Bernoullis are standard when entropy is not an obstruction (Q1275686)

In this paper, actions of the group \(G= \sum^\infty_{n=1} \mathbb{Z}/r_n\mathbb{Z}\), \(r_n\neq 1\), are studied. Let \(T_g: (X,{\mathcal F},\mu)\to (X,{\mathcal F},\mu)\) be a free measure-preserving ergodic action of this group defined on a Lebesgue probability space. Denote by \(T\) and \(S\) two such actions \((S_g: Y\to Y)\), then \(T\) is said to be \(r\)-equivalent to \(S\) (where \(r= (r_1,r_2,\dots)\)), if there exists a one-one measure-preserving transformation \(\phi: X\to Y\) such that \(\{\phi(T_g(x)\}_{g\in G_i}= \{S_g(\phi x)\}_{g\in G_i}\) (where \(G_i= \sum^i_{n= 1} \mathbb{Z}/r_n\mathbb{Z}\)). This equivalence relation is stronger than orbit equivalence and arises from the classification of decreasing sequences of \(\sigma\)-algebras, initiated by \textit{A. M. Vershik} [Funkts. Anal. Prilozh. 2, No. 1, 17-21 (1968; Zbl 0186.20203)]. The aim of the paper is to classify actions of this group \(G\). It is shown under certain conditions that all Bernoulli \(G\) actions are \(r\)-equivalent to a natural \(G\)-action called the translation action. \textit{J. W. Kammeyer} and \textit{D. J. Rudolph} (preprint) introduced the notion of zero \(r\) entropy and \(r\) finitely determined, and it is shown that these actions have both of these properties.