一个具有密集共轭类的康托尔图残余集
One residual set of Cantor graphs with dense conjugacy classes
AI总结：
本文研究了康托尔集上闭合图空间中残余同构类和共轭类的性质，证明了具有密集共轭类的元素集合在该空间中为稠密$G_\delta$子集，并探讨了零拓扑熵元素的稠密性。
AI中文摘要：
在本文中，我们考虑了所有在康托集$\mathcal{C}$上闭合图的空间$\mathcal{G}raph(\mathcal{C})$。通过赋予这个空间豪斯多夫度量并运用组合方法，我们发现单值连续映射空间中的残余同构类不再是一个$G_\delta$集，但仍然在$\mathcal{G}raph(\mathcal{C})$中稠密。我们证明了由具有密集共轭类的元素组成的集合是$\mathcal{G}raph(\mathcal{C})$中的稠密$G_\delta$子集。最后，我们证明了由具有零拓扑熵的元素组成的集合是$\mathcal{G}raph(\mathcal{C})$中的稠密$G_\delta$集。
英文摘要：
In this paper, we consider the space $\mathcal{G}raph(\mathcal{C})$ of all closed graphs on the Cantor set $\mathcal{C}$. Equipping this space with Hausdorff metric and employing combinatorial methods, we see that the residual isomorphism classes in the space of single-valued continuous maps on the Cantor set are no longer a $G_δ$ set but still dense in $\mathcal{G}raph(\mathcal{C})$, and we prove that the set consisting of elements with dense conjugacy classes is a dense $G_δ$ subset of $\mathcal{G}raph(\mathcal{C})$. Finally, we prove that the set consisting of elements with zero topological entropy is a dense $G_δ$ set in $\mathcal{G}raph(\mathcal{C}).$
