Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Some dynamical properties of\(F(z) = z^2 - 2\bar z\)

  • 53 Accesses

Abstract

We study in this note some dynamical properties of\(F(z) = z^2 - 2\bar z\),F:ℂ→ℂ. LetK=K (F) denote the set of all points whose orbit is bounded. We prove thatF restricted to ℂ\K behaves as ψ(z)=z 2 does in the complement of the unit disk;K has positive area;F restricted toK is transitive; the set of periodic points ofF is dense inK and the topological entropy of F/K is positive.

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

References

  1. 1.

    J. C. Alexander, B. R. Hunt, I. Kan andJ. A. Yorke Intermingled Basins for the Triangle Map, Ergod. Th. and Dynam. Sys.,16 (1996), 651–662.

  2. 2.

    J. C. Alexander, J. A. Yorke andZhiping You Riddled Basins, International Journal of Bifurcation and Chaos,2 (1992), Number 4, 795–813.

  3. 3.

    J. Banks, J. Brooks, G. Cairns, G. Davis, andP. Stacey On Devaney's Definition of Chaos, American Mathematical Monthly,99 (1992), 332–334.

  4. 4.

    R. Bowen Topological Entropy and Axiom A, Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, R. I.,14, (1970), 23–41.

  5. 5.

    R. Bowen,Entropy for Group Endomorphisms and Homogeneous Spaces, Trans. Amer. Math. Soc.,153 (1971), 401–414.

  6. 6.

    R. L. Devaney,An Introduction to Chaotic Dynamical Systems,Second Edition, Addison Wesley, Redwood City (1989).

  7. 7.

    L. Gardini, R. Abraham, R. Record andD. Fournier-Prunaret,A double logistic map. International Journal of Bifurcation and Chaos.,4 (1994), 145–176.

  8. 8.

    G. Gómez and S. López de Medrano,Iteractiones de transformaciones cuadráticas del plano, 1993, Memorias de Coloquios (Caos y Sistemas Dinámicos), 1994, División de Ciencias Básicas e Ingeniería, UAM Atzacapotzalco, 33–51.

  9. 9.

    J. King, Ph.D. Thesis, in preparation.

  10. 10.

    I. Peterson Basins of Froth, Science News,142 (1992), 329–330.

  11. 11.

    G. Sienra, On the Dynamics of The One Parameter Functions\(F_a (z) = z^2 + 2a\bar z\), Bol. Soc. Mat. Mexicana2 (1996), Number 3, 41–53.

  12. 12.

    P. Walters,An Introduction to Ergodic Theory, Graduate Texts in Math, Springer Verlag, New York,79 (1982).

Download references

Author information

Correspondence to Jefferson King-Dávalos.

Additional information

Research supported in part by PAPIIT grant IN 101700.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

King-Dávalos, J., Méndez-Lango, H. & Sienra-Loera, G. Some dynamical properties of\(F(z) = z^2 - 2\bar z\) . Qual. Th. Dyn. Syst. 5, 101–120 (2004). https://doi.org/10.1007/BF02968132

Download citation

Key words

  • Periodic points
  • transitive mapping
  • topological entropy