On the Garden of Eden theorem for endomorphisms of symbolic algebraic varieties

Let G be an amenable group and let X be an irreducible complete algebraic variety over an algebraically closed field K. Let A denote the set of K-points of X and let T : A^G ---> A^G be an algebraic cellular automaton over (G,X,K), that is, a cellular automaton over the group G and the alphabet A whose local defining map is induced by a morphism of K-algebraic varieties. We introduce a weak notion of pre-injectivity for algebraic cellular automata, namely (*)-pre-injectivity, and prove that T is surjective if and only if it is (*)-pre-injective. In particular, T has the Myhill property, i.e., is surjective whenever it is pre-injective. Our result gives a positive answer to a question raised by Gromov and yields an analogue of the classical Moore-Myhill Garden of Eden theorem.