For several parabolic systems, a technique often used to prove mixing and other strong chaotic properties consists of a geometric shearing argument. In the case of the horocycle flow (both in constant and in variable negative curvature, as well as for its smooth time-changes), this has been done successfully by analysing the action of the horocycle flow on geodesic arcs. The quantitative mixing estimates one can obtain following this approach, however, are not optimal, since, in the constant curvature case, do not match the ones obtained by Ratner.
In this talk, we will present an effective equidistribution result for the horocycle push-forwards of homogeneous arcs which are transverse to the weak-stable leaves of the geodesic flow. As a corollary, we derive a geometric proof of Ratner's quantitative mixing result for the horocycle flow. We will discuss related open problems for its smooth time-changes.