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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:
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