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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

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.

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

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