On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds
Ryuma Orita
TL;DR
This work extends the Conley conjecture for Hamiltonian diffeomorphisms to non-contractible periodic orbits on closed symplectic manifolds under topological constraints. By combining filtered Floer--Novikov homology for non-contractible orbits with augmented action filtrations in monotone settings, the authors prove the existence of infinitely many simple non-contractible periodic orbits in iterated classes α^p for large primes p in prescribed arithmetic progressions, provided ω is aspherical and π1(M) is virtually abelian or an R-group. The results generalize prior works of Ginzburg and Gürel, applying both in the non-monotone (aspherical) and monotone/negative monotone contexts with the same fundamental-group assumptions. The methodology hinges on precise control of lift behaviors, action/augmented-action spectrums, and continuation maps within the Floer--Novikov framework, as well as critical group-theoretic lemmas for virtually abelian and R-group structures. Overall, the paper advances understanding of non-contractible periodic dynamics and provides robust tools for non-contractible Floer theory in broad geometric settings.
Abstract
We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M,ω)$ implies the existence of infinitely many non-contractible simple periodic orbits, provided that the symplectic form $ω$ is aspherical and the fundamental group $π_1(M)$ 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.