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Abstract

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