A Garden of Eden theorem for linear subshifts
Tullio G. Ceccherini-Silberstein, Michel Coornaert · Ergodic Theory and Dynamical Systems · 2011
Abstract Let G be an amenable group and let V be a finite-dimensional vector space over an arbitrary field 𝕂. We prove that if X⊂VG is a strongly irreducible linear subshift of finite type and τ:X→X is a linear cellular automaton, then τ is surjective if and only if it is pre-injective. We also prove that if G is countable and X⊂VG is a strongly irreducible linear subshift, then every injective linear cellular automaton τ:X→X is surjective.