Quantitative equidistribution of horocycle push-forwards of transverse arcs

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