Vol. 247, No. 1, 2010

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Closed orbits of a charge in a weakly exact magnetic field

Will J. Merry

Vol. 247 (2010), No. 1, 189–212

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) ∪{∞} 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.

magnetic flow, twisted geodesic flow, periodic orbits, Mañé critical value
Mathematical Subject Classification 2000
Primary: 37J45, 70H12
Received: 28 June 2009
Accepted: 21 October 2009
Published: 1 July 2010
Will J. Merry
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge CB3 0WB