迭代Minkowski和、Horoball和南北动力学
Iterated Minkowski sums, horoballs and north-south dynamics
摘要：
给定群$Γ$的有限生成集$A$，我们将映射$W \mapsto WA$作为一个拓扑动力系统--$Γ$的子集的紧度量化空间的连续自映射来研究.如果集合$A$生成$Γ$作为半群并且包含恒等式，则恰好有两个不动点，其中一个是吸引的。这支持了最初的印象，即这个地图的动态是相当微不足道的。事实上，至少当$Γ= \mathbb{Z}^d$和$A \subseteq \mathbb{Z}^d$一个有限的正生成集包含自然可逆扩展的地图$W \mapsto W+A$总是拓扑共轭的唯一的“南北”动态康托集。与此相反，我们表明，各种自然的“几何”属性的的有限生成群$(Γ,A)$可以恢复从这个地图的动态，特别是，生长类型和顺从性的$Γ$。当$Γ= \mathbb{Z}^d$时，证明了生成集$A$的凸包的体积也是拓扑共轭不变量.我们的研究介绍，利用和发展的一个特定的凸性结构的子集的组$Γ$，涉及到一个新的概念，我们称之为一个集的庇护壳。我们还将这项研究的结构horoballs在有限生成群，专注于阿贝尔的情况下。
英文摘要：
Given a finite generating set $A$ for a group $Γ$, we study the map $W \mapsto WA$ as a topological dynamical system -- a continuous self-map of the compact metrizable space of subsets of $Γ$. If the set $A$ generates $Γ$ as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when $Γ= \mathbb{Z}^d$ and $A \subseteq \mathbb{Z}^d$ a finite positively generating set containing the natural invertible extension of the map $W \mapsto W+A$ is always topologically conjugate to the unique "north-south" dynamics on the Cantor set. In contrast to this, we show that various natural "geometric" properties of the finitely generated group $(Γ,A)$ can be recovered from the dynamics of this map, in particular, the growth type and amenability of $Γ$. When $Γ= \mathbb{Z}^d$, we show that the volume of the convex hull of the generating set $A$ is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group $Γ$, related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
