[en] We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal -system. 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 -system that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal -systems that enjoy the unique closing parallelepiped property and provide explicit examples.
Disciplines :
Mathematics
Author, co-author :
Cabezas Aros, Christopher ; Université de Liège - ULiège > Mathematics ; Departamento de Ingenieria Matematica and Centro de Modelamiento Matematico, Universidad de Chile AND UMI-CNRS 2807, Santiago, Chile
Donoso, Sebastian ; Universidad de Chile > Departamento de Ingeniería Matemática
Maass, Alejandro ; Departamento de Ingenieria Matematica and Centro de Modelamiento Matematico, Universidad de Chile AND UMI-CNRS 2807, Santiago, Chile
Language :
English
Title :
Directional dynamical cubes for minimal Zd-systems
Acknowledgement. This work was funded by Proyecto/Grant PIA AFB-170001 and Math-AmSud DCS 17 - MATH 01. The second author is also supported by Fondecyt Iniciación Grant 11160061. We thank Wenbo Sun for many helpful discussions. We also thank the anonymous referee for valuable remarks.
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