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Dendrite Flows

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Abstract

Let (G, D) be a flow such that D is a dendrite and G is a finitely generated group. Denote by E(D) the set of endpoints of D. In this paper, it is shown that if E(D) is closed and countable then the following properties are equivalent:

  1. (1)

    (G, D) is pointwise periodic;

  2. (2)

    (G, D) is pointwise almost periodic;

  3. (3)

    The orbit closure relation is closed;

  4. (4)

    (G, D) is equicontinuous.

In addition, we show that if E(D) is countable, then (G, D) is not a minimal flow. We also show that if (G, D) is a pointwise periodic flow and if D does not contain any homeomorphic copy of special dendrites, then the action of G can factor through a finite group action.

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References

  1. Auslander, J.: Minimal Flows and Their Extensions. North Holland, Amsterdam (1988)

    MATH  Google Scholar 

  2. Auslander, J., Glasner, E., Weiss, B.: On recurrence in zero dimensional flows. Forum Mathematicum. 19(1), 107–114 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arevalo, D., Charatonik, W.J., Covarrubias, P.P., Simon, L.: Dendrites with a closed set of end points. Topol. Appl. 115, 1–17 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beardon, A.F.: Iteration of Rational Functions. Springer, New York (1991)

    Book  MATH  Google Scholar 

  5. Barbachine, E.: Sur la conduite des points sous les transformations homéomorphes de l’espace. C.R. (Doklady) Acad. Sci. URSS 51, 3–5 (1946)

    MathSciNet  MATH  Google Scholar 

  6. Balibrea, F.: Studies on dendrites and the periodic-recurrent property. Extracta Mathematicae 25(3), 211–226 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Bowditch, B.H.: Treelike Structures Arising from Continua and Convergence Groups, Mem. Amer. Math. Soc., 662, Amer. Math. Soc., Providence (1999)

  8. Charatonik, J.J., Charatonik, W.J.: Dendrites. Aprotactions Math. 22, 227–253 (1998)

    MathSciNet  MATH  Google Scholar 

  9. Charatonik, J.J.: On sets of periodic and of recuurent points. Publ. Inst. Math. 63(77), 131–142 (1998)

    Google Scholar 

  10. Culler, M., Morgan, J.W.: Group actions on R-trees. Proc. Lond. Math. Soc. s3–55(3), 571–604 (1987)

    Article  MATH  Google Scholar 

  11. Elkacimi, A., Hattab, H., Salhi, E.: Remarque sur certains groupes d’homémorphismes d’espaces métriques. JP J. Geom. Topol. 4(3), 225–242 (2004)

    MathSciNet  Google Scholar 

  12. Epstein, D.A.: A Topology for the Space of Foliations. Lecture Notes in Math., vol. 597, pp. 132–150 (1977)

  13. Glasner, E.: Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, vol. 101. American Mathematical Society, Providence (2003)

    Book  MATH  Google Scholar 

  14. Gottschalk, W.H., Hedlund, G.A.: Topological Dynamics. AMS Colloquium Publications, Providence (1955)

    Book  MATH  Google Scholar 

  15. Hattab, H.: Group of homeomorphisms of a metric space with finite orbits. JP. Geom. Topol. 2, 115–128 (2004)

    MathSciNet  MATH  Google Scholar 

  16. Hattab, H.: Pointwise recurrent one-dimensional flows. Dyn. Syst. Int. J. 26(1), 77–83 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jmel, A.: Pointwise periodic homeomorphisms on dendrites. Dyn. Syst. 30(1), 34–44 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mai, J.H., Shi, E.H.: The nonexistence of expansive commutative group actions on Peano continua having free dendrites. Topol. Appl. 155, 33–38 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mai, J.H., Shi, E.H.: \(\overline{R} = \overline{P}\) for maps of dendrites X with \(Card(End(X)) < c\). Int. J. Bifurcat. Chaos 19(4), 1391–1396 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Malyutin, A.V.: Pretrees and the shadow topology. St. Petersburg Math. J. 26, 225–271 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mayer, J.C., Nikiel, J., Oversteegen, L.G.: Universal spaces for R-trees. Trans. Am. Math. Soc. 334, 411–432 (1992)

    MathSciNet  MATH  Google Scholar 

  22. Naghmouchi, I.: Pointwise recurrent dendrite maps. Ergod. Theory Dyn. Syst. 33, 1115–1123 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nadler, S.B.: Continuum Theory. Marcel Dekker, Inc, New York (1992)

    MATH  Google Scholar 

  24. Niemytzki, V.: Les systèmes dynamiques généraux. C.R. (Doklady) Acad. Sci. URSS 53, 491–494 (1946)

    MathSciNet  MATH  Google Scholar 

  25. Roberson, F.A.: Recurrence and almost periodicity in a generative transformation. Proc. Am. Math. Soc. 32, 596–598 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shi, E.H., Wang, S., Zhou, L.: Minimal group actions on dendrites. Proc. Am. Math. Soc. 138, 217–223 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shi, E.H., Sun, B.Y.: Fixed point properties of nilpotent group actions on 1-arcwise connected continua. Proc. Am. Math. Soc. 137, 771–775 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. de Vries, J.: Element of Topological Dynamics, Mathematics and Its Application, vol. 257. Kluwer, Dordrecht (1993)

    Book  Google Scholar 

  29. Wang, S., Shi, E.H., Zhou, L.: Topological transitivity and chaos of group action on dendrites. Int. J. Bifurcat. Chaos 19(12), 4165–4174 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Whyburn, G.T.: Analytic Topology, vol. 28. Providence, RI: American Mathematical Society; 1942; reprinted with corrections (1971)

  31. Ward, L.E.: On dendritic sets. Duke Math. J. 25, 505–513 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yokoyama, T.: Recurrence, pointwise almost periodicity and orbit closure relation for flows and foliations. Topol. Appl. 160(17), 2196–2206 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zafridou, S.: Universal dendrites for some families of dendrites with a countable set of endpoints. Topol. Appl. 155(17–18), 1935–1946 (2008)

    Article  Google Scholar 

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Acknowledgements

The authors are grateful to the handling editor and anonymous referees for their careful reading and constructive suggestions which lead to truly significant improvement of the paper.

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Authors and Affiliations

  1. Laboratoire des systèmes dynamiques et combinatoires, Faculté des Sciences de Sfax, Sfax, Tunisia

    Aymen Haj Salem & Hawete Hattab

  2. Umm Al-Qura University Makkah, Makkah, Kingdom of Saudi Arabia

    Hawete Hattab

Authors
  1. Aymen Haj Salem
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  2. Hawete Hattab
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Corresponding author

Correspondence to Hawete Hattab.

Additional information

This work is supported by the laboratory of research “Dynamical systems and combinatorics” University of Sfax, Tunisia.

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Haj Salem, A., Hattab, H. Dendrite Flows. Qual. Theory Dyn. Syst. 16, 623–634 (2017). https://doi.org/10.1007/s12346-017-0237-0

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  • DOI: https://doi.org/10.1007/s12346-017-0237-0

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