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Difference between revisions of "Skew product (ergodic theory)"

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The skew product of vectors is the same as the [[Pseudo-scalar product|pseudo-scalar product]] of vectors.
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The skew product of vectors is the same as the [[pseudo-scalar product]] of vectors.
  
A skew product in ergodic theory is an [[Automorphism|automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857001.png" /> of a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857002.png" /> (and the thereby generated cascade <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857003.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857004.png" /> is the direct product of two measure spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857005.png" /> and the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857006.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857007.png" /> is related in a special way with this direct product structure. Specifically:
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A skew product in ergodic theory is an [[automorphism]] $T$ of a [[measure space]] $E$ (and the thereby generated [[cascade]] $(T^n)$) such that $E$ is the direct product of two measure spaces $X \times Y$ and the action of $T$ in $E$ is related in a special way with this direct product structure. Specifically:
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$$
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T(x,y) = (R(x), S(x,y))
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$$
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where $R$ is an automorphism of $X$ (the  "base" ) and $S(x,{\cdot})$, with $x \in X$ fixed, is an automorphism of $Y$ (the  "fibre" ). The concept of a skew product carries over directly to the case of endomorphisms, flows and more general groups and semi-groups of transformations.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857008.png" /></td> </tr></table>
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In many examples of geometric and algebraic origin, the phase space $E$ is naturally defined as a skew product in the topological sense (a [[fibre space]]). However, this does not necessitate a generalization of the above definition of a skew product, since from the metric (in the sense of measure theory) point of view there is no distinction between direct products and skew products of spaces.
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s0857009.png" /> is an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s08570010.png" /> (the  "base" ) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s08570011.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s08570012.png" /> fixed, is an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s08570013.png" /> (the  "fibre" ). The concept of a skew product carries over directly to the case of endomorphisms, flows and more general groups and semi-groups of transformations.
 
 
 
In many examples of geometric and algebraic origin, the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085700/s08570014.png" /> is naturally defined as a skew product in the topological sense (a [[Fibre space|fibre space]]). However, this does not necessitate a generalization of the above definition of a skew product, since from the metric (in the sense of measure theory) point of view there is no distinction between direct products and skew products of spaces.
 
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.P. [I.P. Kornfel'd] Cornfel'd,  S.V. Fomin,  Ya.G. Sinai,  "Ergodic theory" , Springer  (1982)  pp. Chapt. 10, §1  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  U. Krengel,  "Ergodic theorems" , de Gruyter  (1985)  pp. 261</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  I.P. [I.P. Kornfel'd] Cornfel'd,  S.V. Fomin,  Ya.G. Sinai,  "Ergodic theory" , Springer  (1982)  pp. Chapt. 10, §1  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  U. Krengel,  "Ergodic theorems" , de Gruyter  (1985)  pp. 261</TD></TR>
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</table>
  
 
A skew product in topology, also called twisted product, is an outdated name for a [[Fibre space|fibre space]] with a structure group.
 
A skew product in topology, also called twisted product, is an outdated name for a [[Fibre space|fibre space]] with a structure group.
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Revision as of 21:17, 10 December 2017

The skew product of vectors is the same as the pseudo-scalar product of vectors.

A skew product in ergodic theory is an automorphism $T$ of a measure space $E$ (and the thereby generated cascade $(T^n)$) such that $E$ is the direct product of two measure spaces $X \times Y$ and the action of $T$ in $E$ is related in a special way with this direct product structure. Specifically: $$ T(x,y) = (R(x), S(x,y)) $$ where $R$ is an automorphism of $X$ (the "base" ) and $S(x,{\cdot})$, with $x \in X$ fixed, is an automorphism of $Y$ (the "fibre" ). The concept of a skew product carries over directly to the case of endomorphisms, flows and more general groups and semi-groups of transformations.

In many examples of geometric and algebraic origin, the phase space $E$ is naturally defined as a skew product in the topological sense (a fibre space). However, this does not necessitate a generalization of the above definition of a skew product, since from the metric (in the sense of measure theory) point of view there is no distinction between direct products and skew products of spaces.


Comments

References

[a1] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) pp. Chapt. 10, §1 (Translated from Russian)
[a2] U. Krengel, "Ergodic theorems" , de Gruyter (1985) pp. 261

A skew product in topology, also called twisted product, is an outdated name for a fibre space with a structure group.

How to Cite This Entry:
Skew product (ergodic theory). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skew_product_(ergodic_theory)&oldid=42474
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article