# Existence/Non-existence of limiting distributions for horocycle flows on compact surfaces of constant negative curvature

# Existence/Non-existence of limiting distributions for horocycle flows on compact surfaces of constant negative curvature

A few years ago we have proved in collaboration with L. Flaminio that some non-trivial limit distributions for the horocycle flow must have compact support. In this talk we will refine that result and describe an existence/non-existence result for limiting distributions. In fact it turns out that whether limiting distributions exist or not depends on the geometry of the surface (via the eigenvalues of the Laplace operator) and on the observable under consideration. The main new idea is to express the precise results on the asymptotics of ergodic averages for the horocycle flow in terms of a dynamically defined cocycle which has the correct scaling property under the dynamics of the geodesic flow. Such cocycles are closely related to the invariant distributions of Flaminio-Forni and are analogous to the coycles constructed by A. Bufetov for asymptotic foliations of a Markov compactum (and in particular for area-preserving flows on higher genus surfaces).This is joint work with A. Bufetov.