Generically there is but one self homeomorphism of the Cantor set
  We describe a self-homeomorphism $R$ of the Cantor set $X$ and then show that its conjugacy class in the Polish group $H(X)$ of all homeomorphisms of $X$ forms a dense $G_\delta$ subset of $H(X)$. We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class. 
Ethan Akin, Eli Glasner and Benjamin Weiss