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Minimal sets for group actions on dendrites


Authors: Habib Marzougui and Issam Naghmouchi
Journal: Proc. Amer. Math. Soc. 144 (2016), 4413-4425
MSC (2010): Primary 37B20, 37E99
DOI: https://doi.org/10.1090/proc/13103
Published electronically: April 25, 2016
MathSciNet review: 3531191
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Abstract: Let $ G$ be a group acting by homeomorphisms on a dendrite $ X$. First, we show that any minimal set $ M$ of $ G$ is either a finite orbit or a Cantor set (resp. a finite orbit) when the set of endpoints of $ X$ is closed (resp. countable). Furthermore, we prove, regardless of the type of the dendrite $ X$, that if the action of $ G$ on $ X$ has at least two minimal sets, then necessarily it has a finite orbit (and even an orbit consisting of one or two points). Second, we explore the topological and geometrical properties of infinite minimal sets when the action of $ G$ has a finite orbit. We show in this case that any infinite minimal set $ M$ is a Cantor set which is the set of endpoints of its convex hull $ [M]$ and there is no other infinite minimal set in $ [M]$. On the other hand, we consider the family $ \mathcal {M}$ of all minimal sets in the hyperspace $ 2^{X}$ (endowed with the Hausdorff metric). We prove that $ \mathcal {M}$ is closed in $ 2^{X}$ and that the family $ \mathcal {F}$ of all finite orbits (when it is non-empty) is dense in $ \mathcal {M}$. As a consequence, the union of all minimal sets of $ G$ is closed in $ X$.


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Habib Marzougui
Affiliation: Faculty of Sciences of Bizerte, Department of Mathematics, University of Carthage, Jarzouna, 7021, Tunisia
Email: hmarzoug@ictp.it; habib.marzougui@fsb.rnu.tn

Issam Naghmouchi
Affiliation: Faculty of Sciences of Bizerte, Department of Mathematics, University of Carthage, Jarzouna, 7021, Tunisia
Email: issam.nagh@gmail.com; issam.naghmouchi@fsb.rnu.tn

DOI: https://doi.org/10.1090/proc/13103
Keywords: Dendrite, dendrite map, minimal set, group action, orbit.
Received by editor(s): March 13, 2015
Received by editor(s) in revised form: September 24, 2015, and December 23, 2015
Published electronically: April 25, 2016
Communicated by: Nimish A. Shah
Article copyright: © Copyright 2016 American Mathematical Society