# Relatively perfect σ-algebras for flows

Studia Mathematica (1995)

- Volume: 114, Issue: 1, page 71-85
- ISSN: 0039-3223

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topBlanchard, F., and Kamiński, B.. "Relatively perfect σ-algebras for flows." Studia Mathematica 114.1 (1995): 71-85. <http://eudml.org/doc/216180>.

@article{Blanchard1995,

abstract = {We show that for every ergodic flow, given any factor σ-algebra ℱ, there exists a σ-algebra which is relatively perfect with respect to ℱ. Using this result and Ornstein's isomorphism theorem for flows, we give a functorial definition of the entropy of flows.},

author = {Blanchard, F., Kamiński, B.},

journal = {Studia Mathematica},

keywords = {entropy; flow; principal factor; relatively excellent σ-algebra; relatively perfect σ-algebra; factor -algebra; ergodic flow; Abramov formula},

language = {eng},

number = {1},

pages = {71-85},

title = {Relatively perfect σ-algebras for flows},

url = {http://eudml.org/doc/216180},

volume = {114},

year = {1995},

}

TY - JOUR

AU - Blanchard, F.

AU - Kamiński, B.

TI - Relatively perfect σ-algebras for flows

JO - Studia Mathematica

PY - 1995

VL - 114

IS - 1

SP - 71

EP - 85

AB - We show that for every ergodic flow, given any factor σ-algebra ℱ, there exists a σ-algebra which is relatively perfect with respect to ℱ. Using this result and Ornstein's isomorphism theorem for flows, we give a functorial definition of the entropy of flows.

LA - eng

KW - entropy; flow; principal factor; relatively excellent σ-algebra; relatively perfect σ-algebra; factor -algebra; ergodic flow; Abramov formula

UR - http://eudml.org/doc/216180

ER -

## References

top- [A1] L. M. Abramov, The entropy of a derived automorphism, Dokl. Akad. Nauk SSSR 128 (1959), 647-650 (in Russian). Zbl0094.10001
- [A2] L. M. Abramov, On the entropy of a flow, ibid., 873-875 (in Russian).
- [AK] W. Ambrose and S. Kakutani, Structure and continuity of measurable flows, Duke Math. J. 9 (1942), 25-42. Zbl0063.00065
- [B1] F. Blanchard, Partitions extrémales des flots d'entropie infinie, Z. Wahrsch. Verw. Gebiete 36 (1976), 129-136. Zbl0319.28012
- [B2] F. Blanchard, K-flots et théorème de renouvellement, ibid., 345-358. Zbl0328.60036
- [CFS] I. P. Cornfeld, S. V. Fomin and Y. G. Sinai, Ergodic Theory, Springer, 1982.
- [G1] B. M. Gurevič, Some existence conditions for K-decompositions for special flows, Trans. Moscow Math. Soc. 17 (1967), 99-126.
- [G2] B. M. Gurevič, Perfect partitions for ergodic flows, Functional Anal. Appl. 11 (1977), 20-23.
- [K1] B. Kamiński, The theory of invariant partitions for ${\mathbb{Z}}^{d}$-actions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), 349-362.
- [K2] B. Kamiński, An axiomatic definition of the entropy of a ${\mathbb{Z}}^{d}$-action on a Lebesgue space, Studia Math. 46 (1990), 135-144.
- [O1] D. S. Ornstein, Imbedding Bernoulli shifts in flows, in: Contributions to Ergodic Theory and Probability (Columbus, 1970), Lecture Notes in Math. 160, Springer, 1970, 178-218.
- [O2] D. S. Ornstein, The isomorphism theorem for Bernoulli flows, Adv. in Math. 10 (1973), 124-142. Zbl0265.28011
- [Ro] V. A. Rokhlin, An axiomatic definition of the entropy of a transformation with invariant measure, Dokl. Akad. Nauk SSSR 148 (1963), 779-781 (in Russian).
- [Ru] D. Rudolph, A two-step coding for ergodic flows, Math. Z. 150 (1976), 201-220. Zbl0325.28019

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