Difference between revisions of "Skew product (ergodic theory)"
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− | The skew product of vectors is the same as the [[ | + | The skew product of vectors is the same as the [[pseudo-scalar product]] of vectors. |
− | A skew product in ergodic theory is an [[ | + | 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. | |
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− | In many examples of geometric and algebraic origin, the phase space | ||
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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> | + | <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> | ||
A skew product in topology, also called twisted product, is an outdated name for a [[Fibre space|fibre space]] with a structure group.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.
Skew product (ergodic theory). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skew_product_(ergodic_theory)&oldid=42474