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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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