Horocycle flow on a surface of negative curvature is separating (Q1057558)

Let M be a compact connected oriented surface equipped with a Riemannian metric of negative curvature of class \(C^ 3\). A flow on M is separating if there exists \(\epsilon >0\) such that the distance between each pair of points moving along distinct trajectories becomes greater that \(\epsilon\) at some instant T. The author recalls the definition of the horocycle flow on M; then he proves that horocycle flows are separating.