The spectrum of completely positive entropy actions of countable amenable groups
This paper uses a result from the remarkable paper of \textit{D. J. Rudolph} and \textit{B. Weiss} [Ann. Math., II. Ser. 151, 1119-1150 (2000; Zbl 0957.37003)] that allows (conditional) entropy to be preserved by orbit equivalences (measurable with respect to a given algebra) in order to prove the following general result on the spectrum of actions with completely positive entropy: Any ergodic free action of a countable discrete amenable group with completely positive entropy has countable Lebesgue spectrum. This extends earlier results for amenable groups satisfying special conditions due to \textit{B. Kaminski} [Bull. Acad. Pol. Sci., Sér. Sci. Math. 29, 349-362 (1981; Zbl 0479.28016)], \textit{B. Kaminski} and \textit{P. Liardet} [Stud. Math. 108, 77-85 (1994; Zbl 0824.28011)] and \textit{V. Ya. Golodets} and \textit{S. D. Synel'shchykov} [Ergodic Theory Dyn. Syst. 22, 1-25 (2002; Zbl 1071.37004)]. A consequence of the result is a description of the zero-infinity dichotomy for the entropy of actions of countable discrete abelian groups in terms of the associated spectral measure, extending a special case due to \textit{M. Lemánczyk} [Fundam. Math. 157, 277-286 (1998; Zbl 0917.28016)].
scientific article; zbMATH DE number 1888362
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