Relatively perfect σ-algebras for flows

• Autorzy: F. Blanchard , B. Kamiński
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

entropy; flow; principal factor; relatively excellent σ-algebra; relatively perfect σ-algebra

Kategorie tematyczne

• 28D15: General groups of measure-preserving transformations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1995
• Tom: 114
• Numer: 1
• Strony: 71-85

Daty

wydano

1995

otrzymano

1994-03-10

poprawiono

1994-12-30

Twórcy

autor

F. Blanchard

• Centre National, De la Recherche Scientifique, Laboratoire de Mathématiques Discrètes, UPR 9016-163, Avenue de Luminy-Case 930,13288 Marseille Cedex 9, France

autor

B. Kamiński

• Wydział Matematyki i Informatyki, Uniwersytet Mikołaja Kopernika, ul. Chopina 12/18, 87-100 Toruń, Poland

Bibliografia

• [A1] L. M. Abramov, The entropy of a derived automorphism, Dokl. Akad. Nauk SSSR 128 (1959), 647-650 (in Russian).
• [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.
• [B1] F. Blanchard, Partitions extrémales des flots d'entropie infinie, Z. Wahrsch. Verw. Gebiete 36 (1976), 129-136.
• [B2] F. Blanchard, K-flots et théorème de renouvellement, ibid., 345-358.
• [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 $ℤ^d$-actions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), 349-362.
• [K2] B. Kamiński, An axiomatic definition of the entropy of a $ℤ^d$-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.
• [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.