2 7 O ct 2 00 4 Topologies on the group of homeomorphisms of a Cantor set

Let Homeo(Ω) be the group of all homeomorphisms of a Cantor set Ω. We study topological properties of Homeo(Ω) and its subsets with respect to the uniform (τ) and weak (τ w) topologies. The classes of odometers and periodic, aperiodic, minimal, rank 1 homeomorphisms are considered and the closures of those classes in τ and τ w are found. 0 Introduction The present paper is a continuation of our article [BDK] about topologies on the group Aut(X, B) of all Borel automorphisms of a standard Borel space. In the introduction to that article, we discussed our approach to the study of topologies on groups of transformations of an underlying space. As we mentioned there, we were motivated, first of all, by remarkable results in ergodic theory concerning topological properties of the group of all automorphisms of a standard measure space. We refer to the classical articles of Halmos [H] and Rokhlin [R] where the uniform and weak topologies appeared as " key players " in ergodic theory. The central object of the present paper is the group Homeo(Ω) of all homeomor-phisms of a Cantor set Ω. Although we consider several topologies on Homeo(Ω), this group is mostly studied under two topologies, τ and τ w. These are analogues of the uniform and weak topologies in measurable dynamics. We should remark that