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ORIENTATION OF PIECEWISE POWERS OF A MINIMAL HOMEOMORPHISM
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Journal Article
ORIENTATION OF PIECEWISE POWERS OF A MINIMAL HOMEOMORPHISM
2022
Overview
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
$h \\in \\tau [g]$
.
Publisher
Cambridge University Press
Subject
