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In mathematics, the Ornstein isomorphism theorem is a deep result for ergodic theory. It states that if two different Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic.[1][2] The result, given by Donald Ornstein in 1970, is important because it states that many systems previously believed to be unrelated are in fact isomorphic; these include all finite stationary stochastic processes, subshifts of finite type and Markov shifts, Anosov flows and Sinai's billiards, ergodic automorphisms of the n-torus, and the continued fraction transform

Discussion

The theorem is actually a collection of related theorems. The first theorem states that if two different Bernoulli shifts have the same Kolmogorov entropy, then they are isomorphic as dynamical systems. The third theorem extends this result to flows: namely, that there exists a flow \( T_t \) such that \( T_1 \) is a Bernoulli shift. The fourth theorem states that, for a given fixed entropy, this flow is unique, up to a constant rescaling of time. The fifth theorem states that there is a single, unique flow (up to a constant rescaling of time) that has infinite entropy. The phrase "up to a constant rescaling of time" means simply that if \(T_t \) and \(S_t \) are two flows with the same entropy, then \( S_t = T_{ct} \) for some constant c.

A corollary of these results is that a Bernoulli shift can be factored arbitrarily: So, for example, given a shift T, there is another shift \( \sqrt{T} \) that is isomorphic to it.
History

The question of isomorphism dates to von Neumann, who asked if the two Bernoulli schemes BS(1/2, 1/2) and BS(1/3, 1/3, 1/3) were isomorphic or not. In 1959, Ya. Sinai and Kolmogorov replied in the negative, showing that two different schemes cannot be isomorphic if they do not have the same entropy. Specifically, they showed that the entropy of a Bernoulli scheme BS(p1, p2,..., pn) is given by[3][4]

\( H = -\sum_{i=1}^N p_i \log p_i . \)

The Ornstein isomorphism theorem, proved by Donald Ornstein in 1970, states that two Bernoulli schemes with the same entropy are isomorphic. The result is sharp,[5] in that very similar, non-scheme systems do not have this property; specifically, Kolmogorov systems with the same entropy are not isomorphic. Ornstein received the Bôcher prize for this work.

A simplified proof of the isomorphism theorem was given by Michael S. Keane and M. Smorodinsky in 1979.[6][7] However, the original proof remains more powerful, as it provides a simple criterion that can be applied to determine if two different systems are isomorphic or not.

References

Donald Ornstein, "Bernoulli shifts with the same entropy are isomorphic", Advances in Math. 4 (1970), pp. 337–352
Donald Ornstein, "Ergodic Theory, Randomness and Dynamical Systems" (1974) Yale University Press, ISBN 0-300-01745-6
Ya.G. Sinai, (1959) "On the Notion of Entropy of a Dynamical System", Doklady of Russian Academy of Sciences 124, pp. 768–771.
Ya. G. Sinai, (2007) "Metric Entropy of Dynamical System"
Christopher Hoffman, "A K counterexample machine", Trans. Amer. Math. Soc. 351 (1999), pp 4263–4280
M. Keane and M. Smorodinsky, "The finitary isomorphism theorem for Markov shifts",Bull. Amer. Math. Soc. 1 (1979), pp. 436–438

M. Keane and M. Smorodinsky, "Bernoulli schemes of the same entropy are finitarily isomorphic". Annals of Mathematics (2) 109 (1979), pp 397–406.