Skew product (ergodic theory)

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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[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)

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[a2] |  U. Krengel, "Ergodic theorems" , de Gruyter (1985) pp. 261