Closed orbits of a charge in a weakly exact magnetic field

TL;DR

This paper proves that in a weakly exact magnetic system on a closed manifold, almost all energy levels contain closed orbits, with specific results depending on the energy relative to the Mane critical value.

Contribution

It establishes the existence of closed orbits on almost all energy levels for weakly exact magnetic flows, extending previous results to a broader class of systems.

Findings

Existence of closed orbits for all energy levels above the Mane critical value in each free homotopy class.

Almost all subcritical energy levels contain contractible closed orbits.

If the magnetic form is not exact and the fundamental group is amenable, then contractible closed orbits exist on almost every energy level.

Abstract

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected Riemannian manifold and $\sigma$ a weakly exact 2-form. Let $\phi_{t}$ denote the magnetic flow determined by $\sigma$, and let $c$ denote the Mane critical value of the pair $(g,\sigma)$. We prove that if $k>c$, then for every non-trivial free homotopy class of loops on $M$ there exists a closed orbit with energy $k$ whose projection to $M$ belongs to that free homotopy class. We also prove that for almost all $k<c$ there exists a closed orbit with energy $k$ whose projection to $M$ is contractible. In particular, when $c=\infty$ this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that if $\sigma$ is not exact and $M$ has an amenable fundamental group (which implies $c=\infty$) then there exist contractible closed orbits on almost every energy level.