We study a class of Z d -substitution subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of Z d . We prove that any measurable factor and even any homomorphism associated to a matrix commuting with the expansion matrix, induces a continuous one. We also get strong restrictions on the normalizer group, proving that any endomorphism is invertible, the normalizer group is virtually generated by the shift action and the quotient of the normalizer group by the automorphisms is restricted by the digit tile of the substitution.

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.1.1. Organization of the paper. In Section 2 we give some basic material used throughout the paper. In Section 3 we present directional dynamical cubes for a Z d -system (X, T 1 , . . . , T d ), introduce the main terminology and give their general properties. Next, in Section 4 we introduce the associated (T 1 , . . . , T d )regionally proximal relation and establish some general results. In Section 5 we give a characterization of Z d -systems with the unique closing parallelepiped property in the distal case and in Section 6 we show the structure of a minimal distal system with the unique closing parallelepiped property. Section 7 is devoted to the proof of Theorem 1. Then, in Section 8 we study the sets of return times for minimal distal systems with the unique closing parallelepiped property and give a theorem that characterizes these systems using their return time sets. In Section 9 we provide a family of explicit examples of minimal distal Z d -systems with the unique closing parallelepiped property. Preliminaries2.1. Basic notions from topological dynamics. A topological dynamical system is a pair (X, G), where X is a compact metric space and G is a group of homeomorphisms of the space X into itself. We

We study a class of \mathbb{Z}^{d} -substitutive subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of \mathbb{Z}^{d} . We prove that any measurable factor map and even any homomorphism associated to a matrix commuting with the expansion matrix, induces a continuous one. We also get strong restrictions on the normalizer group, proving that any endomorphism is invertible, the normalizer group is virtually generated by the shift action and the quotient of the normalizer group by the automorphisms is restricted by the digit tile of the substitution.