COUNTING GEODESICS ON A RIEMANNIAN MANIFOLD AND TOPOLOGICAL-ENTROPY OF GEODESIC-FLOWS

Let M be a compact C-infinity Riemannian manifold. Given p and q in M and T > 0, define n(T)(p, q) as the number of geodesic segments joinin g p and q with length less than or equal to T. Mane showed in [7] that [GRAPHICS] where h(top) denotes the topological entropy of the geodes ic flow of M. In this paper we exhibit an open set of metrics on the t wo-sphere for which [GRAPHICS] for a positive measure set of (p, q) is an element of M x M. This answers in the negative questions raised by Mane in [7].