View source for Skew product (ergodic theory)

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====
<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A skew product in topology, also called twisted product, is an outdated name for a [[Fibre space|fibre space]] with a structure group.

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