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  "body": "\n## Definition\n\nLet $(X, \\mathcal{B}, \\mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \\colon X \\to X$ be an invertible [[measure-preserving transformation|556WBDAHNP]] whose inverse is also measure-preserving.\nFor a [[sub-σ-algebra|0SV8K0TDGY]] $\\mathcal{K} \\subseteq \\mathcal{B}$ and $n \\in \\Z$, let $T^n \\mathcal{K} = \\set{T^n A : A \\in \\mathcal{K}}$.\nThe map $T$ is a *Kolmogorov automorphism*, or *K-automorphism*, if there is a sub-σ-algebra $\\mathcal{K}$ such that:\n\n1. $T \\mathcal{K} \\subseteq \\mathcal{K}$.\n2. The σ-algebra generated by $\\bigcup_{n \\ge 0} T^{-n} \\mathcal{K}$ is $\\mathcal{B}$, up to sets of measure $0$.\n3. Every set in $\\bigcap_{n \\ge 0} T^{n} \\mathcal{K}$ has measure $0$ or $1$.\n\n## Remarks\n\nExample: let $X = \\set{0, 1}^{\\Z}$ with the product of the measures giving weight $1/2$ to each symbol, and $T$ the left shift, $(Tx)_n = x_{n+1}$.\nLet $\\mathcal{K}$ be generated by the coordinates $x_m$ with $m \\le 0$.\nThen $T^{n} \\mathcal{K}$ is generated by the $x_m$ with $m \\le -n$, for every $n \\in \\Z$.\nSo clause 1 holds, the $T^{-n}\\mathcal{K}$ for $n \\ge 0$ together generate all coordinates (clause 2), and clause 3 is [[Kolmogorov's zero–one law|073Q8Z94KG]] for the independent coordinates.\n\nNon-example: an irrational rotation $x \\mapsto x + \\alpha \\bmod 1$ is not a K-automorphism.\nA K-automorphism has positive [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] unless the space is trivial, and the rotation has entropy $0$.\n\nOn a standard probability space, $T$ is a K-automorphism if and only if every nontrivial factor of $T$ has positive entropy.\n",
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