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

Rudolph, Daniel J.

Ergodic Theory Dyn. Syst. 6, 583-599 (1986).

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

MSC:

28D05

Measure-preserving transformations

Keywords:

cocycle extensions; ergodic measure-preserving map; measure-preserving free action; weak mixing; measure algebras; Rochlin towers

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