Abstract

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.