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Israel Journal of Mathematics

, Volume 41, Issue 1–2, pp 151–153 | Cite as

Spectra of ergodic group actions

  • Klaus Schmidt
Article

Abstract

LetG be a locally compact second countable abelian group, (X, μ) aσ-finite Lebesgue space, and (g, x) →gx a non-singular, properly ergodic action ofG on (X, μ). Let furthermore Γ be the character group ofG and let Sp(G, X) ⊂ Γ denote theL -spectrum ofG on (X, μ). It has been shown in [5] that Sp(G, X) is a Borel subgroup of Γ and thatσ (Sp(G, X))<1 for every probability measureσ on Γ with lim supg→∞Re\(\hat \sigma \)(g)<1, where\(\hat \sigma \) is the Fourier transform ofσ. In this note we prove the following converse: ifσ is a probability measure on Γ with lim supg→∞Re\(\hat \sigma \)(g)<1 (g)=1 then there exists a non-singular, properly ergodic action ofG on (X, μ) withσ(Sp(G, X))=1.

Keywords

Probability Measure Abelian Group Lebesgue Space Character Group Borel Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University 1982

Authors and Affiliations

  • Klaus Schmidt
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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