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## 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.

#### Autorzy

• Christopher HoffmanDepartment of Mathematics
University of Washington
Seattle, WA 98195, U.S.A.
e-mail
• Daniel RudolphDepartment of Mathematics
University of Maryland
College Park, MD 20742, U.S.A.
e-mail

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