Kolmogorov Automorphism
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Kolmogorov automorphism, ''K''-automorphism, ''K''-shift or ''K''-system is an invertible,
measure-preserving In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
defined on a
standard probability space In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin i ...
that obeys
Kolmogorov's zero–one law In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a ''tail event of independent σ-algebras'', will either almost surely happen or almost sure ...
.Peter Walters, ''An Introduction to Ergodic Theory'', (1982) Springer-Verlag All
Bernoulli automorphism In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical sy ...
s are ''K''-automorphisms (one says they have the ''K''-property), but not vice versa. Many
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s 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 In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphism , isomorphic. The result, given by Donald Ornstein in 1970, is ...
does not apply to ''K''-systems, and so the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
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 In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. The result, given by Donald Ornstein in 1970, is important ...
.


Formal definition

Let (X, \mathcal, \mu) be a
standard probability space In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin i ...
, and let T be an invertible,
measure-preserving transformation In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ca ...
. Then T is called a ''K''-automorphism, ''K''-transform or ''K''-shift, if there exists a sub-
sigma algebra Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as ...
\mathcal\subset\mathcal such that the following three properties hold: :\mbox\mathcal\subset T\mathcal :\mbox\bigvee_^\infty T^n \mathcal=\mathcal :\mbox\bigcap_^\infty T^ \mathcal = \ Here, the symbol \vee is the join of sigma algebras, while \cap is
set intersection In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is writt ...
. The equality should be understood as holding
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, that is, differing at most on a set of
measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null s ...
.


Properties

Assuming that the sigma algebra is not trivial, that is, if \mathcal\ne\, then \mathcal\ne T\mathcal. It follows that ''K''-automorphisms are strong mixing. All
Bernoulli automorphism In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical sy ...
s are ''K''-automorphisms, but not ''vice versa''.


References

{{reflist


Further reading

* Christopher Hoffman,
A K counterexample machine
, ''Trans. Amer. Math. Soc.'' 351 (1999), pp 4263–4280. Ergodic theory