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

Part of: Connections with other structures, applications Topological dynamics

Published online by Cambridge University Press:  26 June 2019

CHRISTOPHER CABEZAS Open the ORCID record for CHRISTOPHER CABEZAS [Opens in a new window] ,
SEBASTIÁN DONOSO Open the ORCID record for SEBASTIÁN DONOSO [Opens in a new window]  and
ALEJANDRO MAASS Open the ORCID record for ALEJANDRO MAASS [Opens in a new window]
CHRISTOPHER CABEZAS
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile & UMI-CNRS 2807, Av. Beauchef 851, Santiago, Chile email ccabezas@dim.uchile.cl, amaass@dim.uchile.cl
SEBASTIÁN DONOSO
Affiliation:
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Av. Lib. Bernardo O’Higgins 611, Rancagua, Chile email sebastian.donoso@uoh.cl
ALEJANDRO MAASS
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile & UMI-CNRS 2807, Av. Beauchef 851, Santiago, Chile email ccabezas@dim.uchile.cl, amaass@dim.uchile.cl
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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.

Keywords

topological dynamicsminimal systemscubespacesdirectional cubescommuting transformations

MSC classification

Primary: 54H20: Topological dynamics
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3257 - 3295
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Copyright
© Cambridge University Press, 2019

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