Israel Journal of Mathematics

, Volume 33, Issue 3–4, pp 198–224 | Cite as

Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products

  • Jonathan Aaronson
  • Michael Lin
  • Benjamin Weiss
Article

Abstract

LetT be a Markov operator onL 1(X, Σ,m) withT*=P. We connect properties ofP with properties of all productsP ×Q, forQ in a certain class: (a) (Weak mixing theorem)P is ergodic and has no unimodular eigenvalues ≠ 1 ⇔ for everyQ ergodic with finite invariant measureP ×Q is ergodic ⇔ for everyuL 1 with∝ udm=0 and everyfL we haveN −1Σ n ≠1/N |<u, P nf>|→0. (b) For everyuL 1 with∝ udm=0 we have ‖T nu‖1 → 0 ⇔ for every ergodicQ, P ×Q is ergodic. (c)P has a finite invariant measure equivalent tom ⇔ for every conservativeQ, P ×Q is conservative. The recent notion of mild mixing is also treated.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Aaronson,On the ergodic theory of infinite measure spaces, Ph.D. thesis, The Hebrew University of Jerusalem, 1977 (in Hebrew).Google Scholar
  2. 2.
    A. Beck,Eigenoperators of ergodic transformations, Trans. Amer. Math. Soc.94 (1960), 118–129.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Brunel,New conditions for existence of invariant measures in ergodic theory, Springer Lecture Notes in Math.160, 1970, pp. 7–17.MathSciNetGoogle Scholar
  4. 4.
    J. L. Doob,Stochastic Processes, J. Wiley, New York, 1953.MATHGoogle Scholar
  5. 5.
    N. Dunford and J. T. Schwartz,Linear Operators, Part I, Interscience, New York, 1958; Part II, 1963.Google Scholar
  6. 6.
    E. Flytzanis,Ergodicity of the Cartesian product, Trans. Amer. Math. Soc.186 (1973), 171–176.CrossRefMathSciNetGoogle Scholar
  7. 7.
    S. Foguel,The Ergodic Theory of Markov Processes, Van-Nostrand Reinhold, New York, 1969.MATHGoogle Scholar
  8. 8.
    H. Furstenberg and B. Weiss,The finite multipliers of infinite ergodic transformations, to appear.Google Scholar
  9. 9.
    P. R. Halmos,Lectures on Ergodic Theory, Chelsea, 1956.Google Scholar
  10. 10.
    T. E. Harris and H. Robbins,Ergodic theory of Markov chains with infinite invariant measures, Proc. Nat. Acad. Sci. U.S.A.39 (1953), 860–864.CrossRefMathSciNetGoogle Scholar
  11. 11.
    S. Horowitz,Markov processes on a locally compact space, Israel J. Math.7 (1969), 311–324.MATHMathSciNetGoogle Scholar
  12. 12.
    B. Jamison and S. Orey,Markov chains recurrent in the sense of Harris, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete8 (1967), 41–48.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    L. Jones and M. Lin,Ergodic theorems of weak mixing type, Proc. Amer. Math. Soc.57 (1976), 50–52.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Kakutani and W. Parry,Infinite measure preserving transformations with “mixing”, Bull. Amer. Math. Soc.69 (1963), 752–756.MATHMathSciNetGoogle Scholar
  15. 15.
    T. Kaluza,Über die Koefizienter reziproker potenzreihen, Math. Z.28 (1928), 161–170.CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    U. Krengel and L. Sucheston,On mixing in infinite measure spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete13 (1969), 150–164.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    M. Lin,Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete19 (1971), 231–242.CrossRefMathSciNetGoogle Scholar
  18. 18.
    M. Lin,Mixing of Cartesian squares of positive operators, Israel J. Math.11 (1972), 349–354.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    M. Lin,Strong ratio limit theorems for Markov processes, Ann. Math. Statist.43 (1972), 569–579.MathSciNetMATHGoogle Scholar
  20. 20.
    S. Orey,Strong ratio limit property, Bull. Amer. Math. Soc.67 (1961), 571–579.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Hebrew University 1979

Authors and Affiliations

  • Jonathan Aaronson
    • 1
    • 2
    • 3
  • Michael Lin
    • 1
    • 2
    • 3
  • Benjamin Weiss
    • 1
    • 2
    • 3
  1. 1.University of RennesRennesFrance
  2. 2.Ben Gurion University of the NegevBeer ShevaIsrael
  3. 3.The Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations