K-system(2)

$\{ T ^ {t} \}$

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 [2]).

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 [6], [7], [11]) and for the action of groups other than $\mathbf R$ and $\mathbf Z$( see [8]–[10], [12]).

$K$- 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 [4]). 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 [5]). 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 [2], [3], [5], [13], [14]).

References

[1a]	A.N. Kolmogorov, "A new metric invariant of transitive dynamical systems and of endomorphisms of Lebesgue spaces" Dokl. Akad. Nauk SSSR , 119 : 5 (1958) pp. 861–864 (In Russian)
[1b]	A.N. Kolmogorov, "On the entropy as a metric invariant of automorphisms" Dokl. Akad. Nauk SSSR , 124 : 4 (1959) pp. 754–755 (In Russian)
[2]	I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)
[3]	A.B. Katok, Ya.G. Sinai, A.M. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" J. Soviet Math. , 7 (1977) pp. 974–1065 Itogi Nauk i Tekhn. Mat. Anal. , 13 (1975) pp. 129–262
[4]	Yu.A. Rozanov, "Stationary random processes" , Holden-Day (1967) (Translated from Russian)
[5]	D. Ornstein, "Ergodic theory, randomness, and dynamical systems" , Yale Univ. Press (1974)
[6]	W. Parry, "Ergodic and spectral analysis of certain infinite measure preserving transformations" Proc. Amer. Math. Soc. , 16 : 5 (1965) pp. 960–966
[7]	J.K. Dugdale, "Kolmogorov automorphisms in $\sigma$-finite measure spaces" Publ. Math. Debrecen , 14 (1967) pp. 79–81
[8]	J.P. Conze, "Entropie d'un groupe abélien de transformations" Z. Wahrsch. Verw. Gebiete , 25 : 1 (1972) pp. 11–30
[9]	R.M. Burton, "An asymptotic definition of $K$-groups of automorphisms" Z. Wahrsch. Verw. Gebiete , 47 : 2 (1979) pp. 207–212
[10]	S. Dani, "Kolmogorov automorphisms on homogeneous spaces" Amer. J. Math. , 98 : 1 (1976) pp. 119–163
[11]	U. Krengel, L. Sucheston, "Note on shift-invariant sets" Ann. Math. Statist. , 40 : 2 (1969) pp. 694–696
[12]	B. Kamiński, "A note on $K$-systems" Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. , 26 : 2 (1978) pp. 95–97
[13]	Ya.G. Sinai, et al., "Dynamical systems" , 4 , Springer (1988) (Translated from Russian)
[14]	N.F.G. Martin, J.W. England, "Mathematical theory of entropy" , Addison-Wesley (1981)