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.