A garden of Eden theorem for linear subshifts
type: Item
instance of: scholarly article
zbMATH Open document ID: 1254.37013
DOI: 10.1017/S0143385710000921
author: Michel Coornaert
author: Tullio Ceccherini-Silberstein
published in: Ergodic Theory and Dynamical Systems
publication date: 2012-05-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.3957
Mathematics Subject Classification ID: 37B15
zbMATH DE Number: 6038225
zbMATH Keywords: garden of Eden theorem
zbMATH Keywords: amenable group
zbMATH Keywords: linear subshift
zbMATH Keywords: linear cellular automata
MaRDI profile type: Publication
arXiv ID: 1002.3957
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OpenAlex ID: W2964141740
title: A garden of Eden theorem for linear subshifts
review text: The Moore-Myhill `garden of Eden' theorem is extended to the setting of linear cellular automata over linear shifts of finite type. For an amenable group \(G\) and any field \(K\) it is shown that if \(X\subset (K^d)^G\) for some \(d\geq1\) is a strongly irreducible linear shift of finite type then a linear cellular automaton \(X\to X\) is surjective if and only if it is pre-injective. Some related examples and results are also given, and it is shown that if \(G\) is countable and \(X\subset(K^d)^G\) is a strongly irreducible linear subshift, then any injective linear cellular automaton \(X\to X\) is surjective.
reviewed by: Thomas B. Ward
description: scientific article; zbMATH DE number 6038225