Topologies on the group of homeomorphisms of a Cantor set

Abstract

Let $Homeo(\Omega )$ be the group of all homeomorphisms of a Cantor set $\Omega$. We study topological properties of $Homeo(\Omega )$ 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.

33 pages