Quantitative equidistribution of horocycle push-forwards of transverse arcs

Abstract

Let M = \Gamma \backslash \mathrm {SL}(2,\mathbb R) be a compact quotient of \mathrm {SL}(2,\mathbb R) equipped with the normalized Haar measure vol, and let \{h_t\}_{t \in \mathbb R} denote the horocycle flow on M . Given p \in M and W \in \mathfrak{sl}_2(\mathbb R) \setminus \{0\} not parallel to the generator of the horocycle flow, let \gamma_{p}^W denote the probability measure uniformly distributed along the arc s \mapsto p \mathrm {exp}(sW) for 0\leq s \leq 1 . We establish quantitative estimates for the rate of convergence of [(h_t)_\ast \gamma_{p}^W](f) to vol( f ) for sufficiently smooth functions f . 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.