Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products
- 145 Downloads
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 everyu ∈L 1 with∝ udm=0 and everyf ∈L ∞ we haveN −1Σ n ≠1/N |<u, P nf>|→0. (b) For everyu ∈L 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.
Unable to display preview. Download preview PDF.
- 1.J. Aaronson,On the ergodic theory of infinite measure spaces, Ph.D. thesis, The Hebrew University of Jerusalem, 1977 (in Hebrew).Google Scholar
- 5.N. Dunford and J. T. Schwartz,Linear Operators, Part I, Interscience, New York, 1958; Part II, 1963.Google Scholar
- 8.H. Furstenberg and B. Weiss,The finite multipliers of infinite ergodic transformations, to appear.Google Scholar
- 9.P. R. Halmos,Lectures on Ergodic Theory, Chelsea, 1956.Google Scholar