A construction of non-isomorphic one-sided Markov shifts

Let T_ρ be a one-sided Bernoulli shift, which acts on the product space (X_ρ,μ_ρ) = Π ^∞_n=1=(I,ρ) is a finite or countable state space with a probability distribution ρ = {ρ(i)}iεI, ρ(i) > 0, Σ_{iε ρ}(i) = 1. Let also Y_d = {1,2, • • • d}, d e N. Suppose p is not homogeneous, i.e. ρ(i) σ(i') for some pair t, i' ε I, and let d > 1. For given such d and p, we construct a uncountable family T_λ, λ ε A = A(ρ, d), such that (i) T_λ is isomorphic to a d-extension of T_ρ, i.e. to a skew product on X_ρ × Y _d over T_ρ. (ii) T_λ is an one-sided Markov shift, corresponding to an aperiodic irreducible positively recurrent Markov chain on the infinite countable state space I × N × Y _d. (iii) All the shifts T_λ λ ε A, are pairwise non-isomorphic. (iv) Each of the shifts T_λ λ ε A, is not Isomorphic to one-sided Markov shifts on finite state spaces (even in the case, when I is finite).