Israel Journal of Mathematics

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

Spectra of ergodic group actions

  • Klaus Schmidt


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.


Probability Measure Abelian Group Lebesgue Space Character Group Borel Subgroup 
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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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