The growth rate of closed prime magnetic geodesics on closed contact manifolds

Abstract

In this paper, we prove that for any given closed contact manifold, there exists an infinite-dimensional space of Riemannian metrics which can be identified with the space of bundle metrics on the induced contact distribution. For each such metric, and for all energy levels, the number of embedded periodic orbits of the corresponding magnetic geodesic flow grows at least as fast as the number of geometrically distinct periodic Reeb orbits of period less than $t$. As a corollary, we deduce that for every closed 3-manifold which is not a graph manifold, there exists an open $C^1$-neighborhood of the set of nondegenerate contact forms such that for each contact form in this neighborhood, there exists an infinite-dimensional space of Riemannian metrics as above. For the corresponding magnetic systems, the number of prime closed magnetic geodesics grows at least exponentially on all energy levels. Consequently, the restriction of the magnetic geodesic flow to any energy surface has positive topological entropy.