Word Metric, Stationary Measure and Minkowski’s Question Mark Function

Given a countably infinite group G acting on some space X, an increasing family of finite subsets G_n, x∈ X and a function f over X we consider the sums S_n(f, x) = ∑_g∈Gnf(gx). The asymptotic behaviour of S_n(f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2, ℤ ) acting on the projective line and G_n are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ^∗n.