Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Abstract

Let $G$ be a group and let $V$ be an algebraic variety over an algebraically closed field $K$. Let $A$ denote the set of $K$-points of $V$. We introduce algebraic sofic subshifts $\Sigma \subset A^G$ and study endomorphisms $\tau \colon \Sigma \to \Sigma$. We generalize several results for dynamical invariant sets and nilpotency of $\tau$ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau$ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, is a singleton. If moreover $G$ is infinite, finitely generated and $\Sigma$ is topologically mixing, we show that $\tau$ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.