Let $C$ be the Cantor set. I was wondering about what we know about the topological group $Homeo(C)$ of self-homeomorphisms of $C$. I am looking for example for a classification of its elements (I have very little hope for that), or at least writing it as some limit of a family of well-known groups. Do we know its homology ? I have heard it is related to the Thompson and Higman-Thompson groups.
Do you know some reference that may give me some information about this group ?