Directional Dynamical Cubes For Minimal Z(D)-Systems

We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal Z(d)-system. X; T-1, ...,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 Z(d)-system (X, T-1, ...., T-d) that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal Z(d)-systems that enjoy the unique closing parallelepiped property and provide explicit examples.