Discrete probability measures on 2 × 2 stochastic matrices and a functional equation on [0, 1]

In this paper, we consider the following natural problem: suppose μ1 and μ2 are two probability measures with finite supports S(μ1), S(μ2) respectively, such that |S(μ1)| = |S(μ2)| and S(μ1) U S(μ2) ⊂ 2 × 2 stochastic matrices, and μn1 (the n-th convolution power of μ1 under matrix multiplication), as well as μn 2 , converges weakly to the same probability measure λ, where S(λ) ⊂ 2 × 2 stochastic matrices with rank one. Then when does it follow that μ1 = μ2? What if S(μ1) = S(μ2)? In other words, can two different random walks, in this context, have the same invariant probability measure? Here, we consider related problems.