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Minimal sets and orbit spaces for group actions on local dendrites

  • Habib MarzouguiEmail author
  • Issam Naghmouchi
Article
  • 31 Downloads

Abstract

We consider a group G acting on a local dendrite X (in particular on a graph). We give a full characterization of minimal sets of G by showing that any minimal set M of G (whenever X is different from a dendrite) is either a finite orbit, or a Cantor set, or a circle. This result extends that of the authors for group actions on dendrites. On the other hand, we show that, for any group G acting on a local dendrite X different from a circle, the following properties are equivalent: (1) (GX) is pointwise almost periodic. (2) The orbit closure relation \(R = \{(x, y)\in X\times X: y\in \overline{G(x)}\}\) is closed. (3) Every non-endpoint of X is periodic. In addition, if G is countable and X is a local dendrite, then (GX) is pointwise periodic if and only if the orbit space X / G is Hausdorff.

Keywords

Graph Dendrite Local dendrite Group action Minimal set Almost periodic point Closed relation orbit Orbit space 

Mathematics Subject Classification

37B05 37B45 37E99 

Notes

Acknowledgements

The authors are thankful to the referee for his/her careful reading of the manuscript, helpful remarks and suggestions which improve the presentation of the paper. This work was supported by the research unit: “Dynamical systems and their applications” (UR17ES21), of Higher Education and Scientific Research, Tunisia.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Science of Bizerte, (UR17ES21), “Dynamical Systems and Their Applications”University of CarthageJarzounaTunisia

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