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We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal
. 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
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.
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