Limit theorems for cuspidal orbits on Fuchsian groups

Abstract

Let $\Gamma \subset \mathrm{SL}(2,\mathbb{R})$ be discrete, co-finite with cusps, and let $\mathfrak{b}\in \mathbb{R}\cup \{\infty \}$ be a cusp of $\Gamma$. Let $f:\Gamma \mathfrak{b}\to \mathbb{R}$ be a quantum modular form in the sense that the maps $${\varphi }_{\gamma }(x)=f(x)-f(\gamma x)(x\in \Gamma \mathfrak{b},\gamma \in \Gamma )$$ extend to Hölder-regular functions. Under bounded moments assumptions of the shape ${\sum }_n{n}^{-2}\int |{\varphi }_{{\gamma }^n}(x){|}^{\alpha }dx<\infty$ for some $\alpha >0$ and every fixed parabolic element $\gamma \in \Gamma$, we prove asymptotic formulas for the characteristic functions of the values $f(x)$ for $x\in \Gamma \mathfrak{b}$, ordered by growing denominators. These yield Gaussian limits when $\alpha >2$ and other stable limit laws under suitable further assumptions. This generalizes to Fuchsian groups the work of Baladi-Vallée on rational orbits of the Gauss map. In companion papers, we use this result, and in particular its flexible assumptions on ${\varphi }_{\gamma }$, to deduce limit theorems for automorphic $L$-series at the central point. One ingredient in our proof is a particular cuspidal acceleration of the Bowen-Series map, whose inverse branches admit a convenient parametrization, and for which we describe explicitly a natural extension.