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

TL;DR

This paper proves that under certain conditions on the symplectic manifold's fundamental group and symplectic form, the existence of a single non-contractible periodic orbit guarantees infinitely many such orbits, extending previous results.

Contribution

It generalizes existing theorems by establishing the existence of infinitely many non-contractible periodic orbits for Hamiltonian diffeomorphisms on broader classes of symplectic manifolds with specific fundamental group conditions.

Findings

Presence of one non-contractible orbit implies infinitely many.

Results apply to aspherical, monotone, and negative monotone manifolds.

Utilizes filtered Floer--Novikov homology for proof.

Abstract

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 non-contractible 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 G\"urel. The proof uses the filtered Floer--Novikov homology for non-contractible periodic orbits.