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Generic results for cocycles with values in a semidirect product. (English) Zbl 0863.28013

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 \((X,\mu)\).

MSC:

28D15 General groups of measure-preserving transformations
28D10 One-parameter continuous families of measure-preserving transformations
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