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

DOI: 10.2140/pjm.2020.306.31

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

Milestones

Received: 29 March 2018

Accepted: 20 December 2019

Published: 14 June 2020