A set $C\subset\mathbb{R}$ is called a Cantor set if it is compact, perfect, and totally disconnected. We may form the group $\operatorname{Homeo}(C):=\{f:C\to C~|~ f$ is a homeomorphism$\}$ under function composition.
Are there any known nontrivial generating sets for this group? This paper gives a rather nice description of a topological generating set, but I have been unable to find any literature for on-the-nose generation.