On the Hofer-Zehnder conjecture

Egor Shelukhin

doi:https://doi.org/10.4007/annals.2022.195.3.1

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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@ARTICLE{Floer-MorseWitten, author = {Floer, Andreas}, 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}, url = {https://doi.org/10.4310/jdg/1214443291}, zblnumber = {0678.58012}, }
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@ARTICLE{Floyd-original, author = {Floyd, E. E.}, title = {On periodic maps and the {E}uler characteristics of associated spaces}, journal = {Trans. Amer. Math. Soc.}, fjournal = {Transactions of the Amer. Math. Soc.}, volume = {72}, year = {1952}, pages = {138--147}, issn = {0002-9947}, mrclass = {56.0X}, mrnumber = {0046039}, mrreviewer = {P. A. Smith}, doi = {10.2307/1990658}, url = {https://doi.org/10.2307/1990658}, zblnumber = {0046.16603}, }
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@ARTICLE{Fortune, author = {Fortune, Barry}, 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}, url = {https://doi.org/10.1007/BF01388770}, zblnumber = {0547.58015}, }
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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}, zblnumber = {0566.58013}, }
[Franks-sphere] $Go to document$ J. Franks, "Geodesics on $S^2$ and periodic points of annulus homeomorphisms," Invent. Math., vol. 108, iss. 2, pp. 403-418, 1992.

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@ARTICLE{Franks-sphere, author = {Franks, John}, title = {Geodesics on {$S^2$} and periodic points of annulus homeomorphisms}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {108}, year = {1992}, number = {2}, pages = {403--418}, issn = {0020-9910}, mrclass = {58F20 (58E10)}, mrnumber = {1161099}, mrreviewer = {Vladislav S. Medvedev}, doi = {10.1007/BF02100612}, url = {https://doi.org/10.1007/BF02100612}, zblnumber = {0766.53037}, }
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@ARTICLE{Franks-NY, author = {Franks, John}, title = {Area preserving homeomorphisms of open surfaces of genus zero}, journal = {New York J. Math.}, fjournal = {New York Journal of Mathematics}, volume = {2}, year = {1996}, pages = {1--19}, mrclass = {58F20}, mrnumber = {1371312}, mrreviewer = {James F. Reineck}, url = {http://nyjm.albany.edu:8000/j/1996/2_1.html}, zblnumber = {0891.58033}, }
[FranksHandel-growthrate] $Go to document$ J. Franks and M. Handel, "Periodic points of Hamiltonian surface diffeomorphisms," Geom. Topol., vol. 7, pp. 713-756, 2003.

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@ARTICLE{FranksHandel-growthrate, author = {Franks, John and Handel, Michael}, title = {Periodic points of {H}amiltonian surface diffeomorphisms}, journal = {Geom. Topol.}, fjournal = {Geometry and Topology}, volume = {7}, year = {2003}, pages = {713--756}, issn = {1465-3060}, mrclass = {37J10 (37C25 37E30 37E45 57M60)}, mrnumber = {2026545}, mrreviewer = {Eijirou Hayakawa}, doi = {10.2140/gt.2003.7.713}, url = {https://doi.org/10.2140/gt.2003.7.713}, zblnumber = {1034.37028}, }
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@ARTICLE{Frauenfelder, author = {Frauenfelder, Urs}, title = {The {A}rnold-{G}ivental conjecture and moment {F}loer homology}, journal = {Int. Math. Res. Not.}, fjournal = {International Mathematics Research Notices}, year = {2004}, number = {42}, pages = {2179--2269}, issn = {1073-7928}, mrclass = {53D40 (37J05)}, mrnumber = {2076142}, mrreviewer = {Cheol-Hyun Cho}, doi = {10.1155/S1073792804133941}, url = {https://doi.org/10.1155/S1073792804133941}, zblnumber = {1088.53058}, }
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@ARTICLE{FOOO-polydiscs, author = {Fukaya, Kenji and Oh, Yong-Geun and Ohta, Hiroshi and Ono, Kaoru}, title = {Displacement of polydisks and {L}agrangian {F}loer theory}, journal = {J. Symplectic Geom.}, fjournal = {The Journal of Symplectic Geometry}, volume = {11}, year = {2013}, number = {2}, 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}, zblnumber = {1292.53056}, }
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@ARTICLE{FukayaOno1, author = {Fukaya, Kenji and Ono, Kaoru}, title = {Arnold conjecture and {G}romov-{W}itten invariant}, journal = {Topology}, fjournal = {Topology. An International Journal of Mathematics}, volume = {38}, year = {1999}, number = {5}, pages = {933--1048}, issn = {0040-9383}, mrclass = {53D45 (37J10 37J45 53D40 57R17 57R58)}, mrnumber = {1688434}, mrreviewer = {David E. Hurtubise}, doi = {10.1016/S0040-9383(98)00042-1}, url = {https://doi.org/10.1016/S0040-9383(98)00042-1}, zblnumber = {0946.53047}, }
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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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@ARTICLE{Ghrist, author = {Ghrist, Robert}, title = {Barcodes: the persistent topology of data}, 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}, zblnumber = {1391.55005}, }
[Ginzburg-CC] $Go to document$ V. L. Ginzburg, "The Conley conjecture," Ann. of Math. (2), vol. 172, iss. 2, pp. 1127-1180, 2010.

