Directional dynamical cubes for minimal -systems

• Published in: Ergodic theory and dynamical systems Vol. 40; no. 12; pp. 3257 - 3295
• Main Authors: CABEZAS, CHRISTOPHER, DONOSO, SEBASTIÁN, MAASS, ALEJANDRO
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 01.12.2020
• Subjects: Cubes; Parallelepipeds; Quotients
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2019.33

We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal $\mathbb{Z}^{d}$ -system $(X,T_{1},\ldots ,T_{d})$ . We study the structural properties of systems that satisfy the so-called unique closing parallelepiped property and we characterize them in several ways. In the distal case, we build the maximal factor of a $\mathbb{Z}^{d}$ -system $(X,T_{1},\ldots ,T_{d})$ that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal $\mathbb{Z}^{d}$ -systems that enjoy the unique closing parallelepiped property and provide explicit examples.