Description

• Title:
  The topological full group of a Cantor minimal system is dense in the full group
• Creator:
  Bezuglyi, Sergey;
  Kwiatkowski, Jan
• Description:
  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 $x$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2000.12.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 16, No 2 (December 2000); 371 - 397;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2000.046/9971
• Subject:
  Cantor set;
  minimal homeomorphism;
  full group;
  37B05;
  37A20
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2000.046
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other