ON THE STRUCTURE OF INVARIANT MEASURES RELATED TO NONCOMMUTATIVE RANDOM PRODUCTS

Let G = SL(R, n) be the group of mappings of the real projective space P^n-1 onto itself. There is introduced the notion of a boundary measure v on P^n-1 for a probability measure μ on G, and its relation to the unique invariant measure on P^n-1 with respect to the operator π(x, A) = μ{g ∊ G: gx ∊ A} is found. It is established that the Markov chain generated by the transition probability π(x, A) and the invariant boundary measure v is a factor-automorphism of an automorphism of a certain Bernoulli space. A limit theorem for random mappings of a segment of the line into itself is proved. Bibliography: 6 items.