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

Abstract

It is shown that if ${\mathcal N}$ and ${\mathcal N}'$ are natural families of factors (in the sense of [E. Glasner, M. K. Mentzen and A. Siemaszko, A natural family of factors for minimal flows, Contemp. Math. 215 (1998), 19–42]) for minimal flows $(X,T)$ and $(X',T')$, respectively, then $\{R\otimes R'\colon R\in{\mathcal N},R'\in{\mathcal N}'\}$ is a natural family of factors for the product $\mathbb{Z}^2$-action on $X\times X'$ generated by $T$ and $T'$. An example is given showing the existence of topologically disjoint minimal flows $(X,T)$ and $(X',T')$ for which the family of factors of the flow $(X\times X',T\times T')$ is strictly bigger than the family of factors of the product $\mathbb{Z}^2$-action on $X\times X'$ generated by $T$ and $T'$. There is also an example of a minimal distal system with no nontrivial compact subgroups in the group of its automorphisms.