-05301-9},
      url = {https://doi.org/10.1215/S0012-7094-86-05301-9},
      zblnumber = {0607.58016},
      }
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    @ARTICLE{Floer2,
      author = {Floer, Andreas},
      title = {Morse theory for fixed points of symplectic diffeomorphisms},
      journal = {Bull. Amer. Math. Soc. (N.S.)},
      fjournal = {Amer. Math. Soc.. Bulletin. New Series},
      volume = {16},
      year = {1987},
      number = {2},
      pages = {279--281},
      issn = {0273-0979},
      mrclass = {58E05 (58C30 58F05)},
      mrnumber = {0876964},
      mrreviewer = {Frans Cantrijn},
      doi = {10.1090/S0273-0979-1987-15517-0},
      url = {https://doi.org/10.1090/S0273-0979-1987-15517-0},
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      }
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      author = {Floer, Andreas},
      title = {Symplectic fixed points and holomorphic spheres},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {120},
      year = {1989},
      number = {4},
      pages = {575--611},
      issn = {0010-3616},
      mrclass = {58F05 (70H05)},
      mrnumber = {0987770},
      mrreviewer = {Yong-Geun Oh},
      doi = {10.1007/BF01260388},
      url = {https://doi.org/10.1007/BF01260388},
      zblnumber = {0755.58022},
      }
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      title = {Witten's complex and infinite-dimensional {M}orse theory},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {30},
      year = {1989},
      number = {1},
      pages = {207--221},
      issn = {0022-040X},
      mrclass = {58E05 (58F05)},
      mrnumber = {1001276},
      mrreviewer = {I. Vaisman},
      doi = {10.4310/jdg/1214443291},
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      }
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      fjournal = {Transactions of the Amer. Math. Soc.},
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      issn = {0002-9947},
      mrclass = {56.0X},
      mrnumber = {0046039},
      mrreviewer = {P. A. Smith},
      doi = {10.2307/1990658},
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      title = {A symplectic fixed point theorem for {${\bf C}{\rm P}^n$}},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {81},
      year = {1985},
      number = {1},
      pages = {29--46},
      issn = {0020-9910},
      mrclass = {58F05 (49A40 58C30)},
      mrnumber = {0796189},
      mrreviewer = {Frans Cantrijn},
      doi = {10.1007/BF01388770},
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      }
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    @ARTICLE{FortuneWeinstein,
      author = {Fortune, Barry and Weinstein, Alan},
      title = {A symplectic fixed point theorem for complex projective spaces},
      journal = {Bull. Amer. Math. Soc. (N.S.)},
      fjournal = {Amer. Math. Soc.. Bulletin. New Series},
      volume = {12},
      year = {1985},
      number = {1},
      pages = {128--130},
      issn = {0273-0979},
      mrclass = {58F05 (58E99)},
      mrnumber = {0766969},
      mrreviewer = {Michèle Audin},
      doi = {10.1090/S0273-0979-1985-15314-5},
      url = {https://doi.org/10.1090/S0273-0979-1985-15314-5},
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      }
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      title = {Geodesics on {$S^2$} and periodic points of annulus homeomorphisms},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
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      year = {1992},
      number = {2},
      pages = {403--418},
      issn = {0020-9910},
      mrclass = {58F20 (58E10)},
      mrnumber = {1161099},
      mrreviewer = {Vladislav S. Medvedev},
      doi = {10.1007/BF02100612},
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      }
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      mrclass = {58F20},
      mrnumber = {1371312},
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      url = {http://nyjm.albany.edu:8000/j/1996/2_1.html},
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      author = {Franks, John and Handel, Michael},
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      journal = {Geom. Topol.},
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      mrclass = {37J10 (37C25 37E30 37E45 57M60)},
      mrnumber = {2026545},
      mrreviewer = {Eijirou Hayakawa},
      doi = {10.2140/gt.2003.7.713},
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      mrreviewer = {Cheol-Hyun Cho},
      doi = {10.1155/S1073792804133941},
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      title = {Displacement of polydisks and {L}agrangian {F}loer theory},
