On the deviation of ergodic averages for horocycle flows

Abstract

We give effective bounds on the deviation of ergodic averages for the horocycle flow on the unit tangent bundle of a noncompact hyperbolic surface of finite area. The bounds depend on the small eigenvalues of the Laplacian and on the rate of excursion into cusps for the geodesic corresponding to the given initial point. We also prove Ω-results which show that in a certain sense our bounds are essentially the best possible for any given initial point.