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@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}, year = {2010}, number = {2}, pages = {1127--1180}, 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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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}, zblnumber = {1428.53085}, }
[GromovPseudohol] $Go to document$ M. Gromov, "Pseudo holomorphic curves in symplectic manifolds," Invent. Math., vol. 82, iss. 2, pp. 307-347, 1985.

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@ARTICLE{GromovPseudohol, author = {Gromov, M.}, title = {Pseudo holomorphic curves in symplectic manifolds}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {82}, year = {1985}, number = {2}, pages = {307--347}, issn = {0020-9910}, mrclass = {53C15 (32F25 53C57 57R15)}, mrnumber = {0809718}, mrreviewer = {Yakov Eliashberg}, doi = {10.1007/BF01388806}, url = {https://doi.org/10.1007/BF01388806}, zblnumber = {0592.53025}, }
[Gurel-nc] $Go to document$ B. Z. Gürel, "On non-contractible periodic orbits of Hamiltonian diffeomorphisms," Bull. Lond. Math. Soc., vol. 45, iss. 6, pp. 1227-1234, 2013.

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@ARTICLE{Gurel-nc, author = {Gürel, Ba\c{s}ak Z.}, title = {On non-contractible periodic orbits of {H}amiltonian diffeomorphisms}, journal = {Bull. Lond. Math. Soc.}, fjournal = {Bulletin of the London Mathematical Society}, volume = {45}, year = {2013}, number = {6}, pages = {1227--1234}, issn = {0024-6093}, mrclass = {37J10 (53D40)}, mrnumber = {3138490}, mrreviewer = {Frédéric Bourgeois}, doi = {10.1112/blms/bdt051}, url = {https://doi.org/10.1112/blms/bdt051}, zblnumber = {1283.37061}, }
[Hein-CC] $Go to document$ D. Hein, "The Conley conjecture for irrational symplectic manifolds," J. Symplectic Geom., vol. 10, iss. 2, pp. 183-202, 2012.

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@ARTICLE{Hein-CC, author = {Hein, Doris}, title = {The {C}onley conjecture for irrational symplectic manifolds}, journal = {J. Symplectic Geom.}, fjournal = {The Journal of Symplectic Geometry}, volume = {10}, year = {2012}, number = {2}, pages = {183--202}, issn = {1527-5256}, mrclass = {53D40 (37C27)}, mrnumber = {2926994}, mrreviewer = {Sonja Hohloch}, doi = {10.4310/JSG.2012.v10.n2.a2}, url = {https://doi.org/10.4310/JSG.2012.v10.n2.a2}, zblnumber = {1275.37026}, }
[Hingston-torus] $Go to document$ N. Hingston, "Subharmonic solutions of Hamiltonian equations on tori," Ann. of Math. (2), vol. 170, iss. 2, pp. 529-560, 2009.

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@ARTICLE{Hingston-torus, author = {Hingston, Nancy}, title = {Subharmonic solutions of {H}amiltonian equations on tori}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {170}, year = {2009}, number = {2}, pages = {529--560}, issn = {0003-486X}, mrclass = {53D35 (37J45)}, mrnumber = {2552101}, mrreviewer = {Tobias Ekholm}, doi = {10.4007/annals.2009.170.529}, url = {https://doi.org/10.4007/annals.2009.170.529}, zblnumber = {1180.58009}, }
[HoferMetric] $Go to document$ H. Hofer, "On the topological properties of symplectic maps," Proc. Roy. Soc. Edinburgh Sect. A, vol. 115, iss. 1-2, pp. 25-38, 1990.

