Abstract
Abstract
We show that, given a compact minimal system
$(X,g)$
and an element h of the topological full group
$\tau [g]$
of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of
$(X,G)$
into minimal and periodic parts, where G is any virtually polycyclic subgroup of
$\tau [g]$
. We also use the orientation of orbits to give a refinement of the index map and to describe the role in
$\tau [g]$
of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that
$h \in \tau [g]$
.
Funder
Australian Research Council
Publisher
Cambridge University Press (CUP)
Reference18 articles.
1. Spaces whose all nonempty clopen subspaces are homeomorphic;Terada;Yokohama Math. J.,1993
2. Groupes plein-topologiques (d’après Matui, Juschenko, Monod, …);Cornulier;Astérisque,2014
3. Cantor systems, piecewise translations and simple amenable groups
4. Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces