YOURNA.COM

Kolmogorov automorphism

Jump to: navigation, search

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. 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{B}, \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{K}\subset\mathcal{B} such that the following three properties hold:

\mbox{(1) }\mathcal{K}\subset T\mathcal{K}
\mbox{(2) }\bigvee_{n=0}^\infty T^n \mathcal{K}=\mathcal{B}
\mbox{(3) }\bigcap_{n=0}^\infty T^{-n} \mathcal{K} = \{X,\varnothing\}

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{B}\ne\{X,\varnothing\}, then \mathcal{K}\ne T\mathcal{K}. It follows that K-automorphisms are strong mixing.

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

References

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

Further reading

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

You may also be interested in:

Daniel Carvalho
Sydney Morning Herald
Steven Blum
Mises Institute
Bentota
Toxorhynchites rajah
Paul Bibeault
Tobolsk Kremlin
Jacobo Timerman
Vehicle Area Network
Contact     Terms of Use     Privacy Policy
All Rights Reserved 2012-2014
0.044538