# Kolmogorov automorphism

In mathematics, a **Kolmogorov automorphism**, ** K-automorphism**,

**or**

*K*-shift**is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law.**

*K*-system^{[1]}All Bernoulli automorphisms are

*K*-automorphisms (one says they have the

**), but not vice versa. Many ergodic dynamical systems have been shown to have the**

*K*-property*K*-property, although more recent research has shown that many of these are in fact Bernoulli automorphisms.

Although the definition of the *K*-property seems reasonably general, it stands in sharp distinction to the Bernoulli automorphism. In particular, the Ornstein isomorphism theorem does not apply to *K*-systems, and so the entropy is not sufficient to classify such systems – there exist uncountably many non-isomorphic *K*-systems with the same entropy. In essence, the collection of *K*-systems is large, messy and uncategorized; whereas the *B*-automorphisms are 'completely' described by Ornstein theory.

## Formal definition[edit]

Let be a standard probability space, and let be an invertible, measure-preserving transformation. Then is called a *K*-automorphism, *K*-transform or *K*-shift, if there exists a sub-sigma algebra such that the following three properties hold:

Here, the symbol is the join of sigma algebras, while is set intersection. The equality should be understood as holding almost everywhere, that is, differing at most on a set of measure zero.

## Properties[edit]

Assuming that the sigma algebra is not trivial, that is, if , then It follows that *K*-automorphisms are strong mixing.

All Bernoulli automorphisms are *K*-automorphisms, but not *vice versa*.

## References[edit]

**^**Peter Walters,*An Introduction to Ergodic Theory*, (1982) Springer-Verlag ISBN 0-387-90599-5

## Further reading[edit]

- Christopher Hoffman, "A K counterexample machine",
*Trans. Amer. Math. Soc.***351**(1999), pp 4263–4280.