Let be a compact quotient of equipped with the normalized Haar measure vol, and let denote the horocycle flow on . Given and not parallel to the generator of the horocycle flow, let denote the probability measure uniformly distributed along the arc for . We establish quantitative estimates for the rate of convergence of to vol() for sufficiently smooth functions . Our result is based on the work of Bufetov and Forni, together with a crucial geometric observation. As a corollary, we provide an alternative proof of Ratner's theorem on quantitative mixing for the horocycle flow.
Cite this article
Davide Ravotti, Quantitative equidistribution of horocycle push-forwards of transverse arcs. Enseign. Math. 66 (2020), no. 1/2, pp. 135–150DOI 10.4171/LEM/66-1/2-7