The topological full group of a Cantor minimal system is dense in the full group

Abstract

To every homeomorphism $T$ of a Cantor set $X$ one can associate the full group $[T]$ formed by all homeomorphisms $\gamma$ such that $\gamma(x)=T^{n(x)}(x)$, $ x\in X$. The topological full group $[[T]]$ consists of all homeomorphisms whose associated orbit cocycle $n(x)$ is continuous. The uniform and weak topologies, $\tau_u$ and $\tau_w$, as well as their intersection $\tau_{uw}$ are studied on ${\rm Homeo}(X)$. It is proved that $[[T]]$ is dense in $[T]$ with respect to $\tau_u$. 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 $\tau_{uw}$ 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 $[[T]]_x \subset [[T]]$ formed by homeomorphisms preserving the forward orbit of $x$.