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arXiv · 2607.21018

One residual set of Cantor graphs with dense conjugacy classes

Zhengyu Yin
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Abstract

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 $G_\delta$ 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 $G_\delta$ subset of $\mathcal{G}raph(\mathcal{C})$. Finally, we prove that the set consisting of elements with zero topological entropy is a dense $G_\delta$ set in $\mathcal{G}raph(\mathcal{C}).$

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Zhengyu Yin. 2026-07-23. One residual set of Cantor graphs with dense conjugacy classes. https://arxiv.org/abs/2607.21018

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