Graphical exploration of the connectivity sets of alternated Julia sets

\(\mathcal{M}\), the set of disconnected alternated Julia sets


Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in Danca et al. (Int. J. Bifurc. Chaos, 19:2123–2129, 2009), on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in Danca et al. (Int. J. Bifurc. Chaos, 19:2123–2129, 2009), the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness), which characterizes the standard Julia sets. Via experimental mathematics, we unveil these connectivity domains, which are four-dimensional fractals. The computer graphics results show here, without substituting the proof but serving as a research guide, that for the alternated Julia sets, the Mandelbrot set consists of the set of all parameter values, for which each alternated Julia set is not only connected, but also disconnected.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    Intuitively, a totally disconnected set is a set, which can be broken into two pieces at each of its points, the breakpoint being always “in between” the original set while if a set can be separated into two open and disjoint sets such that neither set is empty and both sets combined give the original set, then the set is called disconnected. A set which is not disconnected is connected.


  1. 1.

    Danca, M.-F., Romera, M., Pastor, G.: Alternated Julia sets and connectivity properties. Int. J. Bifurc. Chaos 19, 2123–2129 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Almeida, J., Peralta-Salas, D., Romera, M.: Can two chaotic systems give rise to order? Physica D 200, 124–132 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Romera, M., Small, M., Danca, M.-F.: Deterministic and random synthesis of discrete chaos. Appl. Math. Comput. 192, 283–297 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Danca, M.-F., Romera, M., Pastor, G., Montoya, F.: Finding attractors of continuous-time systems by parameter switching. Nonlinear Dyn. 67, 2317–2342 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Peitgen, H.-O., Saupe, D.: The Science of Fractal Images, p. 198. Springer, New York (1988)

    MATH  Google Scholar 

  6. 6.

    Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, New York (1977)

    Google Scholar 

  7. 7.

    Fatou, P.: Bull. Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 47, 161–271 (1919). Also in Bull. Soc. Math. Fr. 48, 33–94, 208–314 (1920)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Julia, G.: FrMémoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8, 47–245 (1918)

    Google Scholar 

  9. 9.

    Blanchard, P., Devaney, R., Keen, L.: Complex dynamics and symbolic dynamics. In: Williams, S.G. (ed.) Symbolic Dynamics and Its Applications. Proc. Symp. Appl. Math., vol. 60, pp. 37–60 (2004)

    Google Scholar 

  10. 10.

    Branner, B., Hubbard, J.H.: Iteration of cubic polynomials. Part I: The global topology of parameter space. Acta Math. 160, 143–206 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Branner, B., Hubbard, J.H.: Iteration of cubic polynomials. Part II: Patterns and parapatterns. Acta Math. 169, 229 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Blanchard, P.: Disconnected Julia sets. In: Barnsley, M., Demko, S. (eds.) Chaotic Dynamics and Fractals, pp. 181–201. Academic Press, San Diego (1986)

    Google Scholar 

  13. 13.

    Qiu, W., Yin, Y.: Proof of the Branner–Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A 52, 45–65 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Altmann, S.L.: Rotations, Quaternions, and Double Groups. Clarendon Press, Oxford (1986)

    MATH  Google Scholar 

  15. 15.

    Drebin, R.A., Carpenter, L., Hanrahan, P.: Volume rendering. In: Proceedings of the 15th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’88), vol. 22, pp. 65–74. ACM, New York (1988)

    Google Scholar 

  16. 16.

    Limaye, A.: Drishti: a volume exploration and presentation tool. In: Proceedings SPIE 8506, Developments in X-Ray Tomography VIII, 85060X (2012).

    Google Scholar 

Download references


We thank Robert L. Devaney and Bodil Branner for their useful comments, which helped us to understand better the connectivity aspects presented in [1].

We thank the reviewers for their valuable comments, which helped to considerably improve the quality of the manuscript.

The work was supported by iVEC through the use of advanced computing resources located at the University of Western Australia. The volume rendering is performed using Drishti, a “Volume Exploration and Presentation Tool” developed by Ajay Limaye at the Australia National University [16].

Author information



Corresponding author

Correspondence to Marius-F. Danca.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Danca, MF., Bourke, P. & Romera, M. Graphical exploration of the connectivity sets of alternated Julia sets. Nonlinear Dyn 73, 1155–1163 (2013).

Download citation


  • Alternated Julia sets
  • Connectedness
  • Quadratic maps