Limit theorems for cuspidal orbits on Fuchsian groups

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 $$ ϕ_γ(x) = f(x) - f(γx) \qquad (x\in Γ{\frak b}, γ\in Γ) $$ ex