 By Chuang C.T. (ed.)

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2 for /2n , and by induction choose k-set partitions Q(n) such that Q(0) = P and a) |d(Q(n) ) − d(P¯ )| ≤ δn+1 , b) 0 ≤ H(T¯, P¯ ) − H(T, Q(n) ) ≤ δn+1 , c) |Q(n) − Q(n−1) | ≤ /2n . 46 The limiting partition Q = limn Q(n) then satisfies the three conditions d) d(Q) = d(P¯ ), e) H(T, Q) = H(T¯, P¯ ), f) |Q − P | ≤ . 10)) that {T i Q} is an independent sequence. This completes the proof of the fundamental lemma. 47 CHAPTER 10. THE ISOMORPHISM THEOREM In this chapter, we shall complete the proof that two Bernoulli shifts with the same entropy are isomorphic.

4) a) Q is a generator for T , i b) {T Q} is an independent sequence, c) d(Q) = d(P¯ ). At this point, we mention that Sinai () showed how to find Q such that (b) and (c) hold. His construction is so difficult that it is not easy to show how one might choose Q so that (a) will also hold. Ornstein established a much stronger version of Sinai’s result, showing how one can choose a Q satisfying (b) and (c), ¯ that almost satisfies (b) and (c). Precise which is not too far away from a Q statements of these results are stated here.

12) 0 < H(P ) − H(P |S) ≤ H(P ) − H(P |Q). 11) holds. The set K is compact, and H(P ) − H(P |S) is a continuous non-vanishing function on K, hence it must be bounded away from 0. 2, is true. 4, we obtain the following fundamental result. ORNSTEIN’S COPYING THEOREM. Suppose T¯ is a Bernoulli shift with independent generator P¯ , where P¯ has k sets. Given > 0, there is a δ > 0 such that, if T is any ergodic transformation and P any k set partition such that (i) |d(P ) − d(P¯ )| < δand (ii) |H(P¯ ) − H(T, P )| < δ, then ¯ T¯, P¯ ), (T, P )) ≤ .

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