On linear shifts of finite type and their endomorphisms

TL;DR

This paper investigates the properties of linear subshifts and cellular automata over groups, establishing conditions for finiteness, nilpotency, and characterizations of limit sets in the context of algebraic and dynamical systems.

Contribution

It characterizes when all linear subshifts are of finite type via group algebra properties and provides new insights into the nilpotency and limit sets of linear cellular automata.

Findings

G is of K-linear Markov type iff K[G] is one-sided Noetherian.

A linear cellular automaton is nilpotent iff its limit set is zero or finite-dimensional.

New characterization of the limit set of cellular automata via pre-injectivity.

Abstract

Let $G$ be a group and let $A$ be a finite-dimensional vector space over an arbitrary field $K$. We study finiteness properties of linear subshifts $\Sigma \subset A^G$ and the dynamical behavior of linear cellular automata $\tau \colon \Sigma \to \Sigma$. We say that $G$ is of $K$-linear Markov type if, for every finite-dimensional vector space $A$ over $K$, all linear subshifts $\Sigma \subset A^G$ are of finite type. We show that $G$ is of $K$-linear Markov type if and only if the group algebra $K[G]$ is one-sided Noetherian. We prove that a linear cellular automaton $\tau$ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, reduces to the zero configuration. If $G$ is infinite, finitely generated, and $\Sigma$ is topologically mixing, we show that $\tau$ is nilpotent if and only if its limit set is finite-dimensional. A new characterization of the limit set of $\tau$ in terms of pre-injectivity is also obtained.