Let B(p) and B(q) be Bernoulli shifts on {0,1,... ,d - 1}Z. If h(p) > h(q), it is a classical theorem of Sinai that there is a factor map taking B(p) to B(q). If, in addition, p stochastically dominates q, we can ask whether there is such a factor map Φ which is monotone: Φ(x)i ≤ xi for each coordinate i of almost every point x. Here we show that there is a monotone flnitary code from B(p) to B(q) in the case where B(q) is a shift on two symbols.
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