The mean ratio set for \(ax+b\) valued cocycles
Let \(X=\prod_{i=1}^n {\mathbf Z}_{l(i)}\), where \({\mathbf Z}_{l(i)}\) is the group of the integers modulo \(l(i)\in\mathbf N\), and let \(\Gamma=\bigoplus_{i=1}^n{\mathbf Z}_{l(i)}\) be acting on \(X\). Let \(\mu\) be a non-singular \(G\)-measure on \(X\), that is a non-singular probability measure on \(X\) and a family of ``compatible and normalized'' measurable functions \(\{G_k\}=G\) on \(X\) such that the Radon-Nikodým derivative of \(\mu\) can be expressed in terms of \(\Gamma\)-translates of functions in \(G\) [\textit{G. Brown} and \textit{A. H. Dooley}, Ergodic Theory Dyn. Syst. 11, No. 2, 279-307 (1991; Zbl 0739.58032)]). The first result in the paper is a structure theorem for cocycles with values in the \(ax+b\) group, which generalizes theorems of \textit{V. Ya. Golodets} [Russ. Math. Surv. 24, No. 4, 1-63 (1969; Zbl 0202.13301)] and \textit{K. R. Parthasarathy} and \textit{K. Schmidt} [Monatsh. Math. 84, 37-48 (1977; Zbl 0384.28017)]. Then, for this kind of cocycles, they introduce the notion of mean ratio set which is a generalization of the notion of essential range and in fact coincides with it in the case of a real-valued cocycle and a measure-preserving action. Although not a cohomological invariant, the mean ratio set has the property that if two cocycles are equivalent via integrable functions, then the corresponding mean ratio sets are conjugate as closed subgroups of \([0,\infty]\times[\infty,\infty]\). Lastly, the authors classify the closed subgroups of the group \(ax+b\) and for each of them they construct a cocycle whose mean ratio set is the given subgroup, hence providing a classification of \(ax+b\)-valued cocycles in an \(L^1\)-equivariant way.
scientific article; zbMATH DE number 1002133
scientific article