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Equicontinuous Actions on Semi-Locally Connected and Local Dendrites

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Abstract

We show a necessary and sufficient condition of equicontinuous semi-locally connected flow. Moreover, we give a sufficient condition of the existence of almost automorphic points, for general flow. We further study equicontinuous local dendrites flows with finitely generated group action. Consequently, we obtain a generalization of Morales’s results in Morales (Topol Appl 198: 101–106, 2016) and Theorem of Su and Qin in Su and Qin (J Differ Equ 25: 1744–1754, 2019).

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References

  1. Abdelli, H.: \(\omega \)-limit sets for monotone local dendrite maps. Chaos Solitons Fractals 71, 66–72 (2015)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  3. Auslander, J., Greschonig, G., Nagar, A.: Reflections on equicontinuity. Proc. Amer. Math. Soc. 142, 3129–3137 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Bauer, W., Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79, 81–92 (1975)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  7. Glasner, E.: Regular PI metric flows are equicontinuous. Proc. Am. Math. Soc. 144, 269–277 (1992)

    Article  MathSciNet  Google Scholar 

  8. Ghys, E.: Groups acting on the circle. Enseign. Math. 47, 329–407 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Gottschalk, W.H.: Characterizations of almost periodic transformation groups. Proc. Amer. Math. Soc. 7, 709–712 (1956)

    Article  MathSciNet  Google Scholar 

  10. Haj Salem, A., Hattab, H.: Group action on local dendrites. Topol. Appl. 247, 91–99 (2018)

    Article  MathSciNet  Google Scholar 

  11. Haj Salem, A., Hattab, H.: On recurrence in dendrite flows, Preprint (2020)

  12. Haj Salem, A., Rejeiba, T., Hattab, H.: Equicontinuous local dendrite maps. Appl. Gen. Topol. 22, 67–77 (2020)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Kuratowski, K.: Topology, vol. 2. Academic Press, New York (1968)

    MATH  Google Scholar 

  15. Lee, Y.L.: Some characterizations of semi-locally connected spaces. Proc. Amer. Math. Soc. 16, 1318–1320 (1965)

    Article  MathSciNet  Google Scholar 

  16. 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  Google Scholar 

  17. Malyutin, A.V.: Classification of the group actions on the real line and circle. St. Petersburg Math. J. 19, 279–296 (2008)

    Article  MathSciNet  Google Scholar 

  18. Marzougui, H., Naghmouchi, I.: Minimal sets for group actions on dendrites. Proc. Amer. Math. Soc. 144, 4413–4425 (2016)

    Article  MathSciNet  Google Scholar 

  19. Marzougui, H., Naghmouchi, I.: Minimal sets and orbit space for group actions on local dendrites. Math. Z. 293, 1057–1070 (2019)

    Article  MathSciNet  Google Scholar 

  20. Morales, C.A.: Equicontinuity on semi-locally connected spaces. Topol. Appl. 198, 101–106 (2016)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  22. Naghmouchi, I.: Dynamics of homeomorphisms of regular curves. Colloq. Math. 162, 263–277 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Su, G., Qin, B.: Equicontinuous dendrite flows. J. Differ. Equ. Appl. 25, 1744–1754 (2019)

    Article  MathSciNet  Google Scholar 

  26. Veech, W.A.: The equicontinuous structure relation for minimal Abelian transformation groups. Amer. J. Math. 90, 723–732 (1968)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank the referees for their valuable comments.

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  1. Institut Supérieur de Gestion de Gabès, Gabes, Tunisia

    Aymen Haj Salem

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  1. Aymen Haj Salem
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Haj Salem, A. Equicontinuous Actions on Semi-Locally Connected and Local Dendrites. Qual. Theory Dyn. Syst. 20, 39 (2021). https://doi.org/10.1007/s12346-021-00477-7

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