A Garden of Eden theorem for linear subshifts

TL;DR

This paper extends the Garden of Eden theorem to linear subshifts over amenable groups, establishing conditions under which linear cellular automata are surjective or pre-injective, with implications for dynamics on algebraic structures.

Contribution

It proves a Garden of Eden theorem for linear subshifts over amenable groups, linking surjectivity and pre-injectivity of linear cellular automata in this setting.

Findings

Linear cellular automata are surjective if and only if pre-injective on strongly irreducible linear subshifts.

Injective linear cellular automata are surjective on such subshifts when the group is countable.

The results generalize classical Garden of Eden theorems to linear and algebraic contexts.

Abstract

Let $G$ be an amenable group and let $V$ be a finite-dimensional vector space over an arbitrary field $\K$. We prove that if $X \subset V^G$ is a strongly irreducible linear subshift of finite type and $\tau \colon X \to X$ is a linear cellular automaton, then $\tau$ is surjective if and only if it is pre-injective. We also prove that if $G$ is countable and $X \subset V^G$ is a strongly irreducible linear subshift, then every injective linear cellular automaton $\tau \colon X \to X$ is surjective.