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

Jonathan Aaronson, Michael Lin, Benjamin Weiss

Research output: Contribution to journalArticlepeer-review

56 Scopus citations

Abstract

Let T be a Markov operator on L 1(X, Σ, m) with T*=P. We connect properties of P with properties of all products P ×Q, for Q in a certain class: (a) (Weak mixing theorem)P is ergodic and has no unimodular eigenvalues ≠ 1 ⇔ for every Q ergodic with finite invariant measure P ×Q is ergodic ⇔ for every u ∈L 1 with ∝ udm=0 and every f ∈L we have N -1Σ n ≠1/N |<u, P nf>|→0. (b) For every u ∈L 1 with ∝ udm=0 we have {norm of matrix}T nu{norm of matrix}1 → 0 ⇔ for every ergodic Q, P ×Q is ergodic. (c)P has a finite invariant measure equivalent to m ⇔ for every conservative Q, P ×Q is conservative. The recent notion of mild mixing is also treated.

Original languageEnglish
Pages (from-to)198-224
Number of pages27
JournalIsrael Journal of Mathematics
Volume33
Issue number3-4
DOIs
StatePublished - 1 Sep 1979

ASJC Scopus subject areas

  • Mathematics (all)

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