### Surjective cellular automata far from the Garden of Eden

*Silvio Capobianco, Pierre Guillon, Jarkko Kari*

#### Abstract

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.

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