The topological full group of a Cantor minimal system is dense in the full group - Bezuglyi, Sergey; Kwiatkowski, Jan (Topological Methods in Nonlinear Analysis, 2000)
To every homeomorphism T of a Cantor set X one can associate the full group [T] formed by all homeomorphisms γ such that γ(x)=T^n(x)(x), xX. The topological full group [[T]] consists of all homeomorphisms whose associated orbit cocycle n(x) is continuous. The uniform and weak topologies, τ_u and τ_w, as well as their intersection τ_uw are studied on Homeo(X). It is proved that [[T]] is dense in [T] with respect to τ_u. A Cantor minimal system (X,T) is called saturated if any two clopen sets of ``the same measure’’ are [[T]]-equivalent. We describe the class of saturated Cantor minimal systems. In particular, (X,T) is saturated if and only if the closure of [[T]] in τ_uw is [T] and if and only if every infinitesimal function is a T-coboundary. These results are based on a description of homeomorphisms from [[T]] related to a given sequence of Kakutani-Rokhlin partitions. It is shown that the offered method works for some symbolic Cantor minimal systems. The tool of Kakutani-Rokhlin partitions is used to characterize [[T]]-equivalent clopen sets and the subgroup [[T]]_x [[T]] formed by homeomorphisms preserving the forward orbit of x.