Difference between revisions of "Continuous flow"
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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 | + | 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"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Hopf, "Ergodentheorie" , Springer (1970) {{MR|0024581}} {{ZBL|0185.29001}} </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 {{MR|}} {{ZBL|0194.16302}} </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 {{MR|0143874}} {{ZBL|0178.17203}} </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 {{MR|0281876}} {{ZBL|0216.14902}} {{ZBL|1085.54027}} </TD></TR></table> |
Revision as of 10:30, 27 March 2012
2020 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
| [1] | P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) MR0097489 Zbl 0073.09302 |
| [2] | E. Hopf, "Ergodentheorie" , Springer (1970) MR0024581 Zbl 0185.29001 |
| [3] | 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 |
| [4] | G.W. Mackey, "Point realizations of transformation groups" Illinois J. Math. , 6 : 2 (1962) pp. 327–335 MR0143874 Zbl 0178.17203 |
| [5] | 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=23596
Continuous flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_flow&oldid=23596
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article