[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, 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 first theorem states that if two different Bernoulli shifts have the same Kolmogorov entropy, then they are isomorphic as dynamical systems.
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.
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] The Ornstein isomorphism theorem, proved by Donald Ornstein in 1970, states that two Bernoulli schemes with the same entropy are isomorphic.
A simplified proof of the isomorphism theorem for symbolic Bernoulli schemes was given by Michael S. Keane and M. Smorodinsky in 1979.