Topologies on the group of homeomorphisms of a Cantor set

Abstract

Let ${\rm Homeo}(\Omega)$ be the group of all homeomorphisms of a Cantor set $\Omega$. We study topological properties of ${\rm Homeo}(\Omega)$ and its subsets with respect to the uniform $(\tau)$ and weak $(\tau_w)$ topologies. The classes of odometers and periodic, aperiodic, minimal, rank 1 homeomorphisms are considered and the closures of those classes in $\tau$ and $\tau_w$ are found.