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@ARTICLE{HoferMetric, author = {Hofer, H.}, title = {On the topological properties of symplectic maps}, journal = {Proc. Roy. Soc. Edinburgh Sect. A}, fjournal = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics}, volume = {115}, year = {1990}, number = {1-2}, pages = {25--38}, issn = {0308-2105}, mrclass = {58F05 (70H05)}, mrnumber = {1059642}, mrreviewer = {Yong-Geun Oh}, doi = {10.1017/S0308210500024549}, url = {https://doi.org/10.1017/S0308210500024549}, zblnumber = {0713.58004}, }
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@BOOK{HZ-book, author = {Hofer, Helmut and Zehnder, Eduard}, title = {Symplectic Invariants and {H}amiltonian Dynamics}, series = {Birkhäuser Advanced Texts: {B}asler Lehrbüscher}, publisher = {Birkhäuser Verlag, Basel}, year = {1994}, mrclass = {58-02 (34C25 57R15 58E05 58F05 70H05)}, mrnumber = {1306732}, mrreviewer = {Daniel M. Burns, Jr.}, doi = {10.1007/978-3-0348-8540-9}, url = {https://doi.org/10.1007/978-3-0348-8540-9}, zblnumber = {0805.58003}, }
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@BOOK{Hsiang-Transformation, author = {Hsiang, Wu Yi}, title = {Cohomology Theory of Topological Transformation Groups}, series = {Ergeb. Math. Grenzgeb.}, volume = {85}, publisher = {Springer-Verlag, New York}, year = {1975}, pages = {x+164}, mrclass = {57EXX}, mrnumber = {0423384}, mrreviewer = {Theodore Chang}, doi = {10.1007/978-3-642-66052-8}, url = {https://doi.org/10.1007/978-3-642-66052-8}, zblnumber = {0429.57011}, }
[Jacobson] N. Jacobson, Basic Algebra. II, Second ed., W. H. Freeman and Company, New York, 1989.

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@BOOK{Jacobson, author = {Jacobson, Nathan}, title = {Basic Algebra. {II}}, edition = {Second}, publisher = {W. H. Freeman and Company, New York}, year = {1989}, pages = {xviii+686}, isbn = {0-7167-1933-9}, mrclass = {00A05 (12-01 15-01 16-01)}, mrnumber = {1009787}, mrreviewer = {P. M. Cohn}, zblnumber = {0694.16001}, }
[Kaledin] D. Kaledin, "Cartier isomorphism and Hodge theory in the non-commutative case," in Arithmetic Geometry, Amer. Math. Soc., Providence, RI, 2009, vol. 8, pp. 537-562.

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@INCOLLECTION{Kaledin, author = {Kaledin, Dmitry}, title = {Cartier isomorphism and {H}odge theory in the non-commutative case}, booktitle = {Arithmetic Geometry}, series = {Clay Math. Proc.}, volume = {8}, pages = {537--562}, publisher = {Amer. Math. Soc., Providence, RI}, year = {2009}, mrclass = {14F40 (14A22 18G60 19D55)}, mrnumber = {2498070}, mrreviewer = {Oliver Röndigs}, zblnumber = {1205.19004}, }
[KS-bounds] $Go to document$ A. Kislev and E. Shelukhin, "Bounds on spectral norms and barcodes," Geom. Topol., vol. 25, iss. 7, pp. 3257-3350, 2021.

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@ARTICLE{KS-bounds, author = {Kislev, Asaf and Shelukhin, Egor}, title = {Bounds on spectral norms and barcodes}, journal = {Geom. Topol.}, fjournal = {Geometry \& Topology}, volume = {25}, year = {2021}, number = {7}, pages = {3257--3350}, issn = {1465-3060}, mrclass = {57R17 (53D12 53D40)}, mrnumber = {4372632}, doi = {10.2140/gt.2021.25.3257}, url = {https://doi.org/10.2140/gt.2021.25.3257}, zblnumber = {7483970}, }
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@ARTICLE{Lalonde-McDuff-Energy, author = {Lalonde, François and McDuff, Dusa}, title = {The geometry of symplectic energy}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {141}, year = {1995}, number = {2}, pages = {349--371}, issn = {0003-486X}, mrclass = {58F05 (53C23)}, mrnumber = {1324138}, mrreviewer = {Hansjörg Geiges}, doi = {10.2307/2118524}, url = {https://doi.org/10.2307/2118524}, zblnumber = {0829.53025}, }
[LeCalvez-growthrate1] $Go to document$ P. Le Calvez, "Periodic orbits of Hamiltonian homeomorphisms of surfaces," Duke Math. J., vol. 133, iss. 1, pp. 125-184, 2006.

