Counting geodesics on a Riemannian manifold andtopological entropy of geodesic flows

Abstract

Let $M$ be a compact $C^{\infty}$ Riemannian manifold. Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number of geodesic segments joining $p$ and $q$ with length $\leq T$. Mañé showed in [7] that \[ \lim_{T\rightarrow \infty}\frac{1}{T}\log \int_{M\times M}n_{T}(p,q)\,dp\,dq = h_{\rm top}, \] where $h_{\rm top}$ denotes the topological entropy of the geodesic flow of $M$.

In this paper we exhibit an open set of metrics on the two-sphere for which \[ \limsup_{T\rightarrow\infty}\frac{1}{T}\log n_{T}(p,q)< h_{\rm top}, \] for a positive measure set of $(p,q)\in M\times M$. This answers in the negative questions raised by Mañé in [7].