On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds

We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M, \omega)$ implies the existence of infinitely many noncontractible simple periodic orbits, provided that the symplectic form $\omega$ is aspherical and the fundamental group $\pi_1 (M)$ is either a virtually abelian group or an $\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian diffeomorphisms of closed monotone or negative monotone symplectic manifolds under the same conditions on their fundamental groups. These results generalize some works by Ginzburg and Gürel. The proof uses the filtered Floer–Novikov homology for non-contractible periodic orbits.

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Received 11 July 2017

Accepted 3 March 2018

Published 17 January 2020