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@ARTICLE{LeCalvez-growthrate1, author = {Le Calvez, Patrice}, title = {Periodic orbits of {H}amiltonian homeomorphisms of surfaces}, journal = {Duke Math. J.}, fjournal = {Duke Mathematical Journal}, volume = {133}, year = {2006}, number = {1}, pages = {125--184}, issn = {0012-7094}, mrclass = {37E30 (37C25 37J45)}, mrnumber = {2219272}, mrreviewer = {Shigenori Matsumoto}, doi = {10.1215/S0012-7094-06-13315-X}, url = {https://doi.org/10.1215/S0012-7094-06-13315-X}, zblnumber = {1101.37031}, }
[LeCalvez-growthrate2] $Go to document$ P. Le Calvez, "Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?," Ann. Sci. Éc. Norm. Supér. (4), vol. 41, iss. 1, pp. 141-176, 2008.

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@ARTICLE{LeCalvez-growthrate2, author = {Le Calvez, Patrice}, title = {Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?}, journal = {Ann. Sci. \'{E}c. Norm. Supér. (4)}, fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série}, volume = {41}, year = {2008}, number = {1}, pages = {141--176}, issn = {0012-9593}, mrclass = {37E30 (37B99 37C25)}, mrnumber = {2423312}, mrreviewer = {Shigenori Matsumoto}, doi = {10.24033/asens.2065}, url = {https://doi.org/10.24033/asens.2065}, zblnumber = {1168.37010}, }
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@ARTICLE{LSV-conj, author = {Le Roux, Frédéric and Seyfaddini, Sobhan and Viterbo, Claude}, title = {Barcodes and area-preserving homeomorphisms}, journal = {Geom. Topol.}, fjournal = {Geometry \& Topology}, volume = {25}, year = {2021}, number = {6}, pages = {2713--2825}, issn = {1465-3060}, mrclass = {37E30 (53D05 53D40)}, mrnumber = {4347306}, doi = {10.2140/gt.2021.25.2713}, url = {https://doi.org/10.2140/gt.2021.25.2713}, zblnumber = {07442193}, }
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@ARTICLE{LeclercqZapolsky, author = {Leclercq, Rémi and Zapolsky, Frol}, title = {Spectral invariants for monotone {L}agrangians}, journal = {J. Topol. Anal.}, fjournal = {Journal of Topology and Analysis}, volume = {10}, year = {2018}, number = {3}, pages = {627--700}, issn = {1793-5253}, mrclass = {57R17 (53D12 53D40)}, mrnumber = {3850107}, mrreviewer = {Roman Golovko}, doi = {10.1142/S1793525318500267}, url = {https://doi.org/10.1142/S1793525318500267}, zblnumber = {1398.57037}, }
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@ARTICLE{Liu-assoc, author = {Liu, Gang}, title = {Associativity of quantum multiplication}, journal = {Comm. Math. Phys.}, fjournal = {Communications in Mathematical Physics}, volume = {191}, year = {1998}, number = {2}, pages = {265--282}, issn = {0010-3616}, mrclass = {58D10 (53C15)}, mrnumber = {1604364}, mrreviewer = {Bernd Siebert}, doi = {10.1007/s002200050268}, url = {https://doi.org/10.1007/s002200050268}, zblnumber = {0891.53022}, }
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@ARTICLE{LiuTian, author = {Liu, Gang and Tian, Gang}, title = {Floer homology and {A}rnold conjecture}, journal = {J. Differential Geom.}, fjournal = {Journal of Differential Geometry}, volume = {49}, year = {1998}, number = {1}, pages = {1--74}, issn = {0022-040X}, mrclass = {58E05 (55N35 57R15 57R70 58F05)}, mrnumber = {1642105}, mrreviewer = {Jean-Claude Sikorav}, doi = {10.4310/jdg/1214460936}, url = {https://doi.org/10.4310/jdg/1214460936}, zblnumber = {0917.58009}, }
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@ARTICLE{Markl-ideal, author = {Markl, Martin}, title = {Ideal perturbation lemma}, journal = {Comm. Algebra}, fjournal = {Communications in Algebra}, volume = {29}, year = {2001}, number = {11}, pages = {5209--5232}, issn = {0092-7872}, mrclass = {16E05 (16E45 18D50)}, mrnumber = {1856940}, mrreviewer = {Michel Van den Bergh}, doi = {10.1081/AGB-100106814}, url = {https://doi.org/10.1081/AGB-100106814}, zblnumber = {0994.55012}, }
[McDuffSalamon-BIG] D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, Second ed., Amer. Math. Soc., Providence, RI, 2012, vol. 52.

