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Difference between revisions of "Measurable flow"

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''in a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631901.png" />''
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''in a measure space $(M,\mu)$''
  
A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631902.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631903.png" /> runs over the set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631904.png" />) of automorphisms of the space such that: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631905.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631907.png" />; and 2) the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631908.png" /> taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631909.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m06319010.png" /> is measurable (a measure is introduced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m06319011.png" /> as the direct product of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m06319012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m06319013.png" /> and the Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m06319014.png" />).  "Automorphisms"  here are to be understood in the strict sense of the word (and not modulo 0), that is, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m06319015.png" /> must be bijections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m06319016.png" /> 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]].
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{{MSC|37A10}}
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[[Category:Ergodic theory]]
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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 $(M,\mu)$

2010 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=11332
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article