On non-contractible periodic orbits of Hamiltonian diffeomorphisms (Q2865917)

The paper is devoted to prove the existence of non-contractible periodic orbits of arbitrarily large periods for Hamiltonian diffeomorphisms, provided there is at least a one-periodic orbit of a particular type. This question is related to the Conley conjecture or one of its variants, the HZ conjecture.NEWLINENEWLINEMore concretely, the author proves the following claim. Let \((M^{2n},\omega)\) be a closed atoroidal symplectic manifold (that is, a closed symplectic manifold such that the integral of \(\omega\) over \(v\) vanishes for all maps \(v\) from the \(2\)-torus to \(M\)) and \(\varphi_H\) a Hamiltonian diffeomorphism of \(M\). Let \(P_1(\varphi_H,[\gamma])\) be the collection of one-periodic orbits of \(\varphi_H\) with homology class \([\gamma] \in H_1(M;\mathbb{Z})/\text{Tor}\). Assume that \(\varphi_H\) possesses a non-degenerate one-periodic orbit \(\gamma\) with homology class \([\gamma] \neq 0\) in \(H_1(M;\mathbb{Z})/\text{Tor}\) and that \(P_1(\varphi_H,[\gamma])\) is finite. Then, for every large enough prime \(p\), \(\varphi_H\) has a simple periodic orbit with homology class \(p[\gamma]\) and period either \(p\) or \(p'\), where \(p'\) is the first prime number greater than \(p\).NEWLINENEWLINEManifolds satisfying the hypotheses of the theorem are, for example, Kähler manifolds with negative sectional curvature, like complex hyperbolic spaces or any closed symplectic manifold \(M\) with \(\pi_2(M) = 0\) and \(\pi_1(M)\) a hyperbolic group.NEWLINENEWLINEThe proof of the theorem is based on the use of the filtered Floer homology for non-contractible periodic orbits of Hamiltonian diffeomorphisms. In fact, it is shown that the claim holds if the one-periodic orbit \(\gamma\) above is isolated and homologically non-trival, that is, if the local Floer homology of \(H\) at \(\gamma\) is non-zero.