      journal = {J. Symplectic Geom.},
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      volume = {11},
      year = {2013},
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      pages = {231--268},
      issn = {1527-5256},
      mrclass = {53D40 (53D12 53D37)},
      mrnumber = {3046491},
      mrreviewer = {Jelena Katić},
      doi = {10.4310/JSG.2013.v11.n2.a4},
      url = {https://doi.org/10.4310/JSG.2013.v11.n2.a4},
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      author = {Fukaya, Kenji and Ono, Kaoru},
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      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {38},
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      number = {5},
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      issn = {0040-9383},
      mrclass = {53D45 (37J10 37J45 53D40 57R17 57R58)},
      mrnumber = {1688434},
      mrreviewer = {David E. Hurtubise},
      doi = {10.1016/S0040-9383(98)00042-1},
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      }
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    @INCOLLECTION{FukayaOno2,
      author = {Fukaya, Kenji and Ono, Kaoru},
      title = {Arnold conjecture and {G}romov-{W}itten invariant for general symplectic manifolds},
      booktitle = {The {A}rnoldfest},
      venue = {{T}oronto, {ON},
      1997},
      series = {Fields Inst. Commun.},
      volume = {24},
      pages = {173--190},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1999},
      mrclass = {53D40 (53D45 57R58)},
      mrnumber = {1733575},
      mrreviewer = {David E. Hurtubise},
      doi = {10.1016/s0040-9383(98)00042-1},
      url = {https://doi.org/10.1016/s0040-9383(98)00042-1},
      zblnumber = {1004.53063},
      }
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      journal = {Bull. Amer. Math. Soc. (N.S.)},
      fjournal = {Amer. Math. Soc.. Bulletin. New Series},
      volume = {45},
      year = {2008},
      number = {1},
      pages = {61--75},
      issn = {0273-0979},
      mrclass = {55N35 (62H35 94A08 94A12)},
      mrnumber = {2358377},
      doi = {10.1090/S0273-0979-07-01191-3},
      url = {https://doi.org/10.1090/S0273-0979-07-01191-3},
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      }
  • [Ginzburg-CC] Go to document V. L. Ginzburg, "The Conley conjecture," Ann. of Math. (2), vol. 172, iss. 2, pp. 1127-1180, 2010.
    @ARTICLE{Ginzburg-CC,
      author = {Ginzburg, Viktor L.},
      title = {The {C}onley conjecture},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {172},
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      number = {2},
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      issn = {0003-486X},
      mrclass = {53D40 (37J05 53D35)},
      mrnumber = {2680488},
      mrreviewer = {Hai-Long Her},
      doi = {10.4007/annals.2010.172.1129},
      url = {https://doi.org/10.4007/annals.2010.172.1129},
      zblnumber = {1228.53098},
      }
  • [GG-ai] Go to document V. L. Ginzburg and B. Z. Gürel, "Action and index spectra and periodic orbits in Hamiltonian dynamics," Geom. Topol., vol. 13, iss. 5, pp. 2745-2805, 2009.
    @ARTICLE{GG-ai,
      author = {Ginzburg, Viktor L. and Gürel, Ba\c{s}ak Z.},
      title = {Action and index spectra and periodic orbits in {H}amiltonian dynamics},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {13},
      year = {2009},
      number = {5},
      pages = {2745--2805},
      issn = {1465-3060},
      mrclass = {53D40 (37J10 37J45)},
      mrnumber = {2529945},
      mrreviewer = {Sonja Hohloch},
      doi = {10.2140/gt.2009.13.2745},
      url = {https://doi.org/10.2140/gt.2009.13.2745},
      zblnumber = {1172.53052},
      }
  • [GG-local-gap] Go to document V. L. Ginzburg and B. Z. Gürel, "Local Floer homology and the action gap," J. Symplectic Geom., vol. 8, iss. 3, pp. 323-357, 2010.
    @ARTICLE{GG-local-gap,
      author = {Ginzburg, Viktor L. and Gürel, Ba\c{s}ak Z.},
      title = {Local {F}loer homology and the action gap},
      journal = {J. Symplectic Geom.},
      fjournal = {The Journal of Symplectic Geometry},
      volume = {8},
      year = {2010},
      number = {3},
      pages = {323--357},
      issn = {1527-5256},
      mrclass = {53D40},
      mrnumber = {2684510},
      mrreviewer = {Vincent Humilière},
      doi = {10.4310/JSG.2010.v8.n3.a4},
      url = {https://doi.org/10.4310/JSG.2010.v8.n3.a4},
      zblnumber = {1206.53087},
      }
  • [GG-negmon] Go to document V. L. Ginzburg and B. Z. Gürel, "Conley conjecture for negative monotone symplectic manifolds," Int. Math. Res. Not. IMRN, iss. 8, pp. 1748-1767, 2012.
