A class of multipliers for \({\mathcal W}^\perp\)
The notion of disjoint ergodic transformations was introduced by \textit{H. Furstenberg} in his paper [``Disjointness in ergodic theory, minimal sets and diophantine approximation'', Math. Syst. Theory 1, 1--49 (1967; Zbl 0146.28502)]. It expresses the fact that the transformations are highly non-similar. In particular, two disjoint transformations have no nontrivial common factors. In the paper under review, the authors are interested in a question that arises from Furstenberg's paper, which is the description of the class \(W^{\perp}\) of ergodic transformations that are disjoint from every weakly mixing transformation. The authors denote by \(M(W^{\perp})\) the set of ergodic transformations whose all ergodic joinings with any element of \(W^{\perp}\) are also in \(W^{\perp}\). Let \(T\) be an ergodic measure preserving transformation of a standard probability space \((X,\mathcal{B}_X,\mu)\), let \(S=(S_g)_{g\in G}\) be measure preserving action of a noncompact locally compact second countable group \(G\) on a standard probability space \((Y,\mathcal{B}_Y,\nu)\) and let \(\phi:X\to G\) be a Borel map. Then \(T_{\phi}\) and \(T_{\phi, S}\) are the measure preserving transformations of the product spaces \((X\times G, \mu\times \lambda_{G})\) and \((X\times Y, \mu\times \nu)\) respectively, defined by \(T_{\phi}(x,g)=(Tx,\phi(x)g)\) and \(T_{\phi, S}(x,g)=(Tx,S_{\phi(x)}y)\), where \(\lambda_G\) is the left Haar measure on \(G\). The main result of this paper is the following: Theorem: Let \(T\) be an ergodic transformation with pure point spectrum and let \(G\) be an amenable locally compact second countable group wit no nontrivial compact normal subgroup. Assume that \(S\) is a mildly mixing action of \(G\). If \(\phi:X\to G\) is an ergodic cocycle of \(T\) for which \(e(T_{\phi})\) is countable, then \(T_{\phi, S}\) belongs to \(M(W^{\perp})\).
scientific article; zbMATH DE number 5003408
scientific article