Kolmogorov automorphism explained

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.^[1] 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, l{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 l{K}\subset l{B} such that the following three properties hold:(1) l{K}\subset T l{K} (2) v ee ^{inf ty} _{n =0} T ^{n} l{K}= l{B} (3) c ap ^{inf ty} _{n =0} T ^{-n} l{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

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

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

Kolmogorov automorphisms are precisely the natural extensions of exact endomorphisms,^[2] i.e. mappings

T for which c ap ^{inf ty} _{n =0} T ^{-n} l{M} consists of measure-zero sets or their complements, where l{M} is the sigma-algebra of measureable sets,.

Further reading

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

Notes and References

1. Peter Walters, An Introduction to Ergodic Theory, (1982) Springer-Verlag
2. V. A. Rohlin, Exact endomorphisms of Lebesgue spaces, Amer. Math. Soc. Transl., Series 2, 39 (1964), 1-36.