Relatively perfect σ-algebras for flows

F. Blanchard; B. Kamiński

Studia Mathematica (1995)

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

Abstract

top
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.

How to cite

top

Blanchard, 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
  1. [A1] L. M. Abramov, The entropy of a derived automorphism, Dokl. Akad. Nauk SSSR 128 (1959), 647-650 (in Russian). Zbl0094.10001
  2. [A2] L. M. Abramov, On the entropy of a flow, ibid., 873-875 (in Russian). 
  3. [AK] W. Ambrose and S. Kakutani, Structure and continuity of measurable flows, Duke Math. J. 9 (1942), 25-42. Zbl0063.00065
  4. [B1] F. Blanchard, Partitions extrémales des flots d'entropie infinie, Z. Wahrsch. Verw. Gebiete 36 (1976), 129-136. Zbl0319.28012
  5. [B2] F. Blanchard, K-flots et théorème de renouvellement, ibid., 345-358. Zbl0328.60036
  6. [CFS] I. P. Cornfeld, S. V. Fomin and Y. G. Sinai, Ergodic Theory, Springer, 1982. 
  7. [G1] B. M. Gurevič, Some existence conditions for K-decompositions for special flows, Trans. Moscow Math. Soc. 17 (1967), 99-126. 
  8. [G2] B. M. Gurevič, Perfect partitions for ergodic flows, Functional Anal. Appl. 11 (1977), 20-23. 
  9. [K1] B. Kamiński, The theory of invariant partitions for ℤ d -actions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), 349-362. 
  10. [K2] B. Kamiński, An axiomatic definition of the entropy of a ℤ d -action on a Lebesgue space, Studia Math. 46 (1990), 135-144. 
  11. [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. 
  12. [O2] D. S. Ornstein, The isomorphism theorem for Bernoulli flows, Adv. in Math. 10 (1973), 124-142. Zbl0265.28011
  13. [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). 
  14. [Ru] D. Rudolph, A two-step coding for ergodic flows, Math. Z. 150 (1976), 201-220. Zbl0325.28019

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.