Résume | Let S be a hyperbolic surface of genus g at least 2 with n cusps. It is well-known that the cardinality of the set of all geodesics in S of length at most T grows exponentially when T tends to infinity. On the other hand, the number N(T, c) of geodesics of length at most T which are in the mapping class group orbit of a simple closed geodesic c grows polynomially. In fact, if c is a simple closed curve then Mirzakhani proved that the limit of N(T, c)/T6g−6+2n exists and is positive. In this talk I will describe some related results for non-simple closed curves. This is joint work with Viveka Erlandsson. |