On the asymptotics of ergodic averages for horocycle flows

We will present joint results with Flaminio and separately with Bufetov on the asymptotics of ergodic averages of a sufficiently smooth function, for the horocycle flow on the unit tangent bundle of a compact surface of constant negative curvature. These results can be interpreted and in fact are derived as statements on the asymptotics of the correlations for the corresponding geodesic flow. The method of proof is based on non-commutative harmonic analysis, that is, on the theory of unitary representations for the group $SL(2,\mathbb R)$.