Random Walks in the Hyperbolic Plane and the Minkowski Question Mark Function

Consider $$G=SL_2(\mathbb {Z})/\{\pm I\}$$ G = S L 2 ( Z ) / { ± I } acting on the complex upper half plane H by $$h_M(z)=\frac{az\,+\,b}{cz\,+\,d}$$ h M ( z ) = a z + b c z + d for $$M \in G$$ M ∈ G . Let $$D=\{z \in H: |z|\ge 1, |\mathfrak {R}(z)|\le 1/2\}$$ D = { z ∈ H : | z | ≥ 1 , | R ( z ) | ≤ 1 / 2 } . We consider the set $${\mathcal {E}} \subset G$$ E ⊂ G with the nine elements M, different from the identity, such that $$\mathrm{tr\,}(MM^T)\le 3$$ tr ( M M T ) ≤ 3 . We equip the tiling of H defined by $$\mathbb {D}=\{h_M(D){:}\, M \in G\}$$ D = { h M ( D ) : M ∈ G } with a graph structure where the neighbours are defined by $$h_M(D) \cap h_{M'}(D) \ne \emptyset $$ h M ( D ) ∩ h M ′ ( D ) ≠ ∅ , equivalently $$M^{-1}M' \in {\mathcal {E}}$$ M - 1 M ′ ∈ 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 $$S_2 W^{S_1}$$ S 2 W S 1 , where $$S_1,S_2,W$$ S 1 , S 2 , W are independent with $$\Pr (S_i=\pm 1)=1/2$$ Pr ( S i = ± 1 ) = 1 / 2 and where W is valued in (0, 1) with distribution $$\Pr (W