Closed geodesics and the first Betti number

Abstract

We prove that, on any closed manifold of dimension at least two with non-trivial first Betti number, a $C^{\infty }$ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We derive this existence result combining a theorem of Mañé together with the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable $C^{\infty }$-close Riemannian metric.

19 pages, final version: minor corrections