If the $[T, {\rm Id}]$ automorphism is Bernoulli then the $[T, {\rm Id}]$ endomorphism is standard
Tom 155 / 2003
Studia Mathematica 155 (2003), 195-206
MSC: 28D05, 37A20.
DOI: 10.4064/sm155-3-1
Streszczenie
For any 1-1 measure preserving map $T$ of a probability space we can form the $[T, {\rm Id}]$ and $[T, T^{-1}]$ automorphisms as well as the corresponding endomorphisms and decreasing sequence of $\sigma $-algebras. In this paper we show that if $T$ has zero entropy and the $ [T, {\rm Id}]$ automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of $\sigma $-algebras generated by the $[T, {\rm Id}]$ endomorphism is standard. We also show that if $T$ has zero entropy and the $[T^2, {\rm Id}]$ automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of $\sigma $-algebras generated by the $[T,T^{-1}]$ endomorphism is standard.
