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 $Σ\subset A^G$ and study endomorphisms $τ\colon Σ\to Σ$. We generalize several results for dynamical invariant sets and nilpotency of $τ$ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $τ$ 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 $Σ$ is topologically mixing, we show that $τ$ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.