Skip to main content
SpringerLink
Log in

Equicontinuity of maps on a dendrite with finite branch points

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let (T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω(x, f) and P(f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω(x, f) = {y| there exist a sequence of points x k ∈ T and a sequence of positive integers n 1 < n 2 < ··· such that lim k→∞ x k = x and lim k→∞ \(f^{n_{k}}\) (x k ) = y}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω(x, f) = Ω(x, f) for any x ∈ T. (3) ∩ ∞ n=1 f n(T) = P(f), and ω(x, f) is a periodic orbit for every x ∈ T and map h: x → ω(x, f) (x ∈ T) is continuous. (4) Ω(x, f) is a periodic orbit for any x ∈ T.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acosta, G., Eslami, P.: On open maps between dendrites. Houston J. Math., 33, 753–770 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Baldwin, S.: Continuous itinerary functions and dendrite maps. Topology Appl., 154, 2889–2938 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balibrea, F., Hric, R., Snoha, L.: Minimal sets on graphs and dendrites. Internat. J. Bifurcation Chaos, 13, 1721–1725 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beardon, A. F.: Iteration of Rational Functions, Springer-Verlag, New York, 1991

    Book  MATH  Google Scholar 

  5. Blokh, A. M.: The set of all iterates is nowhere dense in C([0, 1], [0, 1]). Trans. Amer. Math. Soc., 333, 787–798 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Boyce, W.: Γ-compact maps on an interval and fixed points. Trans. Amer. Math. Soc., 160, 87–102 (1971)

    MathSciNet  MATH  Google Scholar 

  7. Bruckner, A. M., Hu, T.: Equicontinuity of iterates of an interval map. Tamkang J. Math., 21, 287–294 (1990)

    MathSciNet  MATH  Google Scholar 

  8. Cano, J.: Common fixed points for a class of commuting mappings on an interval. Trans. Amer. Math. Soc., 86, 336–338 (1982)

    MathSciNet  MATH  Google Scholar 

  9. Efremova, L. S., Makhrova, E. N.: The dynamics of monotone maps of dendrites. Sb. Math., 192, 807–821 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Efremova, L. S., Makhrova, E. N.: On the center of continuous maps of dendrites. J. Difference Equ. Appl., 9, 381–392 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gu, R.: Equicontinuity of maps on figure-eight space. Southeast Asian Bull. Math., 25, 413–419 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kato, H.: A note on periodic points and recurrent points of maps of dendrites. Bull. Austral. Math. Soc., 51, 459–461 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mai, J.: The structure of equicontinuous maps. Trans. Amer. Math. Soc., 355, 4125–4136 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mai, J.: Pointwise recurrent graph maps. Ergodic Theory Dynam. Systems, 25, 629–637 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mai, J., Shi, E.: \(\bar R = \bar P\) for maps of dendrites X with Card(End(X)) < c. Internat. J. Bifur. Chaos, 19, 1391–1396 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nadler Jr, Sam B.: Continuum Theory: An Introduction, Marcel Dekker, Inc., NY, 1992

    MATH  Google Scholar 

  17. Naghmouchi, I.: Dynamical properties of monotone dendrite maps. Topology Appl., 159, 144–149 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Naghmouchi, I.: Pointwise-recurrent dendrite maps. Ergodic Theory Dynam. Systems, 33, 1115–1123 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Peitgen, H. O., Richter, P. H.: The Beauty of Fractals, Springer-Verlag, Berlin, 1986

    Book  MATH  Google Scholar 

  20. Sun, T.: Equicontinuity of tree maps (in Chinese). J. Math. Study, 34, 125–130 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Sun, T., Zhang, Y., Zhang, X.: Equicontinuity of a graph map. Bull. Austral. Math. Soc., 71, 61–67 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, T., He, Q., Xi, H.: Intra-orbit separation of dense orbits of dendrite maps. Chaos Solitons Fractals, 57, 89–92 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sun, T., Xi, H.: The centre and the depth of the centre for continuous maps on dendrites with finite branch points. Qual. Theory Dyn. Syst., DOI 10.1007/s12346–016–0204–1 (2016)

    Google Scholar 

  24. Sun, T., Xi, H., He, Q.: Non-wandering sets of dendrite maps. Acta Math. Sin., Engl. Ser., DOI: 10.1007/s10114–016–5578–0 (2016)

    Google Scholar 

Download references

Acknowledgements

We thank the referees for their careful reading of the manuscript and constructive comments and suggestions.

Author information

Authors and Affiliations

  1. College of Information and Statistics, Guangxi University of Finance and Economics, Nanning, 530003, P. R. China

    Tai Xiang Sun, Guang Wang Su & Hong Jian Xi

  2. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, P. R. China

    Xin Kong

Authors
  1. Tai Xiang Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guang Wang Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong Jian Xi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xin Kong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guang Wang Su.

Additional information

Supported by NNSF of China (Grant No. 11461003) and SF of Guangxi Univresity of Finance and Economics (Grant Nos. 2016KY15, 2016ZDKT06 and 2016TJYB06)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, T.X., Su, G.W., Xi, H.J. et al. Equicontinuity of maps on a dendrite with finite branch points. Acta. Math. Sin.-English Ser. 33, 1125–1130 (2017). https://doi.org/10.1007/s10114-017-6289-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-017-6289-x

Keywords

MR(2010) Subject Classification

Search

Navigation