Dajani, Karma Generic results for cocycles with values in a semidirect product. (English) Zbl 0863.28013 Can. J. Math. 45, No. 3, 497-516 (1993). Let \(A\propto B\) be the semidirect product of two locally compact Hausdorff topological groups, where \(A\)’s group operation is written multiplicatively and \(B\)’s is written additively. A \(G\)-cocycle taking values in \(A\propto B\), for a group \(G\) acting nonsingularly, ergodically and freely on a Lebesgue probability space \((X,\mu)\), is a pair \((\psi,f)\), where \(\psi\) is a multiplicative \(A\)-valued cocycle and \(f\) is a \(\psi\)-valued (or twisted \(B\)-valued) cocycle (i.e., for a.e. \(x\in X\), for all \(g,g'\in G\), \(f(g'g,x)=f(g,x)+\psi(g,x)f(g',gx)\) and \(f(e,x)=0\), where \(e\) is the identity in \(A\)). The set of \(\psi\)-cocycles \(f\) is topologized by appropriately extending the topology of convergence in measure. The author then proves that if \(T\) is a nonsingular ergodic automorphism of the Lebesgue probability space, then, for a certain class of \(A\)-valued cocycles \(\psi\) which recur simultaneously with the Radon-Nikodým derivative cocycle, there is a dense \(G_\delta\) set of \(\psi\)-cocycles \(f\) whose essential range contains \(\infty\) and for which the cocycle \((\psi,f)\) is recurrent. This is done first for cocycles of the binary odometer on the Cantor set \(\{0,1\}^{\mathbb{N}}\) with a group action \(\Gamma\) which is orbit equivalent to the integer action of \(T\) on \((X,\mu)\). Reviewer: V.S.Prasad (Lowell, MA) (MR 94e:28009) Cited in 1 ReviewCited in 2 Documents MSC: 28D15 General groups of measure-preserving transformations 28D10 One-parameter continuous families of measure-preserving transformations Keywords:semidirect product; locally compact Hausdorff topological groups; ergodic automorphism; Radon-Nikodým derivative cocycle; essential range; odometer; group action PDFBibTeX XMLCite \textit{K. Dajani}, Can. J. Math. 45, No. 3, 497--516 (1993; Zbl 0863.28013) Full Text: DOI