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

DOI: http://dx.doi.org/10.12775/TMNA.2008.045

#### 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, < i> A natural family of factors for

minimal flows< /i> , Contemp. Math. < b> 215< /b> (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.

families of factors (in the sense of [E. Glasner, M. K. Mentzen and A. Siemaszko, < i> A natural family of factors for

minimal flows< /i> , Contemp. Math. < b> 215< /b> (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.

#### Keywords

Topological dynamics; structure of factors; natural families of factors

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