Floer homology for magnetic fields with at most linear growth on the universal cover

TL;DR

This paper extends Floer homology results to magnetic fields with at most linear growth, showing isomorphism to loop space homology and proving existence of closed orbits and infinitely many periodic orbits under certain conditions.

Contribution

It demonstrates Floer homology isomorphism in the presence of Dirac magnetic monopoles with sublinear growth and proves the Conley conjecture for quadratic Hamiltonians.

Findings

Floer homology is isomorphic to loop space homology for certain magnetic fields.

Existence of closed orbits in specified free homotopy classes for small periods.

Infinitely many periodic orbits for quadratic Hamiltonians with bounded primitives.

Abstract

The Floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold, as proved by Abbondandolo-Schwarz, Salamon-Weber, and Viterbo. In this paper we show that in the presence of a Dirac magnetic monopole which admits a primitive with sublinear growth on the universal cover, the Floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology. As a consequence we prove that for any atoroidal free homotopy class and any sufficiently small T>0, any magnetic flow associated to the Dirac magnetic monopole has a closed orbit of period T belonging to the given free homotopy class. In the case where the Dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the Conley conjecture for Hamiltonians that are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.