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 .
Cite this article
Laurent Bartholdi, Gardens of Eden and amenability on cellular automata. J. Eur. Math. Soc. 12 (2010), no. 1, pp. 241–248DOI 10.4171/JEMS/196