On finitely Bernoulli one-sided Markov shifts and their cofiltrations

The problem under consideration is: when is a Markov endomorphism (onesided shift) T = T_P with transition matrix P, isomorphic to a Bernoulli endomorphism T̃_ρ with an appropriate stationary vector ρ = {ρi}_i∈I? An obvious necessary condition is that there exists an independent complement δ of the measurable partition T^-1ε with distr δ = ρ. In this case the cofiltration (decreasing sequence of measurable partitions) ξ(T) = {T^-nε}^∞_n=1 generated by T is finitely isomorphic to the standard Bernoulli cofiltration ξ(T̃_ρ) = {T̃^-n_ρε}^∞_n=1 and T is called finitely ρ-Bernoulli. We show that every ergodic Markov endomorphism T_P, which is finitely Bernoulli, can be represented as a skew product over T̃_ρ with d-point fibres (d ∈ ℕ). We compute the minimal d = d(T_P) in these skew-product representations by means of the transition matrix, and obtain necessary and sufficient conditions under which d(T_P) = 1, i.e. T_P is isomorphic to T̃_ρ. The cofiltration ξ(T) of any finitely Bernoulli ergodic Markov endomorphism T = T_P is represented as a d-point extension of the standard cofiltration ξ(T̃_ρ), and we show that the minimal d = d_ξ(T) in these extensions is equal to d(T). In particular, d(T) = 1 ⇔ d_ξ(T) = 1, that is, a Markov endomorphism T is isomorphic to T̃_ρ iff ξ(T) is isomorphic to the Bernoulli cofiltration ξ(T̃_ρ).