JournalsjemsVol. 12, No. 1pp. 241–248

Gardens of Eden and amenability on cellular automata

  • Laurent Bartholdi

    Georg-August-Universität Göttingen, Germany
Gardens of Eden and amenability on cellular automata cover
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Abstract

We prove a converse to the “Garden-of-Eden” theorem by Ceccherini-Silberstein, Machì and Scarabotti, and to a theorem by Meyerovitch, yielding two new characterizations of amenable groups. The following are 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.

This solves in particular Conjecture 6.2 (1) in [2].

Cite this article

Laurent Bartholdi, Gardens of Eden and amenability on cellular automata. J. Eur. Math. Soc. 12 (2010), no. 1, pp. 241–248

DOI 10.4171/JEMS/196