---
tag: 8H4GF58E9M
kind: statement
role: theorem
name: Ornstein isomorphism theorem
msc: [37A35, 28D20]
uses: [D33JK3AXRZ, X1D1C7VH6P, 556WBDAHNP, YRHJA7C7HS]
about: [D33JK3AXRZ, X1D1C7VH6P]
sources:
  - url: https://en.wikipedia.org/wiki/Ornstein_isomorphism_theorem
same_as:
  wikidata: Q7103669
  wikipedia: Ornstein isomorphism theorem
---

## 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|D33JK3AXRZ]] they define.
Two invertible [[measure-preserving transformations|556WBDAHNP]] $T$ 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|X1D1C7VH6P]], 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.

## Explanations

### L1 human

Two machines that produce random symbols independently, one symbol per second, might look very different, for instance with different numbers of symbols.
The theorem says that one can be perfectly recoded into the other, in a way that respects time, exactly when both produce the same amount of randomness per second.
Entropy, a single number, is the complete answer.

### L2 human

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|YRHJA7C7HS]], 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.