Quantitative equidistribution of horocycle push-forwards of transverse arcs

TL;DR

This paper provides quantitative estimates on how quickly horocycle push-forwards of transverse arcs become uniformly distributed on compact quotients of SL(2,R), offering an alternative proof of Ratner's mixing theorem.

Contribution

It introduces new quantitative bounds for horocycle equidistribution and offers a novel geometric approach, extending previous work by Bufetov and Forni.

Findings

Established explicit convergence rates for horocycle push-forwards

Provided an alternative proof of Ratner's quantitative mixing theorem

Enhanced understanding of horocycle flow dynamics on compact quotients

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 $[(h_t)_{\ast} \gamma_{p}^W](f)$ 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.