论文标题
迭代的Minkowski总和，Horoballs和North-South Dynamics
Iterated Minkowski sums, horoballs and north-south dynamics
论文作者
论文摘要
鉴于有限的生成集$ a $对于组$γ$，我们研究地图$ w \ mapsto wa $作为拓扑动力学系统 - $γ$的子集的紧凑型迁移空间的连续自动图。如果集合$ a $作为半群产生$γ$并包含该身份，则精确有两个固定点，其中一个吸引了一个固定点。这支持了最初的印象，即该地图的动力学相当微不足道。 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.与此相比，我们表明，有限生成的组$（γ，a）$的各种自然“几何”特性可以从该地图的动力学中恢复，尤其是$γ$的增长类型和不舒张性。当$γ= \ Mathbb {z}^d $时，我们表明生成设置$ a $的凸壳的体积也是拓扑结合的不变性。我们的研究介绍，利用并开发了该组$γ$子集的一定凸结构，这与我们称为套装的庇护所的新概念有关。我们还将这项研究与有限生成的群体中的Horoballs结构联系起来，重点是阿贝尔案。
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.
