Authors: Silvio Capobianco ¹; Pierre Guillon ^2,3; Jarkko Kari ²

• 1 Institute of Cybernetics [Tallinn]
• 2 Department of Mathematics
• 3 Institut de mathématiques de Luminy

One of the first and most famous results of cellular automata theory, Moore's Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure.