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

Let M be a compact C^∞ 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 ≤ T. Mañe showed in [7] that lim _T→∞ 1/T log ∫ _M×M n_T(p,q) dpdq = h_top, where h_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 lim sup _T→∞ 1/T log n_T (p, q) < h_top, for a positive measure set of (p, q) ∈ M × M. This answers in the negative questions raised by Mañé in [7].