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 τ : AG AG 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 preinjectivity for algebraic cellular automata, namely ()-preinjectivity, and prove that τ is surjective if and only if it is ()-preinjective. In particular, τ 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
Authors
Tullio Ceccherini-Silberstein
Dipartimento di Ingegneria
Università del Sannio
Benevento
Italy
Michel Coornaert
CNRS, IRMS UMR 7501
Université de Strasbourg
Strasbourg
France
Xuan Kien Phung
CNRS, IRMA UMR 7501
Université de Strasbourg
Strasbourg
France