Closed geodesics and the first Betti number

TL;DR

This paper proves that on closed manifolds with non-trivial first Betti number, generic smooth metrics have infinitely many closed geodesics of arbitrarily large length, combining existing theorems with new results on minimal geodesics and horseshoes.

Contribution

It introduces a new theorem linking minimal closed geodesics to transverse homoclinics, enhancing understanding of geodesic flow complexity.

Findings

Generic metrics have infinitely many closed geodesics.

Existence of minimal closed geodesics implies horseshoe dynamics.

Results apply to manifolds with non-trivial 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 Ma\~n\'e 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.