×

zbMATH — the first resource for mathematics

The Conley conjecture. (English) Zbl 1228.53098
The Conley conjecture states that a Hamiltonian diffeomorphism of a closed symplectically aspherical manifold has infinitely many periodic points. The main result of this paper is a proof of this conjecture. More precisely, the author proves that a Hamiltonian diffeomorphism of a closed symplectically aspherical manifold whose fixed points are isolated admits simple periodic points of arbitrarily large period.

MSC:
53D35 Global theory of symplectic and contact manifolds
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
53D40 Symplectic aspects of Floer homology and cohomology
57R58 Floer homology
PDF BibTeX XML Cite
Full Text: DOI Link arXiv
References:
[1] V. I. Arnold, Mathematical Methods of Classical Mechanics, New York: Springer-Verlag, 1974.
[2] V. Bangert, ”Closed geodesics on complete surfaces,” Math. Ann., vol. 251, iss. 1, pp. 83-96, 1980. · Zbl 0422.53024 · doi:10.1007/BF01420283 · eudml:163426
[3] V. Bangert and W. Klingenberg, ”Homology generated by iterated closed geodesics,” Topology, vol. 22, iss. 4, pp. 379-388, 1983. · Zbl 0525.58015 · doi:10.1016/0040-9383(83)90033-2
[4] P. Biran, L. Polterovich, and D. Salamon, ”Propagation in Hamiltonian dynamics and relative symplectic homology,” Duke Math. J., vol. 119, iss. 1, pp. 65-118, 2003. · Zbl 1034.53089 · doi:10.1215/S0012-7094-03-11913-4 · arxiv:math/0108134
[5] K. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Boston, MA: Birkhäuser, 1993, vol. 6. · Zbl 0779.58005
[6] K. Cieliebak, V. L. Ginzburg, and E. Kerman, ”Symplectic homology and periodic orbits near symplectic submanifolds,” Comment. Math. Helv., vol. 79, iss. 3, pp. 554-581, 2004. · Zbl 1073.53118 · doi:10.1007/s00014-004-0814-0 · arxiv:math/0210468
[7] C. Conley, Lecture at the University of Wisconsin, April 6, 1984.
[8] C. Conley and E. Zehnder, ”Morse-type index theory for flows and periodic solutions for Hamiltonian equations,” Comm. Pure Appl. Math., vol. 37, iss. 2, pp. 207-253, 1984. · Zbl 0559.58019 · doi:10.1002/cpa.3160370204
[9] A. Floer, ”Morse theory for Lagrangian intersections,” J. Differential Geom., vol. 28, iss. 3, pp. 513-547, 1988. · Zbl 0674.57027 · projecteuclid.org
[10] A. Floer, ”The unregularized gradient flow of the symplectic action,” Comm. Pure Appl. Math., vol. 41, iss. 6, pp. 775-813, 1988. · Zbl 0633.53058 · doi:10.1002/cpa.3160410603
[11] A. Floer, ”Cuplength estimates on Lagrangian intersections,” Comm. Pure Appl. Math., vol. 42, iss. 4, pp. 335-356, 1989. · Zbl 0683.58017 · doi:10.1002/cpa.3160420402
[12] A. Floer, ”Witten’s complex and infinite-dimensional Morse theory,” J. Differential Geom., vol. 30, iss. 1, pp. 207-221, 1989. · Zbl 0678.58012 · projecteuclid.org
[13] A. Floer, ”Symplectic fixed points and holomorphic spheres,” Comm. Math. Phys., vol. 120, iss. 4, pp. 575-611, 1989. · Zbl 0755.58022 · doi:10.1007/BF01260388 · projecteuclid.org
[14] A. Floer and H. Hofer, ”Symplectic homology. I. Open sets in \({\mathbf C}^n\),” Math. Z., vol. 215, iss. 1, pp. 37-88, 1994. · Zbl 0810.58013 · doi:10.1007/BF02571699 · eudml:174598
[15] A. Floer, H. Hofer, and D. Salamon, ”Transversality in elliptic Morse theory for the symplectic action,” Duke Math. J., vol. 80, iss. 1, pp. 251-292, 1995. · Zbl 0846.58025 · doi:10.1215/S0012-7094-95-08010-7
