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 ∈ Ω2(M) a weakly exact 2-form. Let ϕt: TM → TM denote the magnetic flow
determined by σ, and let c(g,σ) ∈ ℝ ∪{∞} denote the Mañé critical value of the
pair (g,σ). We prove that if k > c(g,σ), then for every nontrivial free homotopy class
of loops on M there exists a closed orbit of ϕt with energy k whose projection to M
belongs to that free homotopy class. We also prove that for almost all k < c(g,σ)
there exists a closed orbit of ϕt with energy k whose projection to M is contractible.
In particular, when c(g,σ) = ∞ this implies that almost every energy level has a
contractible closed orbit. As a corollary we deduce that a weakly exact magnetic
flow with [σ]≠0 on a manifold with amenable fundamental group (which
implies c(g,σ) = ∞) has contractible closed orbits on almost every energy
level.
Keywords
magnetic flow, twisted geodesic flow, periodic orbits, Mañé
critical value
