Kolmogorov automorphism

Summary

A Kolmogorov automorphism, or K-automorphism, is an invertible measure-preserving transformation with a sub-σ-algebra that plays the role of "the past". This σ-algebra grows under the map, it generates everything, and its remote part is trivial. K-automorphisms are exactly the systems of completely positive entropy: every nontrivial factor has positive entropy. They are mixing, indeed mixing of all orders. Every Bernoulli shift is a K-automorphism, but there are K-automorphisms that are not isomorphic to Bernoulli shifts.