Difference between revisions of "Measurable flow"
From Encyclopedia of Mathematics
(MSC|37A10 Category:Ergodic theory) |
(TeX) |
||
| Line 1: | Line 1: | ||
| − | ''in a measure space | + | {{TEX|done}} |
| + | ''in a measure space $(M,\mu)$'' | ||
{{MSC|37A10}} | {{MSC|37A10}} | ||
| Line 5: | Line 6: | ||
[[Category:Ergodic theory]] | [[Category:Ergodic theory]] | ||
| − | A family | + | A family $\{T^t\}$ ($t$ runs over the set of real numbers $\mathbf R$) of automorphisms of the space such that: 1) $T^t(T^s(x))=T^{t+s}(x)$ for all $t,s\in\mathbf R$, $x\in M$; and 2) the mapping $M\times\mathbf R\to M$ taking $(x,t)$ to $T^tx$ is measurable (a measure is introduced on $M\times\mathbf R$ as the direct product of the measure $\mu$ in $M$ and the Lebesgue measure in $\mathbf R$). "Automorphisms" here are to be understood in the strict sense of the word (and not modulo 0), that is, the $T^t$ must be bijections $M\to M$ carrying measurable sets to measurable sets of the same measure. In using automorphisms modulo 0, it turns out to be expedient to replace condition 2) by a condition of a different character, which leads to the concept of a [[Continuous flow|continuous flow]]. Measurable flows are used in [[Ergodic theory|ergodic theory]]. |
Latest revision as of 16:21, 19 August 2014
in a measure space $(M,\mu)$
2020 Mathematics Subject Classification: Primary: 37A10 [MSN][ZBL]
A family $\{T^t\}$ ($t$ runs over the set of real numbers $\mathbf R$) of automorphisms of the space such that: 1) $T^t(T^s(x))=T^{t+s}(x)$ for all $t,s\in\mathbf R$, $x\in M$; and 2) the mapping $M\times\mathbf R\to M$ taking $(x,t)$ to $T^tx$ is measurable (a measure is introduced on $M\times\mathbf R$ as the direct product of the measure $\mu$ in $M$ and the Lebesgue measure in $\mathbf R$). "Automorphisms" here are to be understood in the strict sense of the word (and not modulo 0), that is, the $T^t$ must be bijections $M\to M$ carrying measurable sets to measurable sets of the same measure. In using automorphisms modulo 0, it turns out to be expedient to replace condition 2) by a condition of a different character, which leads to the concept of a continuous flow. Measurable flows are used in ergodic theory.
How to Cite This Entry:
Measurable flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_flow&oldid=21665
Measurable flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_flow&oldid=21665
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article