[16] A. Floer, H. Hofer, and K. Wysocki, ”Applications of symplectic homology. I,” Math. Z., vol. 217, iss. 4, pp. 577-606, 1994. · Zbl 0869.58012 · doi:10.1007/BF02571962 · eudml:174708
[17] J. Franks and M. Handel, ”Periodic points of Hamiltonian surface diffeomorphisms,” Geom. Topol., vol. 7, pp. 713-756, 2003. · Zbl 1034.37028 · doi:10.2140/gt.2003.7.713 · emis:journals/UW/gt/GTVol7/paper20.abs.html · eudml:123400 · arxiv:math/0303296
[18] U. Frauenfelder and F. Schlenk, ”Hamiltonian dynamics on convex symplectic manifolds,” Israel J. Math., vol. 159, pp. 1-56, 2007. · Zbl 1126.53056 · doi:10.1007/s11856-007-0037-3 · arxiv:math/0305146
[19] V. L. Ginzburg, ”Coisotropic intersections,” Duke Math. J., vol. 140, iss. 1, pp. 111-163, 2007. · Zbl 1129.53062 · doi:10.1215/S0012-7094-07-14014-6 · euclid:dmj/1190730776 · arxiv:math/0605186
[20] V. L. Ginzburg and B. Z. Gürel, ”Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles,” Duke Math. J., vol. 123, iss. 1, pp. 1-47, 2004. · Zbl 1066.53138 · doi:10.1215/S0012-7094-04-12311-5
[21] V. L. Ginzburg and B. Z. Gürel, Local Floer homology and the action gap, 2007.
[22] 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. · Zbl 1172.53052 · doi:10.2140/gt.2009.13.2745 · arxiv:0810.5170
[23] D. Gromoll and W. Meyer, ”On differentiable functions with isolated critical points,” Topology, vol. 8, pp. 361-369, 1969. · Zbl 0212.28903 · doi:10.1016/0040-9383(69)90022-6
[24] B. Z. Gürel, ”Totally non-coisotropic displacement and its applications to Hamiltonian dynamics,” Commun. Contemp. Math., vol. 10, iss. 6, pp. 1103-1128, 2008. · Zbl 1161.53077 · doi:10.1142/S0219199708003198
[25] N. Hingston, ”On the growth of the number of closed geodesics on the two-sphere,” Internat. Math. Res. Notices, iss. 9, pp. 253-262, 1993. · Zbl 0809.53053 · doi:10.1155/S1073792893000285
[26] N. Hingston, ”Subharmonic solutions of Hamiltonian equations on tori,” Ann. of Math., vol. 170, iss. 2, pp. 529-560, 2009. · Zbl 1180.58009 · doi:10.4007/annals.2009.170.529 · annals.princeton.edu
[27] H. Hofer and E. Zehnder, ”A new capacity for symplectic manifolds,” in Analysis, et Cetera, Boston, MA: Academic Press, 1990, pp. 405-427. · Zbl 0702.58021
[28] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Basel: Birkhäuser, 1994. · Zbl 0805.58003
[29] E. Kerman, ”Squeezing in Floer theory and refined Hofer-Zehnder capacities of sets near symplectic submanifolds,” Geom. Topol., vol. 9, pp. 1775-1834, 2005. · Zbl 1090.53074 · doi:10.2140/gt.2005.9.1775 · eudml:126155 · arxiv:math/0502448
[30] E. Kerman and F. Lalonde, ”Length minimizing Hamiltonian paths for symplectically aspherical manifolds,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 53, iss. 5, pp. 1503-1526, 2003. · Zbl 1113.53056 · doi:10.5802/aif.1986 · numdam:AIF_2003__53_5_1503_0 · eudml:116079 · arxiv:math/0206220
[31] F. Lalonde and D. McDuff, ”Hofer’s \(L^\infty\)-geometry: energy and stability of Hamiltonian flows. I, II,” Invent. Math., vol. 122, iss. 1, pp. 1-33, 35, 1995. · Zbl 0844.58020 · doi:10.1007/BF01231437 · eudml:144313
[32] P. Le Calvez, ”Periodic orbits of Hamiltonian homeomorphisms of surfaces,” Duke Math. J., vol. 133, iss. 1, pp. 125-184, 2006. · Zbl 1101.37031 · doi:10.1215/S0012-7094-06-13315-X
