We study a class of $\mathbb{Z}^{d}$-substitution subshifts, including large family constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms $\mathbb{Z}^{d}$. prove that any measurable factor even homomorphism associated to matrix commuting with the expansion matrix, induced continuous one. also get strong restrictions on normalizer group, proving endomor...

Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in Zd. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of the way they are glued together in the image by a substitution. Two problems can arise when defining a substitution in such a way: it can fail to be consistent, and the p...

The-calculus is a-calculus with a local operator closely related to normal-isation procedures in classical logic and control operators in functional programming. We introduce exp, an explicit substitution calculus for , show it preserves strong normali-sation and that its simply typed version is strongly normalising. Interestingly, exp is the rst example for which the decency method of showing ...

In this paper, we introduce a C∗-algebra associated with a proper primitive substitution. We show that the C∗-algebra is simple and purely infinite and contains the associated CuntzKrieger algebra and the crossed product C∗-algebra of the corresponding Cantor minimal system. And we calculate the K-groups.