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arXiv · 1809.09509

Directional dynamical cubes for minimal $\mathbb{Z}^{d}$-systems

Christopher Cabezas·Sebastián Donoso·Alejandro Maass
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Abstract

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.

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Christopher Cabezas, Sebastián Donoso, Alejandro Maass. 2018-09-25. Directional dynamical cubes for minimal $\mathbb{Z}^{d}$-systems. https://arxiv.org/abs/1809.09509

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