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Ornstein isomorphism theorem
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.〔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〕 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 such that 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 and are two flows with the same entropy, then 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 that is isomorphic to it.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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