A natural family of factors for product \({\mathbb Z}^2\)-actions (Q1013007)

\textit{E. Glasner, M. Mentzen} and \textit{A. Siemaszko} [Contemp. Math. 215, 19--42 (1998; Zbl 0916.54024)] introduced a natural family of factors for minimal flows. Here it is shown that if \((X_1,T_1)\) and \((X_2,T_2)\) are minimal flows with natural families \(\mathcal{N}_1\) and \(\mathcal{N}_2\) respectively, then \(\{A_1\otimes A_2\mid A_1\in\mathcal{N}_1,A_2\in\mathcal{N}_2\}\) is a natural family of factors for the product action of \(\mathbb Z^2\) on \(X_1\times X_2\) defined by~\((n,m):(x_1,x_2)\mapsto(T_1^nx_1,T_2^mx_2)\). Examples are found to show that the family of factors of the single transformation \(T_1\times T_2\) is strictly larger than that of the product \(\mathbb Z^2\)-action, and an example is found of a minimal distal system whose automorphism group contains no nontrivial compact subgroups.