Strictly ergodic models and topological mixing for \(Z^ 2\)-action (Q1097362)

A \({\mathbb{Z}}^ 2\) action by homeomorphisms on a compact metric space X is said to be topologically mixing if for any non-empty open sets U, V in X, there exists a finite set \(K\subset {\mathbb{Z}}^ 2\) such that for any g not in K \((gU\cap V)\neq 0.\) The author proves the following theorem: Suppose we are given a free ergodic measure preserving \({\mathbb{Z}}\) 2-action on a Lebesgue space \((Y,B,\lambda).\) Then there exists a strictly ergodic \({\mathbb{Z}}^ 2 \)action of a compact metric space X with unique invariant measure \(\nu\) that is topologically mixing and measure isomorphic to \((Y,B,\lambda).\) This extends a result of \textit{E. Lehrer} [Isr. J. Math. 57, 239-255 (1987; preceding review)], who dealt with the case \(G={\mathbb{Z}},\) however Lehrer's method does not generalize to \(G={\mathbb{Z}}^ 2\) whereas the author's method extends to any discrete amenable group.