Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Let G be a group and let V be an algebraic variety over an algebraically closed field K . Let A denote the set of K -points of V . We introduce algebraic sofic subshifts Σ ⊂ AG and study endomorphisms τ Σ → Σ . We generalize several results for dynamical invariant sets and nilpotency of τ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that τ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and Σ is topologically mixing, we show that τ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.