On injective endomorphisms of symbolic schemes
Tullio G. Ceccherini-Silberstein, Michel Coornaert, Xuan Kien Phung · Communications in Algebra · 2019
Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties, we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular automata previously introduced by the first two named authors. We extend known results to this more general setting. We also establish several new ones regarding the closed image property, surjunctivity, reversibility, and invertibility for cellular automata over algebraic varieties with coefficients in an algebraically closed field. As a byproduct, we obtain a negative answer to a question previously raised by the first two named authors on the existence of a bijective complex affine algebraic cellular automaton τ:AZ→AZ whose inverse is not algebraic.