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

Artur Siemaszko

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.

Keywords


Topological dynamics; structure of factors; natural families of factors

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