Gardens of Eden and amenability on cellular automata
type: Item
instance of: scholarly article
title: Gardens of Eden and amenability on cellular automata
zbMATH Open document ID: 1185.37020
DOI: 10.4171/JEMS/196
author: Laurent Bartholdi
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2010-01-28T00:00:00Z
review text: Two new characterizations of amenable groups were yielded by proof a converse to the Garden-of-Eden theorem by Ceccherini-Silberstein, Machi and Scarabotti and to a theorem of \textit{T.~Meywrovitch} [``Finite entropy for multidimensional cellular automata'', Ergodic Theory Dynam. Systems 28, 1243--1260 (2008; Zbl 1152.37009)]. In particular, the following is proved to be equivalent: the group \(G\) is amenable, all cellular automata living on \(G\) that admit mutually erasable patterns also admit gardens of Eden; all cellular automata living on \(G\) that do not preserve Bernoulli measure admit gardens of Eden.
Mathematics Subject Classification ID: 37B15
Mathematics Subject Classification ID: 68Q80
Mathematics Subject Classification ID: 43A07
zbMATH DE Number: 5663855
zbMATH Keywords: amenable groups
zbMATH Keywords: Garden-of-Eden theorem
zbMATH Keywords: multidimensional cellular automata
zbMATH Keywords: endomorphisms
reviewed by: Denis Sidorov
MaRDI profile type: Publication
cites work: Q5636767
cites work: Amenable groups and cellular automata
cites work: The Garden of Eden theorem for linear cellular automata
cites work: Endomorphisms of irreducible subshifts of finite type
cites work: A conjecture concerning a limit of non-Cayley graphs
cites work: The amenability and non-amenability of skew fields
cites work: Endomorphisms of symbolic algebraic varieties
cites work: Endomorphisms and automorphisms of the shift dynamical system
cites work: Q3485031
cites work: Garden of Eden Configurations for Cellular Automata on Cayley Graphs of Groups
cites work: Finite entropy for multidimensional cellular automata
cites work: Transversal theory. An account of some aspects of combinatorial mathematics
cites work: Q5333602
cites work: Shorter Note: The Converse of Moore's Garden-of-Eden Theorem
cites work: Q3806816
description: scientific article; zbMATH DE number 5663855