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ORIENTATION OF PIECEWISE POWERS OF A MINIMAL HOMEOMORPHISM

Part of: Semigroups Topological dynamics Permutation groups

Published online by Cambridge University Press:  15 November 2021

COLIN D. REID Open the ORCID record for COLIN D. REID [Opens in a new window]
COLIN D. REID*
Affiliation:
School of Information and Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW 2308 Australia
*
email: colin@reidit.net
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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 $h \in \tau [g]$ .

Keywords

Cantor minimal systemstopological full group

MSC classification

Primary: 37B99: None of the above, but in this section
Secondary: 20B99: None of the above, but in this section 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 2 , October 2022 , pp. 226 - 256
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Anthony Henderson

Research funded by ARC project FL170100032.

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