## Counting geodesics on a Riemannian manifold and topological entropy of geodesic flows

**Authors: Keith Burns and
Gabriel Paternain **
**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 that the exponential growth rate of the integral of
$n_{T}(p,q)$ over $M \times M$ is the topological entropy of the geodesic flow of $M$.
In the present paper we exhibit an open set of metrics on the two-sphere for which
the exponential growth rate of $n_{T}(p,q$ is less than the topological entropy of the geodesic flow for a positive measure set of $(p,q)\in M\times M$.
This answers in the negative questions raised by Mañé.

This article is in Ergodic Theory and Dynamical Systems, 17(1997), 1043-1059.

A closely related article is

**Authors' addresses:**
Keith Burns
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
U.S.A.
Gabriel Paternain
DPMMS
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
England

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