## 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
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 [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 [3] V. Bangert and W. Klingenberg, ”Homology generated by iterated closed geodesics,” Topology, vol. 22, iss. 4, pp. 379-388, 1983. · Zbl 0525.58015 [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 [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 [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 [9] A. Floer, ”Morse theory for Lagrangian intersections,” J. Differential Geom., vol. 28, iss. 3, pp. 513-547, 1988. · Zbl 0674.57027 [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 [11] A. Floer, ”Cuplength estimates on Lagrangian intersections,” Comm. Pure Appl. Math., vol. 42, iss. 4, pp. 335-356, 1989. · Zbl 0683.58017 [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 [13] A. Floer, ”Symplectic fixed points and holomorphic spheres,” Comm. Math. Phys., vol. 120, iss. 4, pp. 575-611, 1989. · Zbl 0755.58022 [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 [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 [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 [17] J. Franks and M. Handel, ”Periodic points of Hamiltonian surface diffeomorphisms,” Geom. Topol., vol. 7, pp. 713-756, 2003. · Zbl 1034.37028 [18] U. Frauenfelder and F. Schlenk, ”Hamiltonian dynamics on convex symplectic manifolds,” Israel J. Math., vol. 159, pp. 1-56, 2007. · Zbl 1126.53056 [19] V. L. Ginzburg, ”Coisotropic intersections,” Duke Math. J., vol. 140, iss. 1, pp. 111-163, 2007. · Zbl 1129.53062 [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 [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 [23] D. Gromoll and W. Meyer, ”On differentiable functions with isolated critical points,” Topology, vol. 8, pp. 361-369, 1969. · Zbl 0212.28903 [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 [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 [26] N. Hingston, ”Subharmonic solutions of Hamiltonian equations on tori,” Ann. of Math., vol. 170, iss. 2, pp. 529-560, 2009. · Zbl 1180.58009 [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 [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 [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 [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 [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 [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 [35] M. Mazzucchelli, The Lagrangian Conley conjecture, 2008. · Zbl 1209.37067 [36] D. McDuff and D. Salamon, Introduction to Symplectic Topology, New York: The Clarendon Press Oxford University Press, 1995. · Zbl 0844.58029 [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 [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 [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 [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 [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 [47] C. Viterbo, ”Symplectic topology as the geometry of generating functions,” Math. Ann., vol. 292, iss. 4, pp. 685-710, 1992. · Zbl 0735.58019 [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 [49] A. Weinstein, ”Symplectic manifolds and their Lagrangian submanifolds,” Advances in Math., vol. 6, pp. 329-346 (1971), 1971. · Zbl 0213.48203 [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
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