Alcuin's sequence: Difference between revisions
Jump to navigation
Jump to search
en>Gandalf61 use citation templates |
en>K9re11 mNo edit summary |
||
| Line 1: | Line 1: | ||
In [[mathematics]], a '''Kolmogorov automorphism''', '''''K''-automorphism''', '''''K''-shift''' or '''''K''-system''' is an invertible, [[measure-preserving]] [[automorphism]] defined on a [[standard probability space]] that obeys [[Kolmogorov's zero-one law]].<ref>Peter Walters, ''An Introduction to Ergodic Theory'', (1982) Springer-Verlag ISBN 0-387-90599-5</ref> All [[Bernoulli automorphism]]s are ''K''-automorphisms (one says they have the '''''K''-property'''), but not vice versa. Many [[ergodic]] [[dynamical system]]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]] does not apply to ''K''-systems, and so the [[Kolmogorov entropy|entropy]] 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]]. | |||
==Formal definition== | |||
Let <math>(X, \mathcal{B}, \mu)</math> be a [[standard probability space]], and let <math>T</math> be an invertible, [[measure-preserving transformation]]. Then <math>T</math> is called a ''K''-automorphism, ''K''-transform or ''K''-shift, if there exists a sub-[[sigma algebra]] <math>\mathcal{K}\subset\mathcal{B}</math> such that the following three properties hold: | |||
:<math>\mbox{(1) }\mathcal{K}\subset T\mathcal{K}</math> | |||
:<math>\mbox{(2) }\bigvee_{n=0}^\infty T^n \mathcal{K}=\mathcal{B}</math> | |||
:<math>\mbox{(3) }\bigcap_{n=0}^\infty T^{-n} \mathcal{K} = \{X,\varnothing\}</math> | |||
Here, the symbol <math>\vee</math> is the [[join (sigma algebra)|join of sigma algebras]], while <math>\cap</math> 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 <math>\mathcal{B}\ne\{X,\varnothing\}</math>, then <math>\mathcal{K}\ne T\mathcal{K}.</math> It follows that ''K''-automorphisms are [[strong mixing]]. | |||
All [[Bernoulli automorphism]]s are ''K''-automorphisms, but not ''vice-versa''. | |||
==References== | |||
{{reflist}} | |||
==Further reading== | |||
* Christopher Hoffman, "[http://www.ams.org/journals/tran/1999-351-10/S0002-9947-99-02446-0/ A K counterexample machine]", ''Trans. Amer. Math. Soc.'' '''351''' (1999), pp 4263–4280. | |||
[[Category:Ergodic theory]] | |||
Latest revision as of 18:00, 27 January 2014
In mathematics, a Kolmogorov automorphism, K-automorphism, K-shift or K-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero-one law.[1] All Bernoulli automorphisms are K-automorphisms (one says they have the K-property), but not vice versa. Many ergodic dynamical systems 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 does not apply to K-systems, and so the entropy 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.
Formal definition
Let be a standard probability space, and let be an invertible, measure-preserving transformation. Then is called a K-automorphism, K-transform or K-shift, if there exists a sub-sigma algebra such that the following three properties hold:
Here, the symbol is the join of sigma algebras, while 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 , then It follows that K-automorphisms are strong mixing.
All Bernoulli automorphisms are K-automorphisms, but not vice-versa.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
Further reading
- Christopher Hoffman, "A K counterexample machine", Trans. Amer. Math. Soc. 351 (1999), pp 4263–4280.
- ↑ Peter Walters, An Introduction to Ergodic Theory, (1982) Springer-Verlag ISBN 0-387-90599-5