Abstract

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 $\tau :A^G\to A^G$ be an algebraic cellular automaton over $\left(G,X,K\right)$, 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 preinjectivity for algebraic cellular automata, namely $\left(\ast \right)$-preinjectivity, and prove that $\tau$ is surjective if and only if it is $\left(\ast \right)$-preinjective. In particular, $\tau$ has the Myhill property, i.e., is surjective whenever it is preinjective. Our result gives a positive answer to a question raised by Gromov  (J. Eur. Math. Soc. 1:2 (1999), 109–197) and yields an analogue of the classical Moore–Myhill Garden of Eden theorem.

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Mathematical Subject Classification 2010

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