# Weakly mixing but not mixing quasi-Markovian processes

Studia Mathematica (2000)

- Volume: 142, Issue: 3, page 235-244
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topKowalski, Zbigniew. "Weakly mixing but not mixing quasi-Markovian processes." Studia Mathematica 142.3 (2000): 235-244. <http://eudml.org/doc/216800>.

@article{Kowalski2000,

abstract = {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.},

author = {Kowalski, Zbigniew},

journal = {Studia Mathematica},

keywords = {invariant measures; quasi-Markovian process; weakly mixing; Bernoulli shift},

language = {eng},

number = {3},

pages = {235-244},

title = {Weakly mixing but not mixing quasi-Markovian processes},

url = {http://eudml.org/doc/216800},

volume = {142},

year = {2000},

}

TY - JOUR

AU - Kowalski, Zbigniew

TI - Weakly mixing but not mixing quasi-Markovian processes

JO - Studia Mathematica

PY - 2000

VL - 142

IS - 3

SP - 235

EP - 244

AB - 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.

LA - eng

KW - invariant measures; quasi-Markovian process; weakly mixing; Bernoulli shift

UR - http://eudml.org/doc/216800

ER -

## References

top- [AD] J. Aaronson and M. Denker, Local limit theorems for Gibbs-Markov maps, preprint. Zbl1039.37002
- [Ba] L. Baggett, On circle-valued cocycles of an ergodic measure-preserving transformation, Israel J. Math. 61 (1988), 29-38. Zbl0652.28004
- [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. Zbl0786.28009
- [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. Zbl0884.28010
- [Ko] Z. S. Kowalski, Quasi-Markovian transformations, Ergodic Theory Dynam. Systems 17 (1997), 885-897. Zbl0894.60063
- [KS] Z. S. Kowalski and P. Sachse, Quasi-eigenfunctions and quasi-Markovian processes, Bull. Polish Acad. Sci. Math. 47 (1999), 131-140. Zbl0933.28007
- [LY] A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. Zbl0298.28015
- [M] M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, I.H.E.S., Publ. Math. 53 (1981) 17-51. Zbl0477.58020
- [Wa] P. Walters, Ergodic Theory-Introductory Lectures, Lecture Notes in Math. 458, Springer, 1975. Zbl0299.28012

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.