Precise counting results for closed orbits of Anosov flows

We study the problem of counting closed geodesics according to their length s and under homological constraints on a compact surface of negative curvat ure. We show how to use Dolgopyat's recent results to obtain a full asympto tic expansion, in addition to the leading term given by Lalley. We first state the properties of the stable and unstable leaves used by Che rnov and Dolgopyat; then we introduce the usual transfer operators and we p rove the result with the help of a dynamical C-function. (C) 2000 Editions scientifiques et medicales Elsevier SAS.