Israel Journal of Mathematics

, Volume 33, Issue 3–4, pp 198–224

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 onL1(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 everyuL1 with∝ udm=0 and everyfL we haveN−1Σn≠1/N|<u, Pnf>|→0. (b) For everyuL1 with∝ udm=0 we have ‖Tnu‖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.

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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

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