On approximation of homeomorphisms of a Cantor set

Konstantin Medynets¹

¹Department of Mathematics Institute for Low Temperature Physics 47 Lenin ave. 61103 Kharkov, Ukraine

Fundamenta Mathematicae, Tome 194 (2007) no. 1, pp. 1-13

Résumé

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 $[T]$.

Détail

DOI : 10.4064/fm194-1-1

Keywords: continue study topological properties group homeo homeomorphisms cantor set respect uniform topology tau which started bezuglyi dooley kwiatkowski medynets prove set periodic homeomorphisms tau dense homeo deduce result topological group homeo tau has rokhlin property there exists homeomorphism whose conjugacy class tau dense homeo homeomorphism topological full group tau dense full group