The topological full group of a Cantor minimal system is dense in the full group
S Bezuglyi, J Kwiatkowski - 2000 - projecteuclid.org
S Bezuglyi, J Kwiatkowski
2000•projecteuclid.orgTo 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), x∈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 uw are studied on
\rmHomeo(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 …
all homeomorphisms γ such that γ(x)=T^n(x)(x), x∈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 uw are studied on
\rmHomeo(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 …
Abstract
To every homeomorphism of a Cantor set one can associate the full group formed by all homeomorphisms such that , . The topological full group consists of all homeomorphisms whose associated orbit cocycle is continuous. The uniform and weak topologies, and , as well as their intersection are studied on . It is proved that is dense in with respect to . A Cantor minimal system is called saturated if any two clopen sets of "the same measure" are -equivalent. We describe the class of saturated Cantor minimal systems. In particular, is saturated if and only if the closure of in is and if and only if every infinitesimal function is a -coboundary. These results are based on a description of homeomorphisms from 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 -equivalent clopen sets and the subgroup formed by homeomorphisms preserving the forward orbit of .
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