Certain Words, Tilings, Their Non-Periodicity, and Substitutions of High Dimension

Abstract

We consider certain partition of a lattice into c (1<C≤∞) parts, and its characteristic word W(A; c) on the lattice, which are determined uniquely by c and a given square matrix A of size s × s with integer entries belonging to a class (Bdd). We give a definition of substitutions for s-dimensional words in a general situation, and then, define special ones, and automata of dimension s together with their conjugates. We show that the word W (A; c) can be described by iterations of a substitution for A belonging to a subclass of (Bdd). The hermitian canonical forms of integer matrices play an important role in some cases for finding substitutions. We give a theorem which discloses a p-adic link with hermitian canonical forms. We give two definitions for periodicity: Ψ-periodicity, arid ∑-periodicity, both for words and for tilings of dimension s. The non-∑-periodicity strongly requires non-periodicity, so that it excludes some non-periodic words in the usual sense. We also consider certain Voronoi tessellations corning from a word W(A; c). We show that some of the words W (A; c), and the tessellations are non-∑-periodic.