Dynamical cubes and a criteria for systems having product extensions (Q906177)

\textit{B. Host} and \textit{B. Kra} [Ann. Math. (2) 161, No. 1, 397--488 (2005; Zbl 1077.37002)] introduced so-called cube structures as a tool in their work on multiple ergodic averages, and Host, Kra and Maass [\textit{B. Host} et al., Adv. Math. 224, No. 1, 103--129 (2010; Zbl 1203.37022)] later introduced cube structures in the setting of topological dynamics. Here cube structures for minimal \(\mathbb{Z}^2\) topological dynamical systems are introduced, along with a variant of the regionally proximal relation, and these are used to characterize product systems and their factors. Other topological analogues of concepts used in the study of non-conventional ergodic averages are defined, and applied to show that the automorphism group of the minimal Robinson tiling has no non-trivial elements (that is, it is generated by the shift action itself).