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].