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

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 to...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2012-04, Vol.32 (2), p.691-705
Main Author: KALIKOW, STEVEN
Format: Article
Language:English
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Summary: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385711000034