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@BOOK{McDuffSalamon-BIG, author = {McDuff, Dusa and Salamon, Dietmar}, title = {{$J$}-Holomorphic Curves and Symplectic Topology}, series = {Amer. Math. Soc. Colloq. Publ.}, volume = {52}, edition = {Second}, publisher = {Amer. Math. Soc., Providence, RI}, year = {2012}, pages = {xiv+726}, isbn = {978-0-8218-8746-2}, mrclass = {53D45 (32Q65 53D35)}, mrnumber = {2954391}, mrreviewer = {Mark Alan Branson}, zblnumber = {1272.53002}, }
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@ARTICLE{McLean-geodesics, author = {McLean, Mark}, title = {Local {F}loer homology and infinitely many simple {R}eeb orbits}, journal = {Algebr. Geom. Topol.}, fjournal = {Algebraic \& Geometric Topology}, volume = {12}, year = {2012}, number = {4}, pages = {1901--1923}, issn = {1472-2747}, mrclass = {53D25 (53C22)}, mrnumber = {2994824}, mrreviewer = {Umberto Leone Hryniewicz}, doi = {10.2140/agt.2012.12.1901}, url = {https://doi.org/10.2140/agt.2012.12.1901}, zblnumber = {1253.53078}, }
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@ARTICLE{Oh-specnorm, author = {Oh, Yong-Geun}, title = {Spectral invariants, analysis of the {F}loer moduli space, and geometry of the {H}amiltonian diffeomorphism group}, journal = {Duke Math. J.}, fjournal = {Duke Mathematical Journal}, volume = {130}, year = {2005}, number = {2}, pages = {199--295}, issn = {0012-7094}, mrclass = {53D40 (37J05 53D35)}, mrnumber = {2181090}, mrreviewer = {Michael J. Usher}, doi = {10.1215/00127094-8229689}, url = {https://doi.org/10.1215/00127094-8229689}, zblnumber = {1113.53054}, }
[OhBook] Y. Oh, Symplectic Topology and Floer Homology. Vol. 1, Cambridge Univ. Press, Cambridge, 2015, vol. 28.

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@BOOK{OhBook, author = {Oh, Yong-Geun}, title = {Symplectic {T}opology and {F}loer {H}omology. {V}ol. 1}, series = {New Math. Monogr.}, volume = {28}, titlenote = {Symplectic Geometry and Pseudoholomorphic Curves}, publisher = {Cambridge Univ. Press, Cambridge}, year = {2015}, pages = {xxiii+395}, isbn = {978-1-107-07245-9}, mrclass = {53Dxx (32Q65 37J05)}, mrnumber = {3443239}, mrreviewer = {Hansjörg Geiges}, zblnumber = {1338.53004}, }
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@ARTICLE{Orita1, author = {Orita, Ryuma}, title = {Non-contractible periodic orbits in {H}amiltonian dynamics on tori}, journal = {Bull. Lond. Math. Soc.}, fjournal = {Bulletin of the London Mathematical Society}, volume = {49}, year = {2017}, number = {4}, pages = {571--580}, issn = {0024-6093}, mrclass = {53D40 (37J10)}, mrnumber = {3725480}, mrreviewer = {Sonja Hohloch}, doi = {10.1112/blms.12054}, url = {https://doi.org/10.1112/blms.12054}, zblnumber = {1372.53089}, }
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@ARTICLE{Orita2, author = {Orita, Ryuma}, title = {On the existence of infinitely many non-contractible periodic orbits of {H}amiltonian diffeomorphisms of closed symplectic manifolds}, journal = {J. Symplectic Geom.}, fjournal = {The Journal of Symplectic Geometry}, volume = {17}, year = {2019}, number = {6}, pages = {1893--1927}, issn = {1527-5256}, mrclass = {53D40 (37J46 53E50)}, mrnumber = {4057730}, mrreviewer = {Manuel de León}, doi = {10.4310/JSG.2019.v17.n6.a9}, url = {https://doi.org/10.4310/JSG.2019.v17.n6.a9}, zblnumber = {1434.53084}, }
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@ARTICLE{OstroverAGT, author = {Ostrover, Yaron}, title = {Calabi quasi-morphisms for some non-monotone symplectic manifolds}, journal = {Algebr. Geom. Topol.}, fjournal = {Algebraic \& Geometric Topology}, volume = {6}, year = {2006}, pages = {405--434}, issn = {1472-2747}, mrclass = {53D35 (53D40 53D45)}, mrnumber = {2220683}, mrreviewer = {Michael J. Usher}, doi = {10.2140/agt.2006.6.405}, url = {https://doi.org/10.2140/agt.2006.6.405}, zblnumber = {1114.53070}, }
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@ARTICLE{PardonHam, author = {Pardon, John}, title = {An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves}, journal = {Geom. Topol.}, fjournal = {Geometry \& Topology}, volume = {20}, year = {2016}, number = {2}, pages = {779--1034}, issn = {1465-3060}, mrclass = {53D35 (37J10 53D37 53D40 53D42 53D45 54B40 57R17)}, mrnumber = {3493097}, mrreviewer = {Sonja Hohloch}, doi = {10.2140/gt.2016.20.779}, url = {https://doi.org/10.2140/gt.2016.20.779}, zblnumber = {1342.53109}, }
[PSS] S. Piunikhin, D. Salamon, and M. Schwarz, "Symplectic Floer-Donaldson theory and quantum cohomology," in Contact and Symplectic Geometry, Cambridge Univ. Press, Cambridge, 1996, vol. 8, pp. 171-200.

