Expansivity and Periodicity in Algebraic Subshifts

Abstract

Abstract A d -dimensional configuration $$c:\mathbb {Z}^d\longrightarrow A$$ c : Z d ⟶ A is a coloring of the d -dimensional infinite grid by elements of a finite alphabet $$A\subseteq \mathbb {Z}$$ A ⊆ Z . The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d -variate formal power series, the annihilator is conveniently expressed as a d -variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c . A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d -dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a $$(d-1)$$ ( d - 1 ) -dimensional linear subspace $$S\subseteq \mathbb {R}^d$$ S ⊆ R d is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S . As a subshift is known to be finite if all $$(d-1)$$ ( d - 1 ) -dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of $$\mathbb {Z}^d$$ Z d by translations of a single tile.