论文标题
万花筒群的拓扑动力学
Topological dynamics of kaleidoscopic groups
论文作者
论文摘要
万花筒组是Duchesne，Monod和Wesolek最近引入的一类置换组。从置换组$γ$开始，万花筒结构产生了另一个置换组$ \ Mathcal {k}（γ）$，该$作用于waêski树突上（一种浓密的树枝状树状的紧凑型空间）。在本文中，我们研究了$ \ MATHCAL {K}（γ）$的拓扑动力学如何用$γ$之一来表示，当$γ$中的一个$γ$之一。通过证明用于装饰有根树的Ramsey定理，我们证明了$ \ Mathcal {k}（k}（γ）$的通用最小流量（UMF）是可分离的，如果F $γ$是寡词，而$γ$的UMF是可分离的。更一般而言，当$ \ Mathcal {k}（γ）$的UMF的UMF时，我们在适当的模型理论框架中给出了具体计算。我们的结果还为具有稳定轨道的不可渗透的UMF提供了大量示例。这些结果扩展了Kwiatkowska和Duchesne关于完整同构群体的先前工作。
Kaleidoscopic groups are a class of permutation groups recently introduced by Duchesne, Monod, and Wesolek. Starting with a permutation group $Γ$, the kaleidoscopic construction produces another permutation group $\mathcal{K}(Γ)$ which acts on a Ważewski dendrite (a densely branching tree-like compact space). In this paper, we study how the topological dynamics of $\mathcal{K}(Γ)$ can be expressed in terms of the one of $Γ$, when the group $Γ$ is transitive. By proving a Ramsey theorem for decorated rooted trees, we show that the universal minimal flow (UMF) of $\mathcal{K}(Γ)$ is metrizable iff $Γ$ is oligomorphic and the UMF of $Γ$ is metrizable. More generally, we give concrete calculations, in an appropriate model-theoretic framework, of the UMF of $\mathcal{K}(Γ)$ when the UMF of a point stabilizer $Γ_c$ has a comeager orbit. Our results also give a large class of examples of non-metrizable UMFs with a comeager orbit. These results extend previous work of Kwiatkowska and Duchesne about the full homeomorphism groups.
