Closed orbits of a charge in a weakly exact magnetic field

الأصل · EN

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 σ a weakly exact 2-form. Let ϕₜ denote the magnetic flow determined by σ, and let c denote the Mane critical value of the pair (g,σ). 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=∞ this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that if σ is not exact and M has an amenable fundamental group (which implies c=∞) then there exist contractible closed orbits on almost every energy level.