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## Studia Mathematica

2000 | 142 | 3 | 235-244
Tytuł artykułu

### Weakly mixing but not mixing quasi-Markovian processes

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
Let (f,α) be the process given by an endomorphism f and by a finite partition $α = {A_i}_{i=1}^{s}$ of a Lebesgue space. Let E(f,α) be the class of densities of absolutely continuous invariant measures for skew products with the base (f,α). We say that (f,α) is quasi-Markovian if $E(f,α) ⊂ { g: ⋁_{{B_i}_{i=1}^s} supp g = ⋃ _{i=1}^{s} A_{i} × B_i}$. We show that there exists a quasi-Markovian process which is weakly mixing but not mixing. As a by-product we deduce that the set of all coboundaries which are measurable with respect to the 'chequer-wise' partition for σ × S, where σ is a Bernoulli shift and S is a weakly mixing automorphism, consists of constants.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
235-244
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-08-10
poprawiono
2000-03-23
Twórcy
autor
• Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Bibliografia
• [AD] J. Aaronson and M. Denker, Local limit theorems for Gibbs-Markov maps, preprint.
• [Ba] L. Baggett, On circle-valued cocycles of an ergodic measure-preserving transformation, Israel J. Math. 61 (1988), 29-38.
• [Bo1] C. Bose, Generalized baker's transformations, Ergodic Theory Dynam. Systems 9 (1989), 1-18.
• [Bo2] C. Bose, Mixing examples in the class of piecewise monotone and continuous maps of the unit interval, Israel J. Math. 83 (1993), 129-152.
• [Fr] N. A. Friedman, Introduction to Ergodic Theory, Van Nostrand-Reinhold, 1970.
• [K K] E. Kowalska and Z. S. Kowalski, Eigenfunctions for quasi-Markovian transformations, Bull. Polish Acad. Sci. Math. 45 (1997), 215-222.
• [Ko] Z. S. Kowalski, Quasi-Markovian transformations, Ergodic Theory Dynam. Systems 17 (1997), 885-897.
• [KS] Z. S. Kowalski and P. Sachse, Quasi-eigenfunctions and quasi-Markovian processes, Bull. Polish Acad. Sci. Math. 47 (1999), 131-140.
• [LY] A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488.
• [M] M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, I.H.E.S., Publ. Math. 53 (1981) 17-51.
• [Wa] P. Walters, Ergodic Theory-Introductory Lectures, Lecture Notes in Math. 458, Springer, 1975.
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Bibliografia
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