Probability measures on 2 × 2 stochastic matrices: some open problems

Let u be a probability measure on 2 × 2 stochastic matrices with finite support such that the sequence μⁿ, the nth convolution power of μ, weakly converges to a probability measure λ whose support consists of 2 × 2 stochastic matrices with identical rows. The probability measure λ can, therefore, be regarded as a measure on the unit interval 0,1]. In this paper, we discuss some open problems regarding when λ is continuous singular or absolutely continuous with respect to the Lebesgue measure on 0,1], and when λ determines μ uniquely.