Skip to main content

Localic transitivity


For a dynamical system (Xf), the notion of topological transitivity has been studied by some researchers. There are several definitions of this property, and it is part of the folklore of dynamical systems that under some hypotheses, they are equivalent. In this paper, our aim is to introduce and study some properties of topological transitivity in pointfree topology, for which we first need to define in a way what makes them conservative extensions of topological transitivity defined by G.D. Birkhoff. We describe the way the different properties are related to each other in pointfree topology.

This is a preview of subscription content, access via your institution.


  1. Acosta, G., Hernández-Gutiérrez, R., Naghmouchi, I., Oprocha, P.: Periodic points and transitivity on dendrites. Ergodic Theory Dyn. Syst. 37(7), 2017–2033 (2017)

    MathSciNet  Article  Google Scholar 

  2. Akin, E., Carlson, J.D.: Conceptions of topologicl transitivity. Topol. Appl. 159(12), 2815–2830 (2012)

    Article  Google Scholar 

  3. Akin, E., Auslander, J., Berg, K.: When is a transitive map chaotic? In: Bergelson, V., March, P., Rosenblatt, J. (eds.), Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), de Gruyter, Berlin, pp. 25–40 (1996)

  4. Akin, E., Auslander, J., Nagar, A.: Variations on the concept of topological transitivity. Stud. Math. 235(3), 225–249 (2016)

    MathSciNet  Article  Google Scholar 

  5. Bayart, F., Matheron, E.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics 179. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  6. Banks, J.: Regular periodic decompositions for topologically transitive maps. Ergodic Theory Dyn. Syst. 17(3), 505–529 (1997)

    MathSciNet  Article  Google Scholar 

  7. Birkhoff, G.D.: Demonstration d’un theoreme elementaire sur les fonctions entieres. C. R. Acad. Sci. Paris 189, 473–475 (1929)

    MATH  Google Scholar 

  8. Block, L.S., Coppel, W.A.: Dynamics in One Dimension. Lecture Notes in Math, vol. 1513. Springer, New York (1992)

  9. Cairns, G., Kolganova, A., Nielsen, A.: Topological transitivity and mixing notions for group actions. Rocky Mount. J. Math. 37(2), 371–397 (2007)

    MathSciNet  Article  Google Scholar 

  10. Degirmenci, S., Kocak, S.: Existence of a dense orbit and topological transitivity: when are they equivalent? Acta Math. Hungar. 99(3), 185–187 (2003)

    MathSciNet  Article  Google Scholar 

  11. He, L., Gao, Y., Yang, F.: Some dynamical properties of continuous semi-flows having topological transitivity. Chaos Solitons Fract. 14(8), 1159–1167 (2002)

    MathSciNet  Article  Google Scholar 

  12. Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)

    MathSciNet  Article  Google Scholar 

  13. Kolyada, S., Snoha, L.: Some aspects of topological transitivity a survey. In: Ludwig, R., Jaroslav, S., Gyorgy, T., (eds.) Iteration Theory (ECIT 94, Opava, Czech. Repub.) Grazer Math. Ber., vol. 334, Karl-Franzens-Univ. Graz, Graz, pp. 3–35 (1997)

  14. Li, J.: Transitive points via Furstenberg family. Topol. Appl. 158(16), 2221–2231 (2011)

    MathSciNet  Article  Google Scholar 

  15. Grosse-Erdmann, K.-G., Peris, A.: Frequently dense orbits. C. R. Math. Acad. Sci. Paris Ser. I 341(2), 123–128 (2005)

    MathSciNet  Article  Google Scholar 

  16. Golzy, M.: Transitivity in point-free topology. Bull. Aust. Math. Soc. 80, 317–323 (2009)

    MathSciNet  Article  Google Scholar 

  17. Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  18. Moothathu, T.K.S.: Diagonal points having dense orbit. Colloq. Math. 120, 127–138 (2010)

    MathSciNet  Article  Google Scholar 

  19. Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics, Birkhauser/Springer, Basel AG, Basel (2012)

  20. Wang, X.Y., Huang, Y.: Recurrence of transitive points in dynamical systems with the specification property. Acta. Math. Sin.-Engl. Ser. 34, 1879–1891 (2018)

Download references


Thanks are due to the referee for helpful comments that have improved the readability of this paper and for providing Proposition 3.2.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ali Akbar Estaji.

Additional information

Communicated by Presented by A. Dow.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Estaji, A.A., Robat Sarpoushi, M. & Barzanouni, A. Localic transitivity. Algebra Univers. 83, 29 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • Topologically transitive
  • Transitive point
  • Sublocale
  • Localic map

Mathematics Subject Classification

  • 06D22
  • 54C05
  • 37B99