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