Floer homology for non-resonant magnetic fields on flat tori

TL;DR

This paper defines and computes Floer homology for non-resonant magnetic fields on flat tori, revealing the existence and multiplicity of contractible solutions in the associated Hamiltonian system.

Contribution

It introduces a novel computation of Novikov Floer homology for non-resonant magnetic fields on flat tori, establishing new existence and multiplicity results for solutions.

Findings

Hamiltonian systems have 2N+1 contractible solutions

Generically, there are 2^{2N} contractible solutions

Existence of non-degenerate non-contractible solutions implies another

Abstract

In this article we define and compute the Novikov Floer homology associated to a non-resonant magnetic field and a mechanical Hamiltonian on a flat torus T^{2N}. As a result, we deduce that this Hamiltonian system always has 2N+1 contractible solutions, and generically even 2^{2N} contractible solutions. Moreover if there exists a non-degenerate non-contractible solution then there necessarily exists another.