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 canassociate the full group $[T]$ formed by all homeomorphisms $γ$ suchthat $γ(x)=T^{n(x)}(x)$, $ x\in X$. The topological full group $[[T]]$consists of all homeomorphisms whose associated orbit cocycle $n(x)$ iscontinuous. The uniform and weak topologies, $τ_u$ and $τ_w$, aswell as their intersection $τ_{uw}$ are studied on $\text{\rm Homeo}(X)$. It is provedthat $[[T]]$ is dense in $[T]$ with respect to $τ_u$. A Cantor minimalsystem $(X,T)$ is called saturated if any two clopen sets of ``the samemeasure'' are $[[T]]$-equivalent. We describe the class of saturated Cantorminimal systems. In particular, $(X,T)$ is saturated if and only if theclosure of $[[T]]$ in $τ_{uw}$ is $[T]$ and if and only if everyinfinitesimal function is a $T$-coboundary. These results are based on adescription of homeomorphisms from $[[T]]$ related to a given sequence ofKakutani-Rokhlin partitions. It is shown that the offered method worksfor some symbolic Cantor minimalsystems. The tool of Kakutani-Rokhlin partitions is used to characterize$[[T]]$-equivalent clopen sets and the subgroup $[[T]]_x \subset [[T]]$formed by homeomorphisms preserving the forward orbit of $x$.