On the growth rate of geodesic chords

Abstract

We show that every forward complete Finsler manifold of infinite fundamental group and not homotopy-equivalent to $S^1$ has infinitely many geometrically distinct geodesics joining any given pair of points $p$ and $q$. In the special case in which $\beta_1(M;\mathbb{Z})\geq 1$ and $M$ is closed, the number of geometrically distinct geodesics between two points grows at least logarithmically.