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

DOI: http://dx.doi.org/10.12775/TMNA.2000.046

#### 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 $\text{\rm Homeo}(X)$. 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 $x$.

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 $\text{\rm Homeo}(X)$. 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 $x$.

#### Keywords

Cantor set; minimal homeomorphism; full group

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