# K-system(2)

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$\{ T ^ {t} \}$

A measurable flow ( $K$- flow) or cascade ( $K$- cascade) in a Lebesgue space such that there is a measurable partition (cf. Measurable decomposition) $\xi$ of the phase space with the following properties: a) it is increasing (formerly called invariant) with respect to $\{ T ^ {t} \}$, that is, $T ^ {t} \xi$ is a refinement $\mathop{\rm mod} 0$ of $\xi$ when $t > 0$; b) it is a two-sided generator for $\{ T ^ {t} \}$, that is, the only measurable partition $\mathop{\rm mod} 0$ that is finer $\mathop{\rm mod} 0$ than all the $T ^ {t} \xi$ is the partition into points; and c) the only measurable partition $\mathop{\rm mod} 0$ that is coarser $\mathop{\rm mod} 0$ than all the $T ^ {t} \xi$ is the trivial partition, whose only element is the whole phase space.

An automorphism of a measure space whose iterations form a $K$- cascade is called a $K$- automorphism. If $\{ T ^ {t} \}$ is a $K$- system, then all $T ^ {t}$ with $t \neq 0$ are $K$- automorphisms. Conversely, given a measurable flow or cascade $\{ T ^ {t} \}$, if just one $T ^ {t}$ is a $K$- automorphism, then $\{ T ^ {t} \}$ is a $K$- system. $K$- systems posses strong ergodic properties: positive entropy (cf. Entropy theory of a dynamical system) and ergodicity; mixing of all degrees and they have a countably-multiple Lebesgue spectrum (see Spectrum of a dynamical system; and also ).

An endomorphism of a Lebesgue space has completely-positive entropy if all its non-trivial quotient endomorphisms have positive entropy. Among these are the $K$- automorphisms (namely, they are just the automorphisms with completely-positive entropy) and also other interesting objects (exact endomorphisms; cf. Exact endomorphism). The notion of a $K$- system can be generalized in other directions: to the case of an infinite invariant measure (see , , ) and for the action of groups other than $\mathbf R$ and $\mathbf Z$( see , ).

$K$- systems are sometimes called Kolmogorov systems (flows, etc.), after their originator (see ), who used the term "quasi-regular dynamical systemquasi-regular" . This emphasizes the analogy with regular random processes (see ). If a random process $\{ X _ {t} \}$, stationary in the narrow sense of the word, is interpreted as a dynamical system, then the values of the process "in the past" define a certain increasing measurable partition $\xi$, which is the smallest with respect to which all the $X _ {t}$ with $t < 0$ are measurable. If $\xi$ has the properties b) and c) above (the "all or nothing" law), then the process is called regular. In particular, this probabilistic form presents the simplest example of a $K$- automorphism: a Bernoulli automorphism.

Given a measurable flow or cascade in a Lebesgue space, if one of the $T ^ {t}$ is isomorphic to a Bernoulli automorphism, then they all are (when $t \neq 0$). In this case the dynamical system is called Bernoullian (see ). There are $K$- systems that are not Bernoullian. $K$- systems (even Bernoullian ones) arise naturally not only in probability theory, but also in problems of an algebraic, geometric and even physical nature (see , , , , ).

How to Cite This Entry:
K-system(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K-system(2)&oldid=50264
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article