On linear shifts of finite type and their endomorphisms

Abstract

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