Gardens of Eden and amenability on cellular automata

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].