One residual set of Cantor graphs with dense conjugacy classes

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 $\mathcal{G}raph(\mathcal{C})\mathrm{.}$