Non-intersecting splitting algebras in a non-Bernoulli transformation

Abstract

Given a measure preserving transformation $T$ on a Lebesgue $\sigma$ algebra, a complete $T$ invariant sub $\sigma$ algebra is said to split if there is another complete $T$ invariant sub $\sigma$ algebra on which $T$ is Bernoulli which is completely independent of the given sub $\sigma$ algebra and such that the two sub $\sigma$ algebras together generate the entire $\sigma$ algebra. It is easily shown that two splitting sub $\sigma$ 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 $\sigma$ algebras for the $T,T^{-1}$ transformation, negatively answering a question posed by Thouvenot in 1975.

arXiv commentThis paper has been accepted to and will appear in the special edition of Ergodic Theory and Dynamical systems in honor of Dan Rudolph