On the reversibility and the closed image property of linear cellular automata

Abstract

When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $τ\colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $τ\colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$ is a non-periodic group and $V$ is an infinite-dimensional vector space, then there exist a linear cellular automaton $τ_1 \colon V^G \to V^G$ which is bijective but not reversible and a linear cellular automaton $τ_2 \colon V^G \to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology.