\({\mathbb{Z}}^ n\) and \({\mathbb{R}}^ n\) cocycle extensions and complementary algebras. (English) Zbl 0625.28008

Let T be an ergodic measure-preserving map on a Lebesgue space X, \(S=\{S_ g\}_{g\in G}^ a \)measure-preserving free action of \(G={\mathbb{Z}}^ n\) or \({\mathbb{R}}^ n\) on a Lebesgue space Y, and let \(f: X\times {\mathbb{Z}}\to G\) be a cocycle. Let \(\hat T\) be the extension of T (by S and f) defined by \(\hat T^ n(x,y)=(T^ nx,S_{f(x,n)}y)\) for \((x,y)\in X\times Y\). The author studies how weak mixing and the K-system property lift from T to \(\hat T\) if \(\hat T\) is ergodic. The technical proofs make use of measure algebras and Rochlin towers.
Reviewer: F.M.Dekking


28D05 Measure-preserving transformations
Full Text: DOI


[1] Furstenberg, Springer (1977)
[2] DOI: 10.1007/BF01692494 · Zbl 0146.28502
[3] Junco, Ergod. Th. & Dynam. Sys. none pp none– (none)
[4] Anosov, Trans. Moscow Math. Soc. 23 pp 1– (1970)
[5] DOI: 10.1112/jlms/s2-5.3.511 · Zbl 0242.28013
[6] DOI: 10.2307/2373350 · Zbl 0183.51503
[7] Kalikow, The
[8] Rudolph, J. d’anal. Math. 34 pp 36– (1978)
[9] Rudolph, null
[10] DOI: 10.1007/BF02757724 · Zbl 0305.28008
[11] DOI: 10.1070/IM1973v007n06ABEH002087 · Zbl 0294.28012
[12] DOI: 10.1007/BF01390069 · Zbl 0467.58016
[13] Rudolph, Com. Math. 26 pp 351– (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.