Dynamical systems and ergodic theory Ergodic theory Theorem

Ornstein isomorphism theorem

Builds on Bernoulli scheme Kolmogorov–Sinai entropy Measure-preserving transformation Kolmogorov automorphism

Statement

Let $p = (p_1, \dots, p_k)$ and $q = (q_1, \dots, q_l)$ be probability vectors, and let $B(p)$ and $B(q)$ be the two-sided Bernoulli schemes they define. Two invertible measure-preserving transformations $T$ on $(X, \mu)$ and $S$ on $(Y, \nu)$ are isomorphic if the following holds. There are sets $X_0 \subseteq X$ , $Y_0 \subseteq Y$ of full measure and a measurable bijection $\varphi \colon X_0 \to Y_0$ with measurable inverse. It satisfies $\nu(\varphi(A)) = \mu(A)$ for measurable $A \subseteq X_0$ , and $\varphi \circ T = S \circ \varphi$ on $X_0$ . Then $B(p)$ and $B(q)$ are isomorphic if and only if they have the same Kolmogorov–Sinai entropy, that is, $-\sum_{i=1}^{k} p_i \log p_i = -\sum_{j=1}^{l} q_j \log q_j.$

Remarks

The direction "isomorphic implies equal entropy" is due to Kolmogorov, who introduced entropy as an isomorphism invariant in 1958. The converse is Ornstein's theorem of 1970. Its proof builds the isomorphism as a limit of better and better finite codes, using the finitely determined property, which every Bernoulli scheme has. Example: $B(1/4, 1/4, 1/4, 1/4)$ and $B(1/2, 1/8, 1/8, 1/8, 1/8)$ both have entropy $\log 4$ , so they are isomorphic, although their alphabets have different sizes. Meshalkin had found an explicit isomorphism in this case before Ornstein's theorem. On the other hand $B(1/2, 1/2)$ and $B(1/3, 1/3, 1/3)$ have entropies $\log 2 \ne \log 3$ , so they are not isomorphic, even though they are spectrally isomorphic. Ornstein's theory extends the theorem to Bernoulli schemes on countable alphabets and to Bernoulli flows.

Going deeper

For a student

Before Ornstein, it was not known whether entropy was the only obstruction. The theorem makes entropy a complete invariant for Bernoulli schemes. Ornstein's methods then showed that many smooth systems are isomorphic to Bernoulli schemes. Examples are ergodic toral automorphisms and geodesic flows on surfaces of negative curvature. Every Bernoulli scheme is a Kolmogorov automorphism, but Ornstein also constructed K-automorphisms that are not isomorphic to any Bernoulli scheme. Compare with Sinai's factor theorem, which gives Bernoulli factors of every system of positive entropy.

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Bernoulli process concept
Infinite product measure concept
Kolmogorov automorphism concept
Kolmogorov–Sinai entropy concept
Measure-preserving transformation concept
Mixing concept
Bernoulli scheme example
Ornstein isomorphism theorem statement

Tag	`8H4GF58E9M`
URI	`m2://8H4GF58E9M`
Kind	statement, theorem
Writing	W0 stub
MSC	37A35, 28D20
Status	This copy of the site has no build data. Run `m2 status 8H4GF58E9M`.
About	Bernoulli scheme, Kolmogorov–Sinai entropy
Uses	Bernoulli scheme, Kolmogorov–Sinai entropy, Measure-preserving transformation, Kolmogorov automorphism
Same as	Wikidata Q7103669 Wikipedia Ornstein isomorphism theorem

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