Approximations and the spectral properties of measure-preserving group actions

Abstract

This paper presents some extensions and applications of the method of approximations of ergodic theory (see [6]). Two notions of approximation are defined which are applicable to arbitrary σ-finite-measure-preserving group actions (see §1). Building upon results of [2], [13] and [6], the speeds of such approximations are related to the questions of spectral multiplicity, spectral type and ergodicity (see §3). For the result on spectral multiplicity, there is first established a general result concerning the spectral decomposition of unitary representations (see §2). The last section is devoted to applications—chiefly to certain classes of cylinder transformations which arise in connection with irregularity of distribution (see [12]). These transformations provide examples (on infinite measure spaces) of approximations of all finite multiplicities. The method of approximations is shown to be a natural tool for the study of their spectral properties.