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Concept# Ornstein isomorphism theorem

Summary

In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. 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, including Markov chains and subshifts of finite type, Anosov flows and Sinai's billiards, ergodic automorphisms of the n-torus, and the continued fraction transform.
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 Bernoulli flows with the same entropy, then for some constant c. The developments also included proofs that factors of Bernoulli shifts are isomorphic to Bernoulli shifts, and gave criteria for a given measure-preserving dynamical system to be isomorphic to a Bernoulli shift.
A corollary of these results is a solution to the root problem for Bernoulli shifts: So, for example, given a shift T, there is another shift that is isomorphic to it.
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,...

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In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical systems. Many important dynamical systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the Markov partition.

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