Word norm, Minkowski's function and random walks on the plane

Let G be a group acting on a space X. A natural question to study is the asymptotical behavior of orbits when elements are chosen using some law.
We first look into a specific lattice in SL(2,R) acting on the real projective line when elemets are chosen using a certain word norm, and show connections to the Minkowski question mark funcion, continued fractions and the stationary measure of a closely related random walk.
We then study the random walk on the real plane, generated by action of subgroups in SL(2,R). By applying carefully chosen scaling suggested by Maucourant and using a recent result of Benoist-Quint we reach a reasonable candidate for solution.