1 results
Directional dynamical cubes for minimal
$\mathbb{Z}^{d}$-systems
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 26 June 2019, pp. 3257-3295
- Print publication:
- December 2020
-
- Article
- Export citation
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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.
