---
tag: YRHJA7C7HS
kind: concept
name: Kolmogorov automorphism
msc: [37A25, 37A35, 28D05]
specializes: [S3RJQ9CX2P]
sources:
  - url: https://en.wikipedia.org/wiki/Kolmogorov_automorphism
same_as:
  wikidata: Q6427427
  wikipedia: Kolmogorov automorphism
---

## Summary

A Kolmogorov automorphism, or K-automorphism, is an invertible [[measure-preserving transformation|556WBDAHNP]] 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|X1D1C7VH6P]]: every nontrivial factor has positive entropy.
They are [[mixing|S3RJQ9CX2P]], indeed mixing of all orders.
Every Bernoulli shift is a K-automorphism, but there are K-automorphisms that are not isomorphic to Bernoulli shifts.

## Explanations

### L1 human

Think of a sequence of independent coin tosses running from the infinite past to the infinite future.
The information about the past grows as time moves on, and together all the pasts describe the whole sequence.
Yet what lies in the very remote past tells you nothing about the present.
A K-automorphism is any measure-preserving system with this structure.
It has the strongest kind of randomness in its long-range behaviour, with no predictable part at all.