Gardens of Eden and amenability on cellular automata

We prove a converse to the "Garden-of-Eden" theorem by Ceccherini-Silberstein, Machi 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].