Kolmogorov System

In mathematics, a Kolmogorov automorphism, ''K''-automorphism, ''K''-shift or ''K''-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law.Peter Walters, ''An Introduction to Ergodic Theory'', (1982) Springer-Verlag All Bernoulli automorphisms are ''K''-automorphisms (one says they have the ''K''-property), but not vice versa. Many ergodic dynamical systems have been shown to have the ''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

Let $(X, \mathcal, \mu)$ be a standard probability space, and let $T$ be an invertible, measure-preserving transformation. Then $T$ is called a ''K''-automorphism, ''K''-transform or ''K''-shift, if there exists a sub-sigma algebra $\mathcal\subset\mathcal$ such that the following three properties hold:

:$\mbox\mathcal\subset T\mathcal$

:$\mbox\bigvee_^\infty T^n \mathcal=\mathcal$

:$\mbox\bigcap_^\infty T^ \mathcal = \$

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

Properties

Assuming that the sigma algebra is not trivial, that is, if $\mathcal\ne\$, then $\mathcal\ne T\mathcal.$ It follows that ''K''-automorphisms are strong mixing.

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

References

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Further reading

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

Ergodic theory