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@INCOLLECTION{PSS, author = {Piunikhin, S. and Salamon, D. and Schwarz, M.}, title = {Symplectic {F}loer-{D}onaldson theory and quantum cohomology}, booktitle = {Contact and Symplectic Geometry}, venue = {{C}ambridge, 1994}, series = {Publ. Newton Inst.}, volume = {8}, pages = {171--200}, publisher = {Cambridge Univ. Press, Cambridge}, year = {1996}, mrclass = {57R57 (58E05 58F05)}, mrnumber = {1432464}, mrreviewer = {Vsevolod V. Shevchishin}, zblnumber = {0874.53031}, }
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@ARTICLE{Poincare1912, author = {Poincar{é}, H.}, title = {Sur un th{é}or{è}me de g{é}om{é}trie}, journal = {Rendiconti del Circolo Matematico di Palermo (1884-1940)}, year = {1912}, volume = {33}, number = {1}, pages = {375--407}, zblnumber = {}, doi = {10.1007/BF03015314}, url = {https://doi.org/10.1007/BF03015314}, jfmnumber = {43.0757.03}, }
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@BOOK{PolterovichRosen, author = {Polterovich, Leonid and Rosen, Daniel}, title = {Function Theory on Symplectic Manifolds}, series = {CRM Monogr. Ser.}, volume = {34}, publisher = {Amer. Math. Soc., Providence, RI}, year = {2014}, pages = {xii+203}, isbn = {978-1-4704-1693-5}, mrclass = {53Dxx (22E65 57R17 57R58 58E05 81P45)}, mrnumber = {3241729}, mrreviewer = {Charles-Michel Marle}, doi = {10.1090/crmm/034}, url = {https://doi.org/10.1090/crmm/034}, zblnumber = {1310.53002}, }
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@ARTICLE{PolShe, author = {Polterovich, Leonid and Shelukhin, Egor}, title = {Autonomous {H}amiltonian flows, {H}ofer's geometry and persistence modules}, journal = {Selecta Math. (N.S.)}, fjournal = {Selecta Mathematica. New Series}, volume = {22}, year = {2016}, number = {1}, pages = {227--296}, issn = {1022-1824}, mrclass = {53D05 (37K05)}, mrnumber = {3437837}, mrreviewer = {Karl Friedrich Siburg}, doi = {10.1007/s00029-015-0201-2}, url = {https://doi.org/10.1007/s00029-015-0201-2}, zblnumber = {1337.53094}, }
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@ARTICLE{PolSheSto, author = {Polterovich, Leonid and Shelukhin, Egor and Stojisavljević, Vukašin}, title = {Persistence modules with operators in {M}orse and {F}loer theory}, journal = {Mosc. Math. J.}, fjournal = {Moscow Mathematical Journal}, volume = {17}, year = {2017}, number = {4}, pages = {757--786}, issn = {1609-3321}, mrclass = {53D40 (58E05)}, mrnumber = {3734662}, mrreviewer = {Jelena Katić}, doi = {10.17323/1609-4514-2016-16-4-757-786}, url = {https://doi.org/10.17323/1609-4514-2016-16-4-757-786}, zblnumber = {1422.53074}, }
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@INCOLLECTION{Pozniak, 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}, volume = {196}, pages = {119--181}, publisher = {Amer. Math. Soc., Providence, RI}, year = {1999}, mrclass = {53D40 (57R58)}, mrnumber = {1736217}, mrreviewer = {David E. Hurtubise}, doi = {10.1090/trans2/196/08}, url = {https://doi.org/10.1090/trans2/196/08}, zblnumber = {0948.57025 }, }
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@ARTICLE{RobbinSalamonIndex, author = {Robbin, Joel and Salamon, Dietmar}, title = {The {M}aslov index for paths}, journal = {Topology}, fjournal = {Topology. An International Journal of Mathematics}, volume = {32}, year = {1993}, number = {4}, pages = {827--844}, issn = {0040-9383}, mrclass = {58F05 (58E05)}, mrnumber = {1241874}, mrreviewer = {J. S. Joel}, doi = {10.1016/0040-9383(93)90052-W}, url = {https://doi.org/10.1016/0040-9383(93)90052-W}, zblnumber = {0798.58018}, }
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@INPROCEEDINGS{Ruan, author = {Ruan, Yongbin}, title = {Virtual neighborhoods and pseudo-holomorphic curves}, booktitle = {Proceedings of 6th {G}ökova {G}eometry-{T}opology {C}onference}, journal = {Turkish J. Math.}, fjournal = {Turkish Journal of Mathematics}, volume = {23}, year = {1999}, number = {1}, pages = {161--231}, issn = {1300-0098}, mrclass = {53D45 (53D40 57R58)}, mrnumber = {1701645}, mrreviewer = {Ignasi Mundet-Riera}, zblnumber = {0967.53055}, }
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@ARTICLE{RuanTian-qh1, author = {Ruan, Yongbin and Tian, Gang}, title = {A mathematical theory of quantum cohomology}, journal = {Math. Res. Lett.}, fjournal = {Mathematical Research Letters}, volume = {1}, year = {1994}, number = {2}, pages = {269--278}, issn = {1073-2780}, mrclass = {58D15 (14N10 55N45 57R35 58F05)}, mrnumber = {1266766}, mrreviewer = {Bruce Hunt}, doi = {10.4310/MRL.1994.v1.n2.a15}, url = {https://doi.org/10.4310/MRL.1994.v1.n2.a15}, zblnumber = {0860.58006}, }
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@ARTICLE{RuanTian-qh2, author = {Ruan, Yongbin and Tian, Gang}, title = {A mathematical theory of quantum cohomology}, journal = {J. Differential Geom.}, fjournal = {Journal of Differential Geometry}, volume = {42}, year = {1995}, number = {2}, pages = {259--367}, issn = {0022-040X}, mrclass = {58D29 (14C05 14N10 58D10)}, mrnumber = {1366548}, mrreviewer = {Bernd Siebert}, doi = {10.4310/jdg/1214457234}, url = {https://doi.org/10.4310/jdg/1214457234}, zblnumber = {0860.58005}, }
[Salamon-lectures] $Go to document$ D. Salamon, "Lectures on Floer homology," in Symplectic Geometry and Topology, Amer. Math. Soc., Providence, RI, 1999, vol. 7, pp. 143-229.

