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

TL;DR

This paper introduces directional dynamical cubes and explores their role in understanding the structure of minimal $ ext{Z}^d$-systems, especially those with the unique closing parallelepiped property, providing new characterizations and examples.

Contribution

It defines directional dynamical cubes and the associated proximal relation, characterizes systems with the unique closing parallelepiped property, and describes the structure of distal systems with this property.

Findings

Characterization of systems with the unique closing parallelepiped property.

Construction of the maximal factor satisfying the property via quotienting.

Complete description and examples of distal systems with the property.

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.