Any two irreducible Markov chains of equal entropy are finitarily Kakutani equivalent

We show here that any two finite state irreducible Markov chains of the same entropy are finitarily Kakutani equivalent. By this we mean they are orbit equivalent by an invertible measure preserving mapping that is almost continuous and monotone in time when restricted to some cylinder set. Smorodinsky and Keane have shown that any two irreducible Markov chains of equal entropy and period are finitarily isomorphic. Hence, all that is necessary to obtain our result is to show that for every entropy h > 0 and period p ∈ ℕ there exists two irreducible Markov chains σ ₁, σ ₂ both of entropy h, where (1) σ ₁ is mixing (2) ς ₂ has period p and (3) σ1 and σ ₂ are finitarily Kakutani equivalent.