On linear shifts of finite type and their endomorphisms

Let G be a group and let A finite-dimensional vector space over an arbitrary field K. We study finiteness properties of linear subshifts Σ⊂AG the dynamical behavior cellular automata τ:Σ→Σ. say that is K-linear Markov type if, for every K, all are finite type. show if only algebra K[G] one-sided Noetherian. prove automaton τ nilpotent its limit set, i.e., intersection images iterates, reduces to zero configuration. If infinite, finitely generated, Σ topologically mixing, we set finite-dimensional. new characterization in terms pre-injectivity also obtained.