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

TL;DR

This paper provides a new proof for the reversibility and closed image properties of linear cellular automata over arbitrary groups and finite-dimensional vector spaces, and explores counterexamples in infinite-dimensional cases.

Contribution

It introduces a novel proof technique using the Mittag-Leffler lemma and investigates the limits of these properties in infinite-dimensional settings.

Findings

Bijective linear cellular automata are reversible in finite-dimensional cases.

The image of linear cellular automata is closed in the prodiscrete topology for finite-dimensional cases.

Counterexamples exist for infinite-dimensional vector spaces where properties fail.

Abstract

When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $\tau \colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $\tau \colon V^G \to V^G$ 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 $\tau_1 \colon V^G \to V^G$ which is bijective but not reversible and a linear cellular automaton $\tau_2 \colon V^G \to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology.