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The structure of skew products with ergodic group automorphisms

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Abstract

We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii’s basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.

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References

  1. H. Anzai,Ergodic skew-product transformations on the torus, Osaka Math. J.3 (1951), 83–99.

    MATH  MathSciNet  Google Scholar 

  2. Nobou Aoki and Haruo Totoki,Ergodic automorphisms of T ∞ are Bernoulli transformations, Publ. Res. Inst. Math. Sci.10 (1975), 535–544.

    MathSciNet  Google Scholar 

  3. K. Berg,Convolution of invariant measures, maximal entropy, Math. Systems Theory3 (1969), 146–151.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. E. Blanksby and H. L. Montgomery,Algebraic integers near the unit circle, Acta Arith.18 (1971), 355–369.

    MATH  MathSciNet  Google Scholar 

  5. N. Bourbaki,General Topology, Addison-Wesley, Reading, 1966.

    Google Scholar 

  6. Rufus Bowen,Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc.153 (1971), 401–414.

    Article  MATH  MathSciNet  Google Scholar 

  7. Rufus Bowen and Brian Marcus,Unique ergodicity for horocycle foliations, Israel J. Math.26 (1977), 43–67.

    MATH  MathSciNet  Google Scholar 

  8. Hsin Chu,Some results on affine transformations of compact groups, to appear.

  9. J. P. Conze,Propriétés ergodiques des extensions de systèmes dynamiques par des groupes compact, Thèse, Université de Paris VI, 1965.

  10. L. Fuchs,Infinite Abelian Groups I, Academic Press, New York, 1970.

    MATH  Google Scholar 

  11. P. R. Halmos,On automorphisms of compact groups, Bull. Amer. Math. Soc.49 (1943), 619–624.

    Article  MATH  MathSciNet  Google Scholar 

  12. K. Hoffman and R. Kunze,Linear Algebra, Prentice-Hall, Englewood Cliffs N.J., 1961.

    Google Scholar 

  13. Y. Katznelson,Ergodic automorphisms of T n are Bernoulli shifts, Israel J. Math.10 (1971), 186–195.

    MATH  MathSciNet  Google Scholar 

  14. L. Kronecker,Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math.53 (1857), 173–175.

    MATH  Google Scholar 

  15. L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley, New York, 1974.

    MATH  Google Scholar 

  16. D. H. Lehmer,Factorization of cyclotomic polynomials, Ann. of Math.34 (1933), 461–479.

    Article  MathSciNet  Google Scholar 

  17. D. A. Lind,Translation invariant sigma algebras on groups, Proc. Amer. Math. Soc.42 (1974), 218–221.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. A. Lind,Ergodic automorphisms of the infinite torus are Bernoulli, Israel J. Math.17 (1974), 162–168.

    MATH  MathSciNet  Google Scholar 

  19. S. MacLane and G. Birkhoff,Algebra, Macmillan, New York, 1967.

    MATH  Google Scholar 

  20. Jean-Claude Marcuard, Thèses, Université de Dijon, 1975.

  21. G. Miles and R. K. Thomas,The breakdown of automorphisms of compact groups, to appear.

  22. G. Miles and R. K. Thomas,On the polynomial uniformity of translations of the n-torus, to appear.

  23. G. Miles and R. K. Thomas,Generalized torus automorphisms are Bernoullian, to appear.

  24. D. S. Ornstein,Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math.5 (1970), 339–348.

    Article  MathSciNet  Google Scholar 

  25. D. S. Ornstein,Factors of Bernoulli shifts are Bernoulli shifts, Advances in Math.5 (1970), 349–364.

    Article  MathSciNet  Google Scholar 

  26. D. S. Ornstein,Factors of Bernoulli shifts, Israel J. Math.21 (1975), 145–153.

    MATH  MathSciNet  Google Scholar 

  27. D. S. Ornstein,Ergodic Theory, Randomness, and Dynamical Systems, Yale University Press, New Haven, 1974.

    MATH  Google Scholar 

  28. D. S. Ornstein and Benjamin Weiss,Finitely determined implies very weak Bernoulli, Israel J. Math.17 (1974), 94–104.

    Article  MATH  MathSciNet  Google Scholar 

  29. W. Parry,Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math.91 (1969), 757–771.

    Article  MATH  MathSciNet  Google Scholar 

  30. W. Parry,Entropy and Generators in Ergodic Theory, Benjamin, 1969.

  31. V. A. Rohlin,Metric properties of endomorphisms of compact commutative groups, Amer. Math. Soc. Transl. (3),64 (1967), 244–252.

    Google Scholar 

  32. C. L. Siegel,Algebraic integers whose conjugates lie in the unit circle, Duke Math. J.11 (1944), 597–602.

    Article  MATH  MathSciNet  Google Scholar 

  33. R. K. Thomas,On affine transformations of locally compact groups, J. London Math. Soc. (2)4 (1972), 599–610.

    Article  MATH  MathSciNet  Google Scholar 

  34. R. K. Thomas,The addition theorem for the entropy of transformations of G-spaces, Trans. Amer. Math. Soc.160 (1971), 119–130.

    Article  MATH  MathSciNet  Google Scholar 

  35. R. K. Thomas,Metric properties of transformations of G-spaces, Trans. Amer. Math. Soc.160 (1971), 103–117.

    Article  MATH  MathSciNet  Google Scholar 

  36. Jean-Paul Thouvenot,Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli, Israel J. Math.21 (1975), 177–207.

    MATH  MathSciNet  Google Scholar 

  37. Jean-Paul Thouvenot,Remarques sur les systèmes dynamiques donnés avec plusiers facteurs, Israel J. Math.21 (1975), 215–232.

    MATH  MathSciNet  Google Scholar 

  38. Benjamin Weiss,The isomorphism problem in ergodic theory, Bull. Amer. Math. Soc.78 (1972), 668–684.

    MATH  MathSciNet  Google Scholar 

  39. B. Weiss and D. Ornstein,Geodesic flows are Bernoullian, Israel J. Math.14 (1973), 184–198.

    MATH  MathSciNet  Google Scholar 

  40. S. A. Yuzvinskii,Calculation of the entropy of a group endomorphism, Sibirsk. Mat. Ž.8 (1967), 230–239 (Russian); Siberian Math. J.8 (1963), 172–178 (English).

    MathSciNet  Google Scholar 

  41. S. A. Yuzvinskii,Metric properties of endomorphisms of compact groups, Izv. Akad. Nauk SSSR Ser. Mat.29 (1965), 1295–1328 (Russian); Amer. Math. Soc. Transl. (2),66 (1968), 63–98.

    MathSciNet  Google Scholar 

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  1. The Institute for Advanced Studies, The Hebrew University of Jerusalem, Jerusalem, Israel

    D. A. Lind

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Lind, D.A. The structure of skew products with ergodic group automorphisms. Israel J. Math. 28, 205–248 (1977). https://doi.org/10.1007/BF02759810

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  • DOI: https://doi.org/10.1007/BF02759810

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