论文标题
可重新分组的康托尔动力学
Cantor dynamics of renormalizable groups
论文作者
论文摘要
如果存在一个自我变为$φ\ colonγ\至γ$，其图像是适当的有限索引子组，则据说$γ$是有限的非Co-Hopfian或可重新分配的。如此适当的自我安装称为$γ$的重新归一化。在这项工作中，我们将动态系统与$γ$的重新归一化$φ$相关联。关联的Cantor Dynamical系统的判别不变$ {\ Mathcal D}_φ$是一个涂鸦组，是对动态系统不对称的度量。如果$ {\ MATHCAL D}_φ$是用于某些翻新的有限组，我们表明$γ/c_φ$几乎是nilpotent，其中$c_φ$是动作映射的内核。我们介绍了（实际上）可纠正的Cantor动作的概念，并表明与可纠正的群体相关的动作实际上是可恢复的。我们研究了几乎可重新分配的cantor作用的特性，并表明虚拟的可重量化性是连续轨道等效性的不变性。此外，可重新分配的cantor动作的判别不变是连续轨道等效的不变性。最后，可纠正的cantor作用的概念与生根树的自我复制群体的概念有关。
A group $Γ$ is said to be finitely non-co-Hopfian, or renormalizable, if there exists a self-embedding $φ\colon Γ\to Γ$ whose image is a proper subgroup of finite index. Such a proper self-embedding is called a renormalization for $Γ$. In this work, we associate a dynamical system to a renormalization $φ$ of $Γ$. The discriminant invariant ${\mathcal D}_φ$ of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If ${\mathcal D}_φ$ is a finite group for some renormalization, we show that $Γ/C_φ$ is virtually nilpotent, where $C_φ$ is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.
