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[[Category:Ergodic theory]]
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A continuous flow in [[Ergodic theory|ergodic theory]] is a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256301.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256302.png" /> ranges over the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256303.png" />) of automorphisms modulo 0 of a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256304.png" /> such that: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256305.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256306.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256307.png" />, except possibly for a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256308.png" /> belonging to an exceptional set of measure 0 (which may depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563010.png" />); in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563011.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563012.png" />; b) for each measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563013.png" /> the measure of the symmetric difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563014.png" /> depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563015.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563016.png" /> be the set of all automorphisms modulo 0 of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563017.png" /> with the usual identification: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563019.png" /> coincide almost-everywhere, then they determine the same element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563021.png" /> is endowed with the weak topology (see ), then b) means that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563022.png" /> that takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563023.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563024.png" /> is continuous.
 
A continuous flow in [[Ergodic theory|ergodic theory]] is a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256301.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256302.png" /> ranges over the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256303.png" />) of automorphisms modulo 0 of a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256304.png" /> such that: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256305.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256306.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256307.png" />, except possibly for a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256308.png" /> belonging to an exceptional set of measure 0 (which may depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c0256309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563010.png" />); in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563011.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563012.png" />; b) for each measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563013.png" /> the measure of the symmetric difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563014.png" /> depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563015.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563016.png" /> be the set of all automorphisms modulo 0 of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563017.png" /> with the usual identification: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563019.png" /> coincide almost-everywhere, then they determine the same element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563021.png" /> is endowed with the weak topology (see ), then b) means that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563022.png" /> that takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563023.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563024.png" /> is continuous.
  
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563025.png" /> is a [[Lebesgue space|Lebesgue space]], then the concept of a continuous flow is practically the same as that of a [[Measurable flow|measurable flow]]: The latter is always a continuous flow (see ), and for any continuous flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563026.png" /> there is a measurable flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563028.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563030.png" /> (see ; a related result is proved in , but see also the correction in ). The converse to any of these results depends on the character of the problem in question and the methods used.
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563025.png" /> is a [[Lebesgue space|Lebesgue space]], then the concept of a continuous flow is practically the same as that of a [[Measurable flow|measurable flow]]: The latter is always a continuous flow (see ), and for any continuous flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563026.png" /> there is a measurable flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563028.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563030.png" /> (see ; a related result is proved in , but see also the correction in ). The converse to any of these results depends on the character of the problem in question and the methods used.
  
In another sense the term "continuous flow" can be used to emphasize that the flow is considered in the context of [[Topological dynamics|topological dynamics]]. In this meaning a continuous flow is a collection of homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563031.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563035.png" />; the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563036.png" /> taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563037.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563038.png" /> is continuous.
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In another sense the term "continuous flow" can be used to emphasize that the flow is considered in the context of [[Topological dynamics|topological dynamics]]. In this meaning a continuous flow is a collection of homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563031.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563035.png" />; the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563036.png" /> taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563037.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025630/c02563038.png" /> is continuous.
  
 
To avoid confusion with 1) it is better to talk in this case of a topological flow and in the case of 1) of a metric continuity.
 
To avoid confusion with 1) it is better to talk in this case of a topological flow and in the case of 1) of a metric continuity.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,   "Lectures on ergodic theory" , Math. Soc. Japan (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"E. Hopf,   "Ergodentheorie" , Springer (1970)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"A.M. Vershik,   "Measurable realization of continuous automorphism groups of a unitary ring" ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''29''' : 1 (1965) pp. 127–136</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.W. Mackey,   "Point realizations of transformation groups" ''Illinois J. Math.'' , '''6''' : 2 (1962) pp. 327–335</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Ramsay,   "Virtual groups and group actions" ''Advances in Math.'' , '''6''' : 3 (1971) pp. 253–322</TD></TR></table>
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{|
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}}
 +
|-
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|valign="top"|{{Ref|Ho}}|| E. Hopf, "Ergodentheorie" , Springer (1970) {{MR|0024581}} {{ZBL|0185.29001}}
 +
|-
 +
|valign="top"|{{Ref|V}}|| A.M. Vershik, "Measurable realization of continuous automorphism groups of a unitary ring" ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''29''' : 1 (1965) pp. 127–136 {{MR|}} {{ZBL|0194.16302}}
 +
|-
 +
|valign="top"|{{Ref|M}}|| G.W. Mackey, "Point realizations of transformation groups" ''Illinois J. Math.'' , '''6''' : 2 (1962) pp. 327–335 {{MR|0143874}} {{ZBL|0178.17203}}
 +
|-
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|valign="top"|{{Ref|R}}|| A. Ramsay, "Virtual groups and group actions" ''Advances in Math.'' , '''6''' : 3 (1971) pp. 253–322 {{MR|0281876}} {{ZBL|0216.14902}} {{ZBL|1085.54027}}
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|}

Latest revision as of 14:44, 11 May 2012

2010 Mathematics Subject Classification: Primary: 37A10 [MSN][ZBL]

A continuous flow in ergodic theory is a family (where ranges over the real axis ) of automorphisms modulo 0 of a measure space such that: a) for any and all , except possibly for a set of belonging to an exceptional set of measure 0 (which may depend on and ); in other words, ; b) for each measurable set the measure of the symmetric difference depends continuously on . Let be the set of all automorphisms modulo 0 of the space with the usual identification: if and coincide almost-everywhere, then they determine the same element of . If is endowed with the weak topology (see ), then b) means that the mapping that takes to is continuous.

If is a Lebesgue space, then the concept of a continuous flow is practically the same as that of a measurable flow: The latter is always a continuous flow (see ), and for any continuous flow there is a measurable flow such that for all (see ; a related result is proved in , but see also the correction in ). The converse to any of these results depends on the character of the problem in question and the methods used.

In another sense the term "continuous flow" can be used to emphasize that the flow is considered in the context of topological dynamics. In this meaning a continuous flow is a collection of homeomorphisms of a topological space such that for all and ; the mapping taking to is continuous.

To avoid confusion with 1) it is better to talk in this case of a topological flow and in the case of 1) of a metric continuity.

References

[Ha] P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) MR0097489 Zbl 0073.09302
[Ho] E. Hopf, "Ergodentheorie" , Springer (1970) MR0024581 Zbl 0185.29001
[V] A.M. Vershik, "Measurable realization of continuous automorphism groups of a unitary ring" Izv. Akad. Nauk. SSSR Ser. Mat. , 29 : 1 (1965) pp. 127–136 Zbl 0194.16302
[M] G.W. Mackey, "Point realizations of transformation groups" Illinois J. Math. , 6 : 2 (1962) pp. 327–335 MR0143874 Zbl 0178.17203
[R] A. Ramsay, "Virtual groups and group actions" Advances in Math. , 6 : 3 (1971) pp. 253–322 MR0281876 Zbl 0216.14902 Zbl 1085.54027
How to Cite This Entry:
Continuous flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_flow&oldid=15434
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article