#### Vol. 306, No. 1, 2020

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

### Tullio Ceccherini-Silberstein, Michel Coornaert and Xuan Kien Phung

Vol. 306 (2020), No. 1, 31–66
##### 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.

##### Keywords
algebraic cellular automaton, complete algebraic variety, projective algebraic variety, amenable group, Krull dimension, algebraic mean dimension, preinjectivity, Garden of Eden theorem
##### Mathematical Subject Classification 2010
Primary: 14A10, 14A15, 37B10, 37B15, 43A07, 68Q80