论文标题
代数sofic偏移的内态不变集和努力
Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts
论文作者
论文摘要
让$ g $为一个组，让$ v $是代数封闭的$ k $的代数品种。令$ a $表示$ v $的$ k $点。我们介绍代数sofic子缩影$σ\ subset a^g $，并研究内态$τ\ colonσ\至σ$。我们概括了几个结果，即$τ$的动态不变集和尼尔氏率，这些结果以有限的字母蜂窝蜂窝自动机而闻名。在温和的假设下，我们证明$τ$在且仅当其限制设置（即其迭代图像的相交）是单身人士时才是nilpotent。如果此外，$ g $是无限的，有限生成的并且$σ$是拓扑混合的，我们表明$τ$在且仅当其限制集由定期配置组成并且具有有限的字母值时才nilpotent。
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 $Σ\subset A^G$ and study endomorphisms $τ\colon Σ\to Σ$. 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, i.e., 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.
