Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

TL;DR

This paper proves the existence of non-contractible periodic orbits in twisted cotangent bundles using Floer homology invariance under symplectic deformations, with applications to magnetic flows and capacity estimates.

Contribution

It establishes Floer homology invariance under symplectic deformations for twisted bundles and computes it explicitly, enabling new existence results for periodic orbits in magnetic systems.

Findings

Existence of periodic orbits in any $\sigma$-atoroidal free homotopy class.

Finiteness of the Biran-Polterovich-Salamon capacity for twisted cotangent bundles.

Periodic orbits on almost every energy level for small magnetic fields.

Abstract

In this paper we establish the existence of periodic orbits belonging to any $\sigma$-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact $2$-form $\sigma$ admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any nontrivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran-Polterovich-Salamon capacity.