Alcuin's sequence

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:

$\text{(1) }\mathcal{𝒦}\subset T\mathcal{𝒦}$

$\text{(2) }\underset{n=0}{\overset{\mathrm{\infty }}{\bigvee }}{T}^n\mathcal{𝒦}=\mathcal{ℬ}$

$\text{(3) }\bigcap _{n=0}^{\mathrm{\infty }}{T}^{-n}\mathcal{𝒦}=\{X,\varnothing \}$

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

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Further reading

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