Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

TL;DR

This paper extends classical results on cellular automata to algebraic sofic shifts over groups, characterizing nilpotency of endomorphisms via their limit sets and mixing properties.

Contribution

It introduces algebraic sofic subshifts and generalizes results on invariant sets and nilpotency for endomorphisms beyond finite alphabets.

Findings

Nilpotency of endomorphisms characterized by singleton limit sets.

In infinite, finitely generated groups with mixing shifts, nilpotency linked to periodic configurations.

Established equivalence between nilpotency and finite alphabet values in the limit set.

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.