Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds

Abstract

We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping, and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

24 pages; minor revisions made, references added; to appear in Compositio Mathematica

References in corpus (5)

• Persistent homology and Floer-Novikov theory
• Forcing theory for transverse trajectories of surface homeomorphisms
• Non-contractible periodic orbits, Gromov invariants, and Floer-theoretic torsions
• On hyperbolic cohomology classes
• On non-contractible hyperbolic periodic orbits and periodic points of symplectomorphisms

Cited by in corpus (12)

• Hamiltonian Pseudo-rotations of Projective Spaces
• Non-contractible periodic orbits in Hamiltonian dynamics on tori
• Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles
• On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds
• Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits
• Heavy subsets and non-contractible trajectories
• Random Chain Complexes
• On the Barcode Entropy of Reeb Flows
• Periodic points of rational area-preserving homeomorphisms
• On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics
• Invariant Sets and Hyperbolic Closed Reeb Orbits
• On the Hofer-Zehnder conjecture for semipositive symplectic manifolds