Advertisement

Israel Journal of Mathematics

, Volume 17, Issue 2, pp 162–168 | Cite as

Ergodic automorphisms of the infinite torus are bernoulli

  • D. A. Lind
Article

Abstract

We show that ergodic algebraic automorphisms of the infinite torus are measure isomorphic to Bernoulli shifts. Using the same techniques, we also show that the existence of such an automorphism with finite entropy is equivalent to an open problem in algebraic number theory.

Keywords

Dual Group Finite Rank Free Abelian Group Algebraic Integer Torsion Group 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. R. Berg,Convolutions of invariant measures, maximal entropy, Math. Systems Theory3 (1969), 146–150.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    P. E. Blanksby and H. L. Montgomery,Algebraic integers near the unit circle, Acta Arith.18 (1971), 355–369.zbMATHMathSciNetGoogle Scholar
  3. 3.
    Rufus Bowen,Entropy for group automorphisms and homogeneous spaces, Trans. Amer. Math. Soc.153 (1971), 401–414.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    P. R. Halmos,On automorphisms of compact groups, Bull. Amer. Math. Soc.49 (1943), 619–624.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Y. Katznelson,Ergodic automorphisms of T n are Bernoulli, Israel J. Math.10 (1971), 186–195.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    L. Kronecker,Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math.53 (1857), 173–175.Google Scholar
  7. 7.
    D. H. Lehmer,Factorization of cyclotomic polynomials, Ann. of Math.34 (1933), 461–479.CrossRefMathSciNetGoogle Scholar
  8. 8.
    D. S. Ornstein,Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math.5 (1970), 339–348.CrossRefMathSciNetGoogle Scholar
  9. 9.
    D. S. Ornstein,Factors of Bernoulli shifts are Bernoulli shifts, Advances in Math.5 (1970), 349–364.CrossRefMathSciNetGoogle Scholar
  10. 10.
    V. A. Rokhlin,Metric properties of endomorphisms of compact commutative groups, Amer. Math. Soc. (2)64 (1967), 244–252.Google Scholar
  11. 11.
    C. L. Siegel,Alegbraic integers whose conjugates lie in the unit circle, Duke Math. J.11 (1944), 597–602.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press 1974

Authors and Affiliations

  • D. A. Lind
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyU. S. A.

Personalised recommendations