Skip to main content
SpringerLink
Log in

NewK-automorphisms and a problem of Kakutani

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

A property is introduced, for 1-1 measure-preserving transformations of probability spaces, calledloose Bernoulliness (LB), which is invariant under taking factors, inducing, and tower-building. It amounts to replacing, in Ornstein’s definition ofvery weak Bernoulli, the Hamming distance on strings by a coarser metric. The main result is the construction of a transformationT 0 which is ergodic and of entropy 0 butnot LB. On the other hand, any irrational rotationis LB. Consequently, the equivalence relation generated by inducing and tower-building (which I callKakutani equivalence, and the Russians callmonotone equivalence) has at least two distinct equivalence classes among the ergodic entropy zero transformations. A similar situation exists for ergodic positive-entropy transformations: on the one hand, any Bernoulli shift is LB, while on the other hand a non LBK-automorphism\(\hat T_0 \) can be made by skewingT 0 over a Bernoulli base.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Adler and P. Shields,Skew products of Bernoulli shifts with rotations II, Israel J. Math.19 (1974), 228–236.

    Article  MathSciNet  Google Scholar 

  2. P. Billingsly,Ergodic Theory and Information, Wiley, New York, 1965.

    Google Scholar 

  3. N. Friedman,Introduction to Ergodic Theory, Van Nostrand Reinhold New York, 1970.

    MATH  Google Scholar 

  4. B. M. Gurevič,Some existence conditions for K-decompositions for special flows, Trans. Moscow Math. Soc.17 (1967), 99–128.

    Google Scholar 

  5. S. Kakutani,Induced measure-preserving transformations, Proc. Imp. Acad. Tokyo19 (1943), 635–641.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. Krieger,On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc.199 (1970), 453–464;Erratum, ibid.168 (1972), 549.

    Article  MathSciNet  Google Scholar 

  7. I. Meilijson,Mixing properties of a class of skew products, Israel J. Math.19 (1974), 266–270.

    Article  MathSciNet  Google Scholar 

  8. D. Ornstein,Ergodic Theory, Randomness and Dynamical Systems, Yale University Press, New Haven, 1970.

    Google Scholar 

  9. D. Ornstein and L. Sucheston,An operator theorem on L 1 convergence to zero with applications to Markov kernels, Ann. Math. Statist.41 (1970), 1631–1639.

    MathSciNet  MATH  Google Scholar 

  10. D. Ornstein and B. Weiss,Finitely determined implies very weak Bernoulli, Israel J. Math.17 (1974), 95–104.

    Article  MathSciNet  Google Scholar 

  11. L. Swanson,Induced automorphisms and factors in ergodic theory, Ph.D. thesis, U. C. Berkeley, 1975 (to be submitted to Advances in Math., under titleInduced automorphisms and Bernoulli shifts).

  12. S. Ulam,Some combinatorial problems studied experimentally on computing machines, inApplications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, 94720, Berkeley, California, U.S.A.

    J. Feldman

Authors
  1. J. Feldman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Feldman, J. NewK-automorphisms and a problem of Kakutani. Israel J. Math. 24, 16–38 (1976). https://doi.org/10.1007/BF02761426

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761426

Keywords

Search

Navigation