On a class of one-sided Markov shifts
  We study one-sided Markov shifts, corresponding to positively recurrent Markov chains with countable (finite or infinite) state spaces. The following classification problem is considered: when two one-sided Markov shifts are isomorphic up to a measure preserving isomorphism In this paper we solve the problem for the class of rho-uniform (or finitely rho-Bernoulli) one-sided Markov shifts considered in Ru_6   We show that every ergodic rho-uniform Markov shift T can be represented in a canonical form T = T_G by means of a canonical (uniquely determined by T) stochastic graph G. In the canonical form, two such shifts T_{G_1} and T_{G_2} are isomorphic if and only if their canonical stochastic graphs G_1 and G_2 are isomorphic. 
Ben-Zion Rubshtein