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 of a measure space (and the thereby generated cascade ) such that is the direct product of two measure spaces and the action of in is related in a special way with this direct product structure. Specifically:

where is an automorphism of (the "base" ) and , with fixed, is an automorphism of (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 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.

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A skew product in topology, also called twisted product, is an outdated name for a fibre space with a structure group.