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Full groups of minimal homeomorphisms and Baire category methods

Published online by Cambridge University Press:  02 October 2014

TOMÁS IBARLUCÍA  and
JULIEN MELLERAY
TOMÁS IBARLUCÍA
Affiliation:
Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France email melleray@math.univ-lyon1.fr
JULIEN MELLERAY
Affiliation:
Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France email melleray@math.univ-lyon1.fr
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Abstract

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space $X$, showing that these groups do not admit a compatible Polish group topology and, in the case of $\mathbb{Z}$-actions, are coanalytic non-Borel inside $\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside $\text{Homeo}(X)$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 2 , April 2016 , pp. 550 - 573
Copyright
© Cambridge University Press, 2014 

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