Israel Journal of Mathematics

, Volume 61, Issue 2, pp 219–224 | Cite as

On measures simultaneously 2- and 3-invariant

  • Russell Lyons


Furstenberg has conjectured that the only continuous probability measure on the circleT=R/Z which is invariant under bothx ↦ 2x andx ↦ 3x is Lebesgue measure. We shall show that under additional hypotheses, this is true. We also discuss related conjectures and theorems.


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  1. 1.
    Gavin Brown, William Moran and Charles E. M. Pearce,Riesz products and normal numbers, J. London Math. Soc. (2)32 (1985), 12–18.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Harry Furstenberg,Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, Math. Systems Theory1 (1967), 1–49.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Russell Lyons,Mixing and asymptotic distribution modulo 1, Ergodic Theory Dynamical Systems (to appear).Google Scholar
  4. 4.
    Russell Lyons,The local structure of some measure-algebra homomorphisms, in preparation.Google Scholar
  5. 5.
    C. E. M. Pearce and M. S. Keane,On normal numbers, J. Austral. Math. Soc. (A)32 (1982), 79–87.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Wolfgang Schmidt,On normal numbers, Pacific J. Math.10 (1960), 661–672.MATHMathSciNetGoogle Scholar
  7. 7.
    H. G. Senge and E. G. Straus,PV-numbers and sets of multiplicity, Per. Math. Hung.3 (1–2) (1973), 93–100.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    C. L. Stewart,On the representation of an integer in two different bases, J. Reine Angew. Math.319 (1980), 63–72.MATHMathSciNetGoogle Scholar

Copyright information

© The Weizmann Sciene Press of Israel 1988

Authors and Affiliations

  • Russell Lyons
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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