    @ARTICLE{GG-negmon,
      author = {Ginzburg, Viktor L. and Gürel, Ba\c{s}ak Z.},
      title = {Conley conjecture for negative monotone symplectic manifolds},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2012},
      number = {8},
      pages = {1748--1767},
      issn = {1073-7928},
      mrclass = {53D40 (53D35)},
      mrnumber = {2920829},
      mrreviewer = {Vincent Humilière},
      doi = {10.1093/imrn/rnr081},
      url = {https://doi.org/10.1093/imrn/rnr081},
      zblnumber = {1242.53100},
      }
  • [GG-hyperbolic] Go to document V. L. Ginzburg and B. Z. Gürel, "Hyperbolic fixed points and periodic orbits of Hamiltonian diffeomorphisms," Duke Math. J., vol. 163, iss. 3, pp. 565-590, 2014.
    @ARTICLE{GG-hyperbolic,
      author = {Ginzburg, Viktor L. and Gürel, Ba\c{s}ak Z.},
      title = {Hyperbolic fixed points and periodic orbits of {H}amiltonian diffeomorphisms},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {163},
      year = {2014},
      number = {3},
      pages = {565--590},
      issn = {0012-7094},
      mrclass = {53D40 (37J10 70H12)},
      mrnumber = {3165423},
      mrreviewer = {Cristian L\u{a}zureanu},
      doi = {10.1215/00127094-2410433},
      url = {https://doi.org/10.1215/00127094-2410433},
      zblnumber = {1408.53112},
      }
  • [GG-nc] Go to document V. L. Ginzburg and B. Z. Gürel, "Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds," Compos. Math., vol. 152, iss. 9, pp. 1777-1799, 2016.
    @ARTICLE{GG-nc,
      author = {Ginzburg, Viktor L. and Gürel, Ba\c{s}ak Z.},
      title = {Non-contractible periodic orbits in {H}amiltonian dynamics on closed symplectic manifolds},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {152},
      year = {2016},
      number = {9},
      pages = {1777--1799},
      issn = {0010-437X},
      mrclass = {53D40 (37J10)},
      mrnumber = {3568939},
      mrreviewer = {Gabriele Benedetti},
      doi = {10.1112/S0010437X16007508},
      url = {https://doi.org/10.1112/S0010437X16007508},
      zblnumber = {1375.53106},
      }
  • [GG-pseudorotations] Go to document V. L. Ginzburg and B. Z. Gürel, "Hamiltonian pseudo-rotations of projective spaces," Invent. Math., vol. 214, iss. 3, pp. 1081-1130, 2018.
    @ARTICLE{GG-pseudorotations,
      author = {Ginzburg, Viktor L. and Gürel, Ba\c{s}ak Z.},
      title = {Hamiltonian pseudo-rotations of projective spaces},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {214},
      year = {2018},
      number = {3},
      pages = {1081--1130},
      issn = {0020-9910},
      mrclass = {53D40 (37J10 37J45)},
      mrnumber = {3878727},
      doi = {10.1007/s00222-018-0818-9},
      url = {https://doi.org/10.1007/s00222-018-0818-9},
      zblnumber = {1447.53073},
      }
  • [GG-revisited] Go to document V. L. Ginzburg and B. Z. Gürel, "Conley conjecture revisited," Int. Math. Res. Not. IMRN, iss. 3, pp. 761-798, 2019.
    @ARTICLE{GG-revisited,
      author = {Ginzburg, Viktor L. and Gürel, Ba\c{s}ak Z.},
      title = {Conley conjecture revisited},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2019},
      number = {3},
      pages = {761--798},
      issn = {1073-7928},
      mrclass = {53D05 (37J45 58E05)},
      mrnumber = {3910472},
      mrreviewer = {Naiara Vergian de Paulo},
      doi = {10.1093/imrn/rnx137},
      url = {https://doi.org/10.1093/imrn/rnx137},
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      }
  • [GromovPseudohol] Go to document M. Gromov, "Pseudo holomorphic curves in symplectic manifolds," Invent. Math., vol. 82, iss. 2, pp. 307-347, 1985.
    @ARTICLE{GromovPseudohol,
      author = {Gromov, M.},
      title = {Pseudo holomorphic curves in symplectic manifolds},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
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Authors

Egor Shelukhin

Université de Montréal, Montréal, Canada