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

TL;DR

This paper proves that the existence of a single non-contractible periodic orbit in certain symplectic manifolds guarantees infinitely many such orbits, using a novel Floer homology filtration based on augmented action.

Contribution

It introduces a new augmented action filtration in Floer homology that extends results on non-contractible periodic orbits to toroidally monotone manifolds.

Findings

Presence of one non-contractible orbit implies infinitely many in specific classes

Augmented action filtration is independent of capping

Results extend to toroidally monotone and negative monotone 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.