Kolmogorov automorphism

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 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 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.