---
tag: 8SFZYANHNC
kind: definition
name: Kolmogorov automorphism
msc: [37A25, 37A35, 28D05]
defines: YRHJA7C7HS
uses: [0NFPJVJJHX, 556WBDAHNP, 0SV8K0TDGY, X1D1C7VH6P, 073Q8Z94KG]
sources:
  - url: https://en.wikipedia.org/wiki/Kolmogorov_automorphism
---

## Definition

Let $(X, \mathcal{B}, \mu)$ 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.
For a [[sub-σ-algebra|0SV8K0TDGY]] $\mathcal{K} \subseteq \mathcal{B}$ and $n \in \Z$, let $T^n \mathcal{K} = \set{T^n A : A \in \mathcal{K}}$.
The map $T$ is a *Kolmogorov automorphism*, or *K-automorphism*, if there is a sub-σ-algebra $\mathcal{K}$ such that:

1. $T \mathcal{K} \subseteq \mathcal{K}$.
2. The σ-algebra generated by $\bigcup_{n \ge 0} T^{-n} \mathcal{K}$ is $\mathcal{B}$, up to sets of measure $0$.
3. Every set in $\bigcap_{n \ge 0} T^{n} \mathcal{K}$ has measure $0$ or $1$.

## Remarks

Example: let $X = \set{0, 1}^{\Z}$ with the product of the measures giving weight $1/2$ to each symbol, and $T$ the left shift, $(Tx)_n = x_{n+1}$.
Let $\mathcal{K}$ be generated by the coordinates $x_m$ with $m \le 0$.
Then $T^{n} \mathcal{K}$ is generated by the $x_m$ with $m \le -n$, for every $n \in \Z$.
So 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.

Non-example: an irrational rotation $x \mapsto x + \alpha \bmod 1$ is not a K-automorphism.
A K-automorphism has positive [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] unless the space is trivial, and the rotation has entropy $0$.

On a standard probability space, $T$ is a K-automorphism if and only if every nontrivial factor of $T$ has positive entropy.