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Non-intersecting splitting σ-algebras in a non-Bernoulli transformation

Published online by Cambridge University Press:  28 April 2011

STEVEN KALIKOW
STEVEN KALIKOW*
Affiliation:
Department of Mathematics, University of Memphis, 3725 Norriswood, Memphis, TN 38152, USA (email: skalikow@memphis.edu)
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Abstract

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Given a measure-preserving transformation T on a Lebesgue σ-algebra, a complete T-invariant sub-σ-algebra is said to split if there is another complete T-invariant sub-σ-algebra on which T is Bernoulli which is completely independent of the given sub-σ-algebra and such that the two sub-σ-algebras together generate the entire σ-algebra. It is easily shown that two splitting sub-σ-algebras with nothing in common imply T to be K. Here it is shown that T does not have to be Bernoulli by exhibiting two such non-intersecting σ-algebras for the T,T−1 transformation, negatively answering a question posed by Thouvenot in 1975.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012, pp. 691 - 705
Copyright
Copyright © Cambridge University Press 2011

References

[1]Ornstein, D. S. and Shields, P. S.. An uncountable family of K-automorphisms. Adv. Math. 10 (1973), 63–88.CrossRefGoogle Scholar
[2]Kalikow, S.. T,T −1 transformation is not loosely Bernoulli. Ann. of Math. (2) 115 (1982), 154–160.CrossRefGoogle Scholar
[3]Thouvenot, J.. Quelques propriétés des systèmes dynamiques qui se decomposent en un produit de deux systèmes dont l’un est un schema de Bernoulli. Israel J. Math. 21 (1975), 2–3.CrossRefGoogle Scholar
[4]Thouvenot, J.. Two facts concerning the transformations which satisfy the weak Pinsker property. Ergod. Th. & Dynam. Sys. 28 (2008).CrossRefGoogle Scholar
[5]Hoffman, C.. The scenery factor of the [T,T −1] transformation is not loosely Bernoulli. Proc. Amer. Math. Soc. 131(12) (2003), 3731–3735 (electronic).CrossRefGoogle Scholar
[6]Matzinger, H.. Reconstructing a 2 color scenery by observing it along a simple random walk path. Ann. Appl. Probab. 15(1B) (2005), 778–815.CrossRefGoogle Scholar
[7]Thouvenot, J.. Remarque sur les systèmes dynamiques donnés avec plusieurs facteurs. Israel J. Math. 21 (1975), 2–3.CrossRefGoogle Scholar