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

In this paper, we establish the existence of periodic orbits belonging to any a-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 a 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 non-trivial 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.

Automated summary

This paper establishes the existence of periodic orbits in a-atoroidal free homotopy classes for Hamiltonian systems in the twisted disc bundle, under certain conditions. The proof relies on the invariance of Floer homology and computation of Floer homology for the cotangent bundle. As an application, it is shown that twisted geodesic flows associated with a small magnetic field have periodic orbits on almost every energy level.