On extremal and perfect σ-algebras for ℤ d -actions on a Lebesgue space

B. Kamiński; Z. Kowalski; P. Liardet

Studia Mathematica (1997)

  • Volume: 124, Issue: 2, page 173-178
  • ISSN: 0039-3223

Abstract

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We show that for every positive integer d there exists a ℤ d -action and an extremal σ-algebra of it which is not perfect.

How to cite

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Kamiński, B., Kowalski, Z., and Liardet, P.. "On extremal and perfect σ-algebras for $ℤ^{d}$-actions on a Lebesgue space." Studia Mathematica 124.2 (1997): 173-178. <http://eudml.org/doc/216406>.

@article{Kamiński1997,
abstract = {We show that for every positive integer d there exists a $ℤ^d$-action and an extremal σ-algebra of it which is not perfect.},
author = {Kamiński, B., Kowalski, Z., Liardet, P.},
journal = {Studia Mathematica},
keywords = {extremal -algebra; perfect -algebra; entropy; Pinsker -algebra; ergodic transformation; finite generator},
language = {eng},
number = {2},
pages = {173-178},
title = {On extremal and perfect σ-algebras for $ℤ^\{d\}$-actions on a Lebesgue space},
url = {http://eudml.org/doc/216406},
volume = {124},
year = {1997},
}

TY - JOUR
AU - Kamiński, B.
AU - Kowalski, Z.
AU - Liardet, P.
TI - On extremal and perfect σ-algebras for $ℤ^{d}$-actions on a Lebesgue space
JO - Studia Mathematica
PY - 1997
VL - 124
IS - 2
SP - 173
EP - 178
AB - We show that for every positive integer d there exists a $ℤ^d$-action and an extremal σ-algebra of it which is not perfect.
LA - eng
KW - extremal -algebra; perfect -algebra; entropy; Pinsker -algebra; ergodic transformation; finite generator
UR - http://eudml.org/doc/216406
ER -

References

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  1. [1] M. Binkowska and B. Kamiński, Entropy increase for ℤ d -actions on a Lebesgue space, Israel J. Math. 88 (1994), 307-318. Zbl0826.28008
  2. [2] J. P. Conze, Entropie d'un groupe abélien de transformations, Z. Wahrsch. Verw. Gebiete 25 (1972), 11-30. Zbl0261.28015
  3. [3] S. Goldstein and O. Penrose, A non-equilibrium entropy for dynamical systems, J. Statist. Phys. 24 (1981), 325-343. Zbl0516.70021
  4. [4] S. A. Kalikow, T , T - 1 transformation is not loosely Bernoulli, Ann. of Math. 115 (1982), 393-409. Zbl0523.28018
  5. [5] B. Kamiński, Mixing properties of two-dimensional dynamical systems with completely positive entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 453-463. Zbl0469.28013
  6. [6] B. Kamiński, The theory of invariant partitions for ℤ d -actions, ibid. 29 (1981), 349-362. Zbl0479.28016
  7. [7] B. Kamiński, A representation theorem for perfect partitions for ℤ 2 -actions with finite entropy, Colloq. Math. 56 (1988), 121-127. Zbl0685.28009
  8. [8] B. Kamiński, Decreasing nets of σ-algebras and their applications to ergodic theory, Tôhoku Math. J. 43 (1991), 263-274. Zbl0752.28010
  9. [9] Z. S. Kowalski, A generalized skew product, Studia Math. 87 (1987), 215-222. Zbl0651.28013
  10. [10] W. Krieger, On generators in exhaustive σ-algebras of ergodic measure-preserving transformations, Z. Wahrsch. Verw. Gebiete 20 (1971), 75-82. Zbl0214.07201
  11. [11] I. Meilijson, Mixing properties of a class of skew-products, Israel J. Math. 19 (1974), 266-270. Zbl0305.28008
  12. [12] V. A. Rokhlin, Lectures on the entropy theory of measure-preserving transformations, Uspekhi Mat. Nauk 22 (5) (1967), 3-56 (in Russian). 
  13. [13] T. Shimano, An invariant of systems in the ergodic theory, Tôhoku Math. J. 30 (1978), 337-350. Zbl0394.28009
  14. [14] T. Shimano, The multiplicity of helices for a regularly increasing sequence of σ-fields, ibid. 36 (1984), 141-148. Zbl0551.28021
  15. [15] T. Shimano, On helices for Kolmogorov system with two indices, Math. J. Toyama Univ. 14 (1991), 213-226. Zbl0768.60031
  16. [16] P. Walters, Some results on the classification of non-invertible measure preserving transformations, in: Lecture Notes in Math. 318, Springer, 1973, 266-276. 

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