If the $[T, {\rm Id}]$ automorphism is Bernoulli then the $[T, {\rm Id}]$ endomorphism is standard

Christopher Hoffman¹ ; Daniel Rudolph²

¹Department of Mathematics University of Washington Seattle, WA 98195, U.S.A.
²Department of Mathematics University of Maryland College Park, MD 20742, U.S.A.

Studia Mathematica, Tome 155 (2003) no. 3, pp. 195-206

Résumé

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.

Détail

DOI : 10.4064/sm155-3-1

Mots-clés : measure preserving map probability space form automorphisms corresponding endomorphisms decreasing sequence sigma algebras paper has zero entropy automorphism isomorphic bernoulli shift decreasing sequence sigma algebras generated endomorphism standard has zero entropy automorphism isomorphic bernoulli shift decreasing sequence sigma algebras generated endomorphism standard