# 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) |

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K-system(2).

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