Quantitative equidistribution of horocycle push-forwards of transverse arcs

Abstract: Let $M = \Gamma \backslash \text{SL}(2,\mathbb{R})$ 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 $W \in \mathfrak{sl}_2(\mathbb{R}) \setminus {0}$ not parallel to the generator of the horocycle flow, let $\gamma{p}^W$ denote the probability measure uniformly distributed along the arc $s \mapsto p \exp(sW)$ for $0\leq s \leq 1$. We establish quantitative estimates for the rate of convergence of $(h_t){\ast} \gamma{p}^W$ 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.