Kolmogorov automorphism

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.

Kolmogorov automorphisms are precisely the natural extensions of exact endomorphisms, i.e. mappings T for which \bigcap_{n=0}^\infty T^{-n} \mathcal{M} consists of measure-zero sets or their complements, where \mathcal{M} is the sigma-algebra of measureable sets,.

References

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.