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Kolmogorov automorphism

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Measure preserving automorphism

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 , B , μ ) {\displaystyle (X,{\mathcal {B}},\mu )} be a standard probability space, and let T {\displaystyle T} be an invertible, measure-preserving transformation. Then T {\displaystyle T} is called a K-automorphism, K-transform or K-shift, if there exists a sub-sigma algebra K ⊂ B {\displaystyle {\mathcal {K}}\subset {\mathcal {B}}} such that the following three properties hold:

(1)  K ⊂ T K {\displaystyle {\mbox{(1) }}{\mathcal {K}}\subset T{\mathcal {K}}}
(2)  ⋁ n = 0 ∞ T n K = B {\displaystyle {\mbox{(2) }}\bigvee _{n=0}^{\infty }T^{n}{\mathcal {K}}={\mathcal {B}}}
(3)  ⋂ n = 0 ∞ T − n K = { X , ∅ } {\displaystyle {\mbox{(3) }}\bigcap _{n=0}^{\infty }T^{-n}{\mathcal {K}}=\{X,\varnothing \}}

Here, the symbol ∨ {\displaystyle \vee } is the join of sigma algebras, while ∩ {\displaystyle \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 B ≠ { X , ∅ } {\displaystyle {\mathcal {B}}\neq \{X,\varnothing \}} , then K ≠ T K . {\displaystyle {\mathcal {K}}\neq T{\mathcal {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, i.e. mappings T {\displaystyle T} for which ⋂ n = 0 ∞ T − n M {\displaystyle \bigcap _{n=0}^{\infty }T^{-n}{\mathcal {M}}} consists of measure-zero sets or their complements, where M {\displaystyle {\mathcal {M}}} is the sigma-algebra of measureable sets,.

References

  1. Peter Walters, An Introduction to Ergodic Theory, (1982) Springer-Verlag ISBN 0-387-90599-5
  2. V. A. Rohlin, Exact endomorphisms of Lebesgue spaces, Amer. Math. Soc. Transl., Series 2, 39 (1964), 1-36.

Further reading

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