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Subharmonic solutions of Hamiltonian equations on tori. (English) Zbl 1180.58009

The author proves the following very interesting result: Let \(T^{2n}\) be the torus equipped with the standard symplectic structure and a periodic Hamiltonian \(H\). If the Hamiltonian flow has only finitely many orbits with the same period as \(H\), then there are subharmonic solutions with arbitrary high period. Thus there are always infinitely many distinct periodic orbits.
This property was conjectured by C. Conley [see the paper of D. Salamon and E. Zehnder, Commun. Pure Appl. Math. 45, No. 10, 1303–1360 (1992; Zbl 0766.58023)] and it was proved by C. Conley and E. Zehnder [Physica A 124, 649–658 (1984; Zbl 0605.58015)] in the nondegenerate case. The author of the present paper studied a similar phenomenon for closed geodesics [Int. Math. Res. Not. 1993, No. 9, 253–262 (1993; Zbl 0809.53053); Proc. Am. Math. Soc. 125, No. 10, 3099–3106 (1997; Zbl 0889.58026)].

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
53C22 Geodesics in global differential geometry
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
57R70 Critical points and critical submanifolds in differential topology
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References:

[1] H. Amann and E. Zehnder, ”Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations,” Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 7, iss. 4, pp. 539-603, 1980. · Zbl 0452.47077
[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
[3] R. Bott, ”On the iteration of closed geodesics and the Sturm intersection theory,” Comm. Pure Appl. Math., vol. 9, pp. 171-206, 1956. · Zbl 0074.17202 · doi:10.1002/cpa.3160090204
[4] K. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Boston, MA: Birkhäuser, 1993. · Zbl 0779.58005
[5] C. Conley and E. Zehnder, ”The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol\cprime d,” Invent. Math., vol. 73, iss. 1, pp. 33-49, 1983. · Zbl 0516.58017 · doi:10.1007/BF01393824
[6] C. Conley and E. Zehnder, ”Subharmonic solutions and Morse theory,” Phys. A, vol. 124, iss. 1-3, pp. 649-658, 1984. · Zbl 0605.58015 · doi:10.1016/0378-4371(84)90282-6
[7] C. Conley and E. Zehnder, ”A global fixed point theorem for symplectic maps and subharmonic solutions of Hamiltonian equations on tori,” in Nonlinear Functional Analysis and its Applications, Providence, RI: Amer. Math. Soc., 1986, vol. 1, pp. 283-299. · Zbl 0607.58035
[8] C. Conley, Isolated Invariant Sets and the Morse Index, Providence, R.I.: Amer. Math. Soc., 1978. · Zbl 0397.34056
[9] A. Floer, ”Proof of the Arnol\cprime d conjecture for surfaces and generalizations to certain Kähler manifolds,” Duke Math. J., vol. 53, iss. 1, pp. 1-32, 1986. · Zbl 0607.58016 · doi:10.1215/S0012-7094-86-05301-9
[10] 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
[11] N. Hingston, Perambulation in the symplectic group, preprint, 1986.
[12] 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
[13] N. Hingston, ”On the lengths of closed geodesics on a two-sphere,” Proc. Amer. Math. Soc., vol. 125, iss. 10, pp. 3099-3106, 1997. · Zbl 0889.58026 · doi:10.1090/S0002-9939-97-04235-4
[14] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Basel: Birkhäuser, 1994. · Zbl 0805.58003
[15] Y. Long, ”Bott formula of the Maslov-type index theory,” Pacific J. Math., vol. 187, iss. 1, pp. 113-149, 1999. · Zbl 0924.58024 · doi:10.2140/pjm.1999.187.113
[16] Y. Long, Index Theory for Symplectic Paths with Applications, Basel: Birkhäuser, 2002. · Zbl 1012.37012
[17] M. Morse, The Calculus of Variations in the Large, Providence, RI: Amer. Math. Soc., 1996. · Zbl 0007.21203
[18] D. McDuff and D. Salamon, Introduction to Symplectic Topology, New York: The Clarendon Press Oxford University Press, 1995. · Zbl 0844.58029
[19] P. H. Rabinowitz, ”On subharmonic solutions of Hamiltonian systems,” Comm. Pure Appl. Math., vol. 33, iss. 5, pp. 609-633, 1980. · Zbl 0425.34024 · doi:10.1002/cpa.3160330504
[20] 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
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