Let p = {p1, , pn} and q = {qi, , qm} be finite probability vectors, each having at least three non-zero components, such that . Let C = {1, , n}, D = {1, , m} and let (Cℤ, pℤ, σ) and (Dℤ, qℤ, σ) be the corresponding Bernoulli shifts. Then there exists an isomorphism φ between these shifts such that for a.a. x ∈ Cℤ σ(x)(0) is determined by finitely many of the future co-ordinates x(0), x(1), and for a. a.y ∈ Dℤφ−1(y)(0) is determined by finitely many of the co-ordinates y(−1), y(0), y(1), .