Random walks in the hyperbolic plane and the question mark function

Consider $G=S{L}_2(\mathbb{Z})\mathrm{/}\{\pm I\}$ acting on the complex upper half plane $H$ by $h_M(z)=\frac{az+b}{cz+d},$ for $M\in G$. Let $D={z \in H: |z|\geq 1, |\Re(z)|\leq 1/2}$\mathrm{.}Weconsidertheset$$\mathcal{E}$ \subset G$withthe$9$elements$M$,differentfromtheidentity,suchthat$(MM^T)\leq 3$\mathrm{.}Weequipthetilingof$H$definedby$$\mathbb{D}$={h_M(D), M \in G}$ with a graph structure where the neighbours are defined by $h_M(D) \cap h_{M’}(D) \neq \emptyset$,equivalently$M^{-1}M’ \in $\mathcal{E}$$. The present paper studies several Markov chains related to the above structure. We show that the simple random walk on the above graph converges a.s. to a point $X$ of the real line with the same distribution of $S_2 W^{S_1}$,where$S_1,S_2,W$areindependentwith$\Pr (S_i=\pm 1)=1/2$ and where $W$ is valued in $(0,1)$ with distribution $\mathrm{Pr}(W<w)=?(w)$. Here $?$ is the Minkowski function. If $K_1,K_2,\dots$ are i.i.d with distribution $\Pr (K_i=n)= 1/2^n$for$n=1,2,\ldots$,then$W= \frac{1}{K_1+\frac {1}{K_2+\ldots}}$: this known result (Isola (2014)) is derived again here.