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@INCOLLECTION{Salamon-lectures, author = {Salamon, Dietmar}, title = {Lectures on {F}loer homology}, booktitle = {Symplectic Geometry and Topology}, venue = {{P}ark {C}ity, {UT}, 1997}, series = {IAS/Park City Math. Ser.}, volume = {7}, pages = {143--229}, publisher = {Amer. Math. Soc., Providence, RI}, year = {1999}, mrclass = {53D40 (37J45 53D45 57R17 57R58)}, mrnumber = {1702944}, mrreviewer = {David E. Hurtubise}, doi = {10.1016/S0165-2427(99)00127-0}, url = {https://doi.org/10.1016/S0165-2427(99)00127-0}, zblnumber = {1031.53118}, }
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@ARTICLE{SalamonZehnder, author = {Salamon, Dietmar and Zehnder, Eduard}, title = {Morse theory for periodic solutions of {H}amiltonian systems and the {M}aslov index}, journal = {Comm. Pure Appl. Math.}, fjournal = {Communications on Pure and Applied Mathematics}, volume = {45}, year = {1992}, number = {10}, pages = {1303--1360}, issn = {0010-3640}, mrclass = {58E05 (34C25 58F22 70H05)}, mrnumber = {1181727}, mrreviewer = {Yong-Geun Oh}, doi = {10.1002/cpa.3160451004}, url = {https://doi.org/10.1002/cpa.3160451004}, zblnumber = {0766.58023}, }
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@ARTICLE{Schwarz:action-spectrum, author = {Schwarz, Matthias}, title = {On the action spectrum for closed symplectically aspherical manifolds}, journal = {Pacific J. Math.}, fjournal = {Pacific Journal of Mathematics}, volume = {193}, year = {2000}, number = {2}, pages = {419--461}, issn = {0030-8730}, mrclass = {53D40 (57R58)}, mrnumber = {1755825}, mrreviewer = {David E. Hurtubise}, doi = {10.2140/pjm.2000.193.419}, url = {https://doi.org/10.2140/pjm.2000.193.419}, zblnumber = {1023.57020}, }
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@ARTICLE{SeidelMCG, author = {Seidel, Paul}, title = {Symplectic {F}loer homology and the mapping class group}, journal = {Pacific J. Math.}, fjournal = {Pacific Journal of Mathematics}, volume = {206}, year = {2002}, number = {1}, pages = {219--229}, issn = {0030-8730}, mrclass = {53D40}, mrnumber = {1924827}, mrreviewer = {Kai Cieliebak}, doi = {10.2140/pjm.2002.206.219}, url = {https://doi.org/10.2140/pjm.2002.206.219}, zblnumber = {1061.53065}, }
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@BOOK{SeidelBook, author = {Seidel, Paul}, title = {Fukaya Categories and {P}icard-{L}efschetz Theory}, series = {Zurich Lect. Adv. Math.}, publisher = {European Mathematical Society (EMS), Zürich}, year = {2008}, pages = {viii+326}, isbn = {978-3-03719-063-0}, mrclass = {53D40 (16E45 32Q65 53D12)}, mrnumber = {2441780}, mrreviewer = {Timothy Perutz}, doi = {10.4171/063}, url = {https://doi.org/10.4171/063}, zblnumber = {1159.53001}, }
[Seidel-pants] $Go to document$ P. Seidel, "The equivariant pair-of-pants product in fixed point Floer cohomology," Geom. Funct. Anal., vol. 25, iss. 3, pp. 942-1007, 2015.

