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We introduce the notions of directional dynamical cubes and directionalregionally proximal relation defined via these cubes for a minimal$\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$. We study the structural propertiesof systems that satisfy the so called unique closing parallelepiped propertyand we characterize them in several ways. In the distal case, we build themaximal factor of a $\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$ that satisfiesthis property by taking the quotient with respect to the directional regionallyproximal relation. Finally, we completely describe distal$\mathbb{Z}^d$-systems that enjoy the unique closing parallelepiped propertyand provide explicit examples.

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