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
        




Click here for other preprints by Keith Burns

Click here for other preprints by Gabriel Paternain