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