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

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 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. Contents1. Introduction 1 2. Main results 3 2.1. Results 3 2.2. Examples 4 3. Preliminaries 4 3.1. Conventions and notation 5 3.2. Floer-Novikov homology 6 4. Lemmas from algebraic topology and group theory 9 4.1. Aspherical cohomology classes and Eilenberg-MacLane spaces 9 4.2. Fundamental groups of free loop spaces 10 5. Proof of Theorem 2.1 13 6. Proof of Theorems 2.2 and 2.3 15 6.1. Augmented action filtered Floer-Novikov homology 15 6.2. Proof of Theorem 2.2 18 6.3. Proof of Theorem 2.3 20 Acknowledgments 21 References 21

Hamiltonian pseudo-rotations of projective spaces

“…Although in this case the Conley conjecture holds and ϕ has infinitely many periodic orbits unconditionally (see [CGG,GG12]), Theorem 6.2 gives additional information about the augmented actions of the orbits or their location and free homotopy classes; cf. [Ba15b,GG16a,Gü13,Or17a,Or17b].…”

On non-contractible periodic orbits and bounded deviations

“…In this work we study the dynamical consequences of the existence or absence of noncontractible periodic orbits in conservative surface dynamics. This is a direction that has been garnering increased attention in recent years, with new results due to mostly the increasingly developed field of Brouwer-homeomorphims like techniques, and that has drawn increased interest due not only to its applications in the study of some relevant classes of torus homeomorphisms, but also to its connection to symplectic dynamics, where the subject has been much more largely exploited (see for example [8] as well as references herein, see also [17]), albeit through very different techniques.…”

On Non-contractible Periodic Orbits and Bounded Deviations

“…In this work we study the dynamical consequences of the existence or absence of non-contractible periodic points in conservative surface dynamics. This is a direction that has been garnering increased attention in recent years, with new results due to mostly the increasingly developed field of Brouwer-homeomorphims like techniques, and that has drawn increased interest due not only to its applications in the study of some relevant classes of torus homeomorphisms, but also to its connection to symplectic dynamics, where the subject has been much more largely exploited (see for example [8] as well as references herein, see also [16]), albeit through very different techniques.…”