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  "body": "\n## Statement\n\nLet $p = (p_1, \\dots, p_k)$ 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.\nTwo invertible [[measure-preserving transformations|556WBDAHNP]] $T$ on $(X, \\mu)$ and $S$ on $(Y, \\nu)$ are isomorphic if the following holds.\nThere 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.\nIt satisfies $\\nu(\\varphi(A)) = \\mu(A)$ for measurable $A \\subseteq X_0$, and $\\varphi \\circ T = S \\circ \\varphi$ on $X_0$.\nThen $B(p)$ and $B(q)$ are isomorphic if and only if they have the same [[Kolmogorov–Sinai entropy|X1D1C7VH6P]], that is,\n$$-\\sum_{i=1}^{k} p_i \\log p_i = -\\sum_{j=1}^{l} q_j \\log q_j.$$\n\n## Remarks\n\nThe direction \"isomorphic implies equal entropy\" is due to Kolmogorov, who introduced entropy as an isomorphism invariant in 1958.\nThe converse is Ornstein's theorem of 1970.\nIts proof builds the isomorphism as a limit of better and better finite codes, using the finitely determined property, which every Bernoulli scheme has.\nExample: $B(1/4, 1/4, 1/4, 1/4)$ 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.\nMeshalkin had found an explicit isomorphism in this case before Ornstein's theorem.\nOn 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.\nOrnstein's theory extends the theorem to Bernoulli schemes on countable alphabets and to Bernoulli flows.\n\n## Explanations\n\n### L1 human\n\nTwo machines that produce random symbols independently, one symbol per second, might look very different, for instance with different numbers of symbols.\nThe 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.\nEntropy, a single number, is the complete answer.\n\n### L2 human\n\nBefore Ornstein, it was not known whether entropy was the only obstruction.\nThe theorem makes entropy a complete invariant for Bernoulli schemes.\nOrnstein's methods then showed that many smooth systems are isomorphic to Bernoulli schemes.\nExamples are ergodic toral automorphisms and geodesic flows on surfaces of negative curvature.\nEvery Bernoulli scheme is a [[Kolmogorov automorphism|YRHJA7C7HS]], but Ornstein also constructed K-automorphisms that are not isomorphic to any Bernoulli scheme.\nCompare with Sinai's factor theorem, which gives Bernoulli factors of every system of positive entropy.\n",
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