AI 中文总结
本文证明了阿贝尔群（混合秩一Z-作用、horocycle流及Cantor空间上的混合作用）存在IC-刚性作用，并引入弱渐近对概念，建立其与拓扑极小自联结的联系。
AI 中文摘要
设$G$为连续作用于紧致度量空间$X$上的群。这自然诱导$G$在$X$的非空紧致子集空间$\mathcal{K}(X)$上的作用。在[KS26]中，Kra和Schmieding定义系统$(X,G)$为IC-刚性，若$\mathcal{K}(X)$上$G$-不变Borel概率测度的支撑之并尽可能小。他们留下了IC-刚性阿贝尔系统存在性的开放问题。我们证明了三类阿贝尔群的IC-刚性作用的存在性：混合秩一$\mathbb{Z}$-作用、horocycle流，以及$G=\bigoplus_{n=1}^{\infty}G_n$在Cantor空间上的某些混合作用，其中$(G_n)_n$为任意非平凡有限阿贝尔群序列。在证明这些结果的过程中，我们引入了弱渐近对的概念，并建立了其与拓扑极小自联结的密切联系。
英文摘要
Let $G$ be a group acting continuously on a compact metric space $X$. This naturally induces an action of $G$ on the space $\mathcal{K}(X)$ of nonempty compact subsets of $X$. In [KS26], Kra and Schmieding define a system $(X,G)$ to be IC-rigid if the union of the supports of the $G$-invariant Borel probability measures on $\mathcal{K}(X)$ is as small as possible. They leave open the question of existence of IC-rigid abelian systems. We show the existence of IC-rigid actions for three classes of abelian groups: mixing rank-one $\mathbb{Z}$-actions, horocycle flows, and certain mixing actions on a Cantor space of $G=\bigoplus_{n=1}^{\infty}G_n$, where $(G_n)_n$ is any sequence of nontrivial finite abelian groups. In the course of proving these results, we introduce the notion of weakly asymptotic pairs and establish its close connection to topological minimal self-joinings.
