Skip to main content
SpringerLink
Log in

A zero-entropy mixing transformation whose product with itself is loosely Bernoulli

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

Abstract

A zero-entropy mixing transformationT is constructed such thatT×T is loosely Bernoulli (LB). Previously known examples were not mixing. The construction is then generalized to yield a zero-entropy mixing transformationT such that then-fold productT × … ×T is LB for each positive integern. Furthermore, a flow with the same properties is obtained.

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. V. Chacon,Transformations with continuous spectrum, J. Math. Mech.16 (1966), 399–415.

    MATH  MathSciNet  Google Scholar 

  2. J. Feldman,New K-automorphisms and a problem of Kakutani, Israel J. Math.24 (1976), 16–38.

    Article  MATH  MathSciNet  Google Scholar 

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

  4. A. Katok,Time change, monotone equivalent and standard dynamical systems Soviet Math. Dokl.16 (1975), 986–990.

    MATH  Google Scholar 

  5. A. Katok,Monotone equivalence in ergodic theory, Math. USSR-Izv.11 (1977), 99–146.

    Article  MATH  Google Scholar 

  6. A. Katok and E. Sataev,Standardness of automorphisms of transpositions of intervals and fluxes on surfaces, Math. Notes20 (1976), 826–830.

    MATH  MathSciNet  Google Scholar 

  7. D. Lind,The isomorphism theorem for multidimensional Bernoulli flows, preprint, to appear in Israel J. Math.

  8. D. Ornstein,Ergodic Theory, Randomness and Dynamical Systems, Yale University Press, 1974.

  9. M. Ratner,Horocycle flows are loosely Bernoulli, Israel J. Math.31 (1978), 122–132.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Ratner,The Cartesian square of the horocycle flow is not loosely Bernoulli, Israel J. Math.34 (1979), 72–96.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Rothstein,Versik processes: first steps, Ph.D. thesis, University of Chicago, 1979.

  12. D. Rudolph,Non-equivalence of measure preserving transformations, Lecture Notes, The Institute for Advanced Studies, The Hebrew University of Jerusalem, 1976.

  13. E. Sataev,An invariant of monotone equivalence determining the quotient of automorphisms monotonely equivalent to a Bernoulli shift, Math. USSR-Izv.11 (1977), 147–169.

    Article  MATH  Google Scholar 

  14. L. Swanson,Loosely Bernoulli Cartesian products, Proc. Amer. Math. Soc.73 (1979), 73–78.

    Article  MATH  MathSciNet  Google Scholar 

  15. L. Swanson,An automorphism all of whose n-fold Cartesian products are loosely Bernoulli, Preliminary Report, Amer. Math. Soc.26 (1979), A-133.

    Google Scholar 

  16. B. Weiss,Equivalence of measure preserving transformations, Lecture Notes, The Institute for Advanced Studies, The Hebrew University of Jerusalem, 1976.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Marlies Gerber

Authors
  1. Marlies Gerber
    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

Gerber, M. A zero-entropy mixing transformation whose product with itself is loosely Bernoulli. Israel J. Math. 38, 1–22 (1981). https://doi.org/10.1007/BF02761843

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761843

Keywords

Search

Navigation