论文标题
关于有限类型的线性偏移及其内态性
On linear shifts of finite type and their endomorphisms
论文作者
论文摘要
让$ g $为一个组，让$ a $为任意字段$ k $的有限维矢量空间。我们研究线性子缩影的有限属性$σ\ subset a^g $以及线性蜂窝自动机$τ\ colonσ\至σ$的动力学行为。我们说，如果对于每种有限维矢量空间$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ g $均为$ k $ g $，则所有线性subsifts $σ\ subset a^g $都是有限类型的。我们证明，$ g $是$ k $ - 线性马尔可夫类型的类型，并且只有当组代数$ k [g] $是单方面的noetherian。我们证明，当且仅当其限制设置（即其迭代的图像的相交）时，线性蜂窝自动机$τ$在其限制集合时才是nilpotent的。如果$ g $是无限的，有限生成的，并且$σ$是拓扑混合的，我们表明$τ$在且仅当其限制设置为有限尺寸时才nilpotent。还获得了$τ$的限制集的新表征。
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 $Σ\subset A^G$ and the dynamical behavior of linear cellular automata $τ\colon Σ\to Σ$. We say that $G$ is of $K$-linear Markov type if, for every finite-dimensional vector space $A$ over $K$, all linear subshifts $Σ\subset A^G$ 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.
