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 ${\Sigma \subset A^G}$ and study endomorphisms $\tau \colon \Sigma \to \Sigma $ . We generalize several results for dynamical invariant sets and nilpotency of $\tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau $ 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 $\Sigma $ is topologically mixing, we show that $\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.

中文翻译：

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代数 sofic 位移的自同态的不变集和幂零性

让G成为一个团体并让V是代数闭域上的代数簇K。让A表示集合K- 点V。我们引入代数 sofic 子移 ${\Sigma \子集A^G}$ 并研究自同态 $\tau \colon \Sigma \到 \Sigma $ 。我们概括了动力学不变集和幂零性的几个结果 $\tau$ 以有限字母元胞自动机而闻名。在温和的假设下，我们证明 $\tau$ 当且仅当其极限设置（即其迭代图像的交集）是单例时，它才是幂零的。如果此外G是无限的，有限生成的并且 $\西格玛$ 是拓扑混合，我们证明 $\tau$ 当且仅当其极限集由周期配置组成并且具有有限的字母值集时，才是幂零的。