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@ARTICLE{Seidel-pants, author = {Seidel, Paul}, title = {The equivariant pair-of-pants product in fixed point {F}loer cohomology}, journal = {Geom. Funct. Anal.}, fjournal = {Geometric and Functional Analysis}, volume = {25}, year = {2015}, number = {3}, pages = {942--1007}, issn = {1016-443X}, mrclass = {53D40 (55N91 57R58)}, mrnumber = {3361776}, mrreviewer = {Darko Milinković}, doi = {10.1007/s00039-015-0331-x}, url = {https://doi.org/10.1007/s00039-015-0331-x}, zblnumber = {1331.53119}, }
[S-Zoll] E. Shelukhin, Viterbo conjecture for Zoll symmetric spaces, 2018.

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@MISC{S-Zoll, author = {Shelukhin, E.}, title = {Viterbo conjecture for {Z}oll symmetric spaces}, arxiv = {1811.05552}, year = {2018}, zblnumber = {}, }
[SZhao-pants] E. Shelukhin and J. Zhao, The ${\Z}/p{\Z}$-equivariant product-isomorphism in fixed point Floer homology.

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@MISC{SZhao-pants, author = {Shelukhin, E. and Zhao, J.}, title = {The {${\Z}/p{\Z}$}-equivariant product-isomorphism in fixed point {F}loer homology}, note = {\emph{J. Symplectic Geom.}, to appear}, arxiv = {1905.03666}, zblnumber = {}, }
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@ARTICLE{Smith-original, author = {Smith, P. A.}, title = {Transformations of finite period}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {39}, year = {1938}, number = {1}, pages = {127--164}, issn = {0003-486X}, mrclass = {DML}, mrnumber = {1503393}, doi = {10.2307/1968718}, url = {https://doi.org/10.2307/1968718}, zblnumber = {0018.33204}, }
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@ARTICLE{Tonkonog-commuting, author = {Tonkonog, Dmitry}, title = {Commuting symplectomorphisms and {D}ehn twists in divisors}, journal = {Geom. Topol.}, fjournal = {Geometry \& Topology}, volume = {19}, year = {2015}, number = {6}, pages = {3345--3403}, issn = {1465-3060}, mrclass = {53D40 (14D05 14F35 37J10 53D05)}, mrnumber = {3447106}, mrreviewer = {Georgios Dimitroglou Rizell}, doi = {10.2140/gt.2015.19.3345}, url = {https://doi.org/10.2140/gt.2015.19.3345}, zblnumber = {1332.53106}, }
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On the Hofer-Zehnder conjecture

Abstract

Authors