Bernoulli flow

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. 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 , subshifts of finite type and Markov shifts, 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 ${\displaystyle T_{t}}$ such that ${\displaystyle T_{1}}$ 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 ${\displaystyle T_{t}}$ and ${\displaystyle S_{t}}$ are two flows with the same entropy, then ${\displaystyle S_{t}=T_{ct}}$ for some constant c.

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