On linear shifts of finite type and their endomorphisms

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.