Symbolic group varieties and dual surjunctivity

Xuan Kien Phung¹ 

¹Université de Montréal, Canada

Groups, geometry, and dynamics, Tome 18 (2024) no. 1, pp. 213-234

Résumé

Let G be a group. Let X be an algebraic group over an algebraically closed field K. Denote by A=X(K) the set of rational points of X. We study algebraic group cellular automata τ:AG→AG whose local defining map is induced by a homomorphism of algebraic groups XM→X, where M is a finite memory. When G is sofic and K is uncountable, we show that if τ is post-surjective, then it is weakly pre-injective. Our result extends the dual version of Gottschalk's conjecture for finite alphabets proposed by Capobianco, Kari, and Taati. When G is amenable, we prove that if τ is surjective, then it is weakly pre-injective, and conversely, if τ is pre-injective, then it is surjective. Hence, we obtain a complete answer to a question of Gromov on the Garden of Eden theorem in the case of algebraic group cellular automata.

Détail

DOI : 10.4171/ggd/749

Classification : 14-XX, 37-XX, 43-XX, 68-XX
Mots-clés : Garden of Eden theorem, sofic group, amenable group, surjunctivity, pre-injectivity, post-surjectivity, cellular automata, algebraic group