### Topologies on the group of homeomorphisms of a Cantor set

DOI: http://dx.doi.org/10.12775/TMNA.2006.013

#### Abstract

Let $\text{\rm Homeo}(\Omega)$ be the group of all homeomorphisms of a Cantor set

$\Omega$. We study topological properties of $\text{\rm Homeo}(\Omega)$ 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.

$\Omega$. We study topological properties of $\text{\rm Homeo}(\Omega)$ 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.

#### Keywords

Cantor set; minimal homeomorphisms; full groups; odometer

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