The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

Abstract

In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713–756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C 1 -Hamiltonian systems on the cotangent bundle of a C 3 -smooth compact manifold M without boundary, of a time 1-periodic C 2 -smooth Hamiltonian H : R × T * M → R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H ( − t , q , − p ) = H ( t , q , p ) for any ( t , q , p ) ∈ R × T * M , it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T * M . If M is C 5 -smooth and dim M > 1 , H is of C 4 class and independent of time t, then for any τ > 0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L : R × T M → R , which is proved to be strongly convex and to have quadratic growth in the velocities yet.