[33] Y. Long, ”Multiple periodic points of the Poincaré map of Lagrangian systems on tori,” Math. Z., vol. 233, iss. 3, pp. 443-470, 2000. · Zbl 0984.37074 · doi:10.1007/PL00004805
[34] 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. · Zbl 1184.53080 · doi:10.1016/j.jfa.2009.01.001 · arxiv:0806.0425
[35] M. Mazzucchelli, The Lagrangian Conley conjecture, 2008. · Zbl 1209.37067 · doi:10.4171/CMH/222 · arxiv:0810.2108
[36] D. McDuff and D. Salamon, Introduction to Symplectic Topology, New York: The Clarendon Press Oxford University Press, 1995. · Zbl 0844.58029 · arxiv:dg-ga/9709021
[37] D. McDuff and D. Salamon, \(J\)-Holomorphic Curves and Symplectic Topology, Providence, RI: Amer. Math. Soc., 2004, vol. 52. · Zbl 1064.53051
[38] D. McDuff and J. Slimowitz, ”Hofer-Zehnder capacity and length minimizing Hamiltonian paths,” Geom. Topol., vol. 5, pp. 799-830, 2001. · Zbl 1002.57056 · doi:10.2140/gt.2001.5.799 · emis:journals/UW/gt/GTVol5/paper25.abs.html · eudml:122298 · arxiv:math/0101085
[39] M. Morse, The Calculus of Variations in the Large, Providence, RI: Amer. Math. Soc., 1996, vol. 18. · Zbl 0011.02802
[40] Y. Oh, ”Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomorphism group,” Asian J. Math., vol. 6, iss. 4, pp. 579-624, 2002. · Zbl 1038.53084 · arxiv:math/0104243
[41] M. Poźniak, ”Floer homology, Novikov rings and clean intersections,” in Northern California Symplectic Geometry Seminar, Providence, RI: Amer. Math. Soc., 1999, vol. 196, pp. 119-181. · Zbl 0948.57025
[42] D. Salamon, ”Morse theory, the Conley index and Floer homology,” Bull. London Math. Soc., vol. 22, iss. 2, pp. 113-140, 1990. · Zbl 0709.58011 · doi:10.1112/blms/22.2.113
[43] D. Salamon, ”Lectures on Floer homology,” in Symplectic Geometry and Topology, Providence, RI: Amer. Math. Soc., 1999, vol. 7, pp. 145-229. · Zbl 1031.53118
[44] D. Salamon and E. Zehnder, ”Morse theory for periodic solutions of Hamiltonian systems and the Maslov index,” Comm. Pure Appl. Math., vol. 45, iss. 10, pp. 1303-1360, 1992. · Zbl 0766.58023 · doi:10.1002/cpa.3160451004
[45] M. Schwarz, Morse Homology, Basel: Birkhäuser, 1993, vol. 111. · Zbl 0806.57020
[46] M. Schwarz, ”On the action spectrum for closed symplectically aspherical manifolds,” Pacific J. Math., vol. 193, iss. 2, pp. 419-461, 2000. · Zbl 1023.57020 · doi:10.2140/pjm.2000.193.419 · pjm.math.berkeley.edu
[47] C. Viterbo, ”Symplectic topology as the geometry of generating functions,” Math. Ann., vol. 292, iss. 4, pp. 685-710, 1992. · Zbl 0735.58019 · doi:10.1007/BF01444643 · eudml:164936
[48] C. Viterbo, ”Functors and computations in Floer homology with applications. I,” Geom. Funct. Anal., vol. 9, iss. 5, pp. 985-1033, 1999. · Zbl 0954.57015 · doi:10.1007/s000390050106
[49] A. Weinstein, ”Symplectic manifolds and their Lagrangian submanifolds,” Advances in Math., vol. 6, pp. 329-346 (1971), 1971. · Zbl 0213.48203 · doi:10.1016/0001-8708(71)90020-X
[50] A. Weinstein, Lectures on Symplectic Manifolds, Providence, R.I.: Amer. Math. Soc., 1977. · Zbl 0406.53031
[51] J. Williamson, ”On the algebraic problem concerning the normal forms of linear dynamical systems,” Amer. J. Math., vol. 58, iss. 1, pp. 141-163, 1936. · Zbl 0013.28401 · doi:10.2307/2371062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.