Limit theorems for cuspidal orbits on Fuchsian groups

Abstract

Let $Γ\subset \operatorname{SL}(2, \mathbb{R})$ be discrete, co-finite with cusps, and let ${\frak b}\in \mathbb{R}\cup\{\infty\}$ be a cusp of $Γ$. Let $f: Γ{\frak b} \to \mathbb{R}$ be a quantum modular form in the sense that the maps $$ ϕ_γ(x) = f(x) - f(γx) \qquad (x\in Γ{\frak b}, γ\in Γ) $$ extend to Hölder-regular functions. Under bounded moments assumptions of the shape $\sum_n n^{-2} \int |ϕ_{γ^n}(x)|^αd x < \infty$ for some $α>0$ and every fixed parabolic element $γ\in Γ$, we prove asymptotic formulas for the characteristic functions of the values $f(x)$ for $x\in Γ{\frak b}$, ordered by growing denominators. These yield Gaussian limits when $α>2$ 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 $ϕ_γ$, 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.