  We survey, and extend, results on the adjoint action of the homeomorphism
group $H(X)$ on the space of surjective continuous maps, $C_s(X)$, where $X$ is
a Cantor set. We look also at the restriction of the action to various
dynamically defined subsets of $C_s(X)$, e. g. the sets of topologically
transitive maps, chain transitive maps, chain mixing maps, etc. In each case,
we consider whether there exist elements with a dense conjugacy class and if
so, what the generic elements look like.
