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A skew product in topology, also called twisted product, is an outdated name for a fibre space with a structure group.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