Weakly mixing but not mixing quasi-Markovian processes

Zbigniew Kowalski

Studia Mathematica (2000)

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

Abstract

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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 s u p p 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.

How to cite

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Kowalski, 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

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  8. [KS] Z. S. Kowalski and P. Sachse, Quasi-eigenfunctions and quasi-Markovian processes, Bull. Polish Acad. Sci. Math. 47 (1999), 131-140. Zbl0933.28007
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