A complete classification of the two-point extensions of a multidimensional Bernoulli shift
The author shows that \({\mathbb{Z}}^ n\)-Bernoulli shifts are characterized by one of the following conditions: (1) finitely determined, (2) very weak Bernoulli, (3) weakly extremal, (4) extremal, (5) strongly extremal. (This extends earlier work by Ornstein and Weiss.) The main part of the paper contains a classification theorem for 2-point extensions of a \({\mathbb{Z}}^ n\)-Bernoulli shift \(T_{\nu}\) \((\nu \in {\mathbb{Z}}^ n)\) on the space X. For \(g_ i(x)\), the identity or the flip map on two points, let \[ T^ g_{\nu}(x,c)=(T_{\nu}(x),g_ 1\circ g_ 1(T_{e_ 1})\circ...\circ g_ n(T_{e_ n}^{\nu_ n- 1}...T^{\nu_ 1}_{e_ 1}(x)(\cdot)))\quad (x\in X). \] If a certain cocycle condition for g is satisfied, the \(T_{\nu}\) are commuting and these transformations are either a Bernoulli shift or one of the \(2^ n\) non weakly mixing extensions where \(g_ i=\) constant.
scientific article; zbMATH DE number 4148462