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" />'' | ''in a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063190/m0631901.png" />'' | ||
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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]]. | 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]]. | ||
Revision as of 15:47, 13 March 2012
in a measure space 
2020 Mathematics Subject Classification: Primary: 37A10 [MSN][ZBL]
A family 
(
runs over the set of real numbers 
) of automorphisms of the space such that: 1) 
for all 
, 
; and 2) the mapping 
taking 
to 
is measurable (a measure is introduced on 
as the direct product of the measure 
in 
and the Lebesgue measure in 
). "Automorphisms" here are to be understood in the strict sense of the word (and not modulo 0), that is, the 
must be bijections 
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
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