On approximation of homeomorphisms of a Cantor set

Abstract

We continue the study of topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology τ , which was started by Bezuglyi, Dooley, Kwiatkowski and Medynets. We prove that the set of periodic homeomorphisms is τ -dense in Homeo(X) and deduce from this result that the topological group (Homeo(X), τ) has the Rokhlin property, i.e., there exists a homeomorphism whose conjugacy class is τ -dense in Homeo(X). We also show that for any homeomorphism T the topological full group [[T ]] is τ -dense in the full group [T ].