Conditions Where the Chaotic Set Has a Non-Empty Residual Julia Set for Two Classes of Meromorphic Functions

Affiliation(s)

Facultad de Ciencias Fsico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico.

Facultad de Ciencias Fsico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico.

ABSTRACT

We define the Fatou and Julia sets for two classes of
meromorphic functions. The Julia set is the chaotic set where the fractals
appear. The chaotic set can have points and components which are buried. The
set of these points and components is called the *residual Julia set*,
denoted by , and is defined to be the subset of those points of the
Julia set, chaotic set, which do not belong to the boundary of any component of
the Fatou set (stable set). The points of are called *buried
points* and the components of are called *buried
components*. In this paper we extend some results related with the residual
Julia set of transcendental meromorphic functions to functions which are
meromorphic outside a compact countable set of essential singularities. We give
some conditions where .

Cite this paper

P. Domínguez and I. Hernández, "Conditions Where the Chaotic Set Has a Non-Empty Residual Julia Set for Two Classes of Meromorphic Functions,"*Applied Mathematics*, Vol. 4 No. 11, 2013, pp. 18-21. doi: 10.4236/am.2013.411A2004.

P. Domínguez and I. Hernández, "Conditions Where the Chaotic Set Has a Non-Empty Residual Julia Set for Two Classes of Meromorphic Functions,"

References

[1] J. F. Gouyet, “Physics and Fractal Structures,” Masson Springer, Paris, New York, 1996.

[2] M. A. Montes de Oca Balderas, G. J. F Sienra Loera and J. E. King Dávalos, “Baker Domains for Period Two for the Family*f*_{λ,μ} =*λ*e^{z} + μ/z ,” International Journal of Bifurcation and Chaos, in Press, 2013.

[3] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Meromorphic Functions II: Examples of Wandering Domains,” Journal of the London Mathematical Society, Vol. 42, No. 2, 1990, pp. 267-278.

http://dx.doi.org/10.1112/jlms/s2-42.2.267

[4] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Meromorphic Functions: I,” Ergodic Theory and Dynamical Systems, Vol. 11, No. 2, 1991, pp. 241-248.

[5] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Meromorphic Functions III. Preperiodic Domains,” Ergodic Theory Dynamical Systems, Vol. 11, 1991, pp. 603-618.

[6] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Meromorphic Functions IV. Critically Finite Functions,” Results in Mathematics, Vol. 22, No. 2-4, 1992, pp. 651656.

[7] B. Bolsch, “Repulsive Periodic Points of Meromorphic Function,” Complex Variables Theory and Application, Vol. 31, No. 1, 1996, pp. 75-79.

http://dx.doi.org/10.1080/17476939608814947

[8] A. Bolsch, “Iteration of Meromorphic Functions with Countably Many essential Singularities,” Technische Universitat Berlin, Berlin, 1997.

[9] A. Bolsch, “Periodic Fatou Components of Meromorphic Functions,” Bulletin of the London Mathematical Society, Vol. 31, No. 5, 1999, pp. 543-555.

http://dx.doi.org/10.1112/S0024609399005950

[10] W. Abikoff, “Some Remarks on Kleinian Groups,” Annals of Mathematics, Vol. 66, 1971, pp. 1-5.

[11] W. Abikoff, “The Residual Limits Sets of Kleinian Groups,” Acta Mathematica, Vol. 130, No. 1, 1973, pp. 127-144. http://dx.doi.org/10.1007/BF02392264

[12] C. McMullen, “Automorphisms of Rational Maps, Holomorphic Functions and Modulii I,” MSRI Publications 10, Springer Verlag, New York, 1988.

[13] I. N. Baker and P. Domínguez, “Residual Julia Sets,” Journal of Analysis, Vol. 8, 2000, pp. 121-137.

[14] J. Y. Qiao, “The Buried Points on the Julia Sets of Rational and Entire Functions,” Science in China Series A, Vol. 38, No. 12, 1995, pp. 1409-1419.

[15] P. Domínguez and N. Fagella, “Residual Julia Sets for Rational and Transcendental Functions,” Cambridge University Press, Cambridge, 2008, pp. 138-164.

http://dx.doi.org/10.1017/CBO9780511735233.008

[16] P. Domínguez and N. Fagella, “Existence of Herman Rings for Meromorphic Functions,” Complex Variables, Vol. 49, No. 12, 2004, pp. 851-870.

http://dx.doi.org/10.1080/02781070412331298589

[17] P. Domínguez, “Residual Julia Sets for Meromorphic Functions with Countably Many Essential Singularities,” Journal of Difference Equations and Applications, Vol. 16, No. 5-6, 2010, pp. 519-522.

http://dx.doi.org/10.1080/10236190903203879

[1] J. F. Gouyet, “Physics and Fractal Structures,” Masson Springer, Paris, New York, 1996.

[2] M. A. Montes de Oca Balderas, G. J. F Sienra Loera and J. E. King Dávalos, “Baker Domains for Period Two for the Family

[3] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Meromorphic Functions II: Examples of Wandering Domains,” Journal of the London Mathematical Society, Vol. 42, No. 2, 1990, pp. 267-278.

http://dx.doi.org/10.1112/jlms/s2-42.2.267

[4] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Meromorphic Functions: I,” Ergodic Theory and Dynamical Systems, Vol. 11, No. 2, 1991, pp. 241-248.

[5] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Meromorphic Functions III. Preperiodic Domains,” Ergodic Theory Dynamical Systems, Vol. 11, 1991, pp. 603-618.

[6] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Meromorphic Functions IV. Critically Finite Functions,” Results in Mathematics, Vol. 22, No. 2-4, 1992, pp. 651656.

[7] B. Bolsch, “Repulsive Periodic Points of Meromorphic Function,” Complex Variables Theory and Application, Vol. 31, No. 1, 1996, pp. 75-79.

http://dx.doi.org/10.1080/17476939608814947

[8] A. Bolsch, “Iteration of Meromorphic Functions with Countably Many essential Singularities,” Technische Universitat Berlin, Berlin, 1997.

[9] A. Bolsch, “Periodic Fatou Components of Meromorphic Functions,” Bulletin of the London Mathematical Society, Vol. 31, No. 5, 1999, pp. 543-555.

http://dx.doi.org/10.1112/S0024609399005950

[10] W. Abikoff, “Some Remarks on Kleinian Groups,” Annals of Mathematics, Vol. 66, 1971, pp. 1-5.

[11] W. Abikoff, “The Residual Limits Sets of Kleinian Groups,” Acta Mathematica, Vol. 130, No. 1, 1973, pp. 127-144. http://dx.doi.org/10.1007/BF02392264

[12] C. McMullen, “Automorphisms of Rational Maps, Holomorphic Functions and Modulii I,” MSRI Publications 10, Springer Verlag, New York, 1988.

[13] I. N. Baker and P. Domínguez, “Residual Julia Sets,” Journal of Analysis, Vol. 8, 2000, pp. 121-137.

[14] J. Y. Qiao, “The Buried Points on the Julia Sets of Rational and Entire Functions,” Science in China Series A, Vol. 38, No. 12, 1995, pp. 1409-1419.

[15] P. Domínguez and N. Fagella, “Residual Julia Sets for Rational and Transcendental Functions,” Cambridge University Press, Cambridge, 2008, pp. 138-164.

http://dx.doi.org/10.1017/CBO9780511735233.008

[16] P. Domínguez and N. Fagella, “Existence of Herman Rings for Meromorphic Functions,” Complex Variables, Vol. 49, No. 12, 2004, pp. 851-870.

http://dx.doi.org/10.1080/02781070412331298589

[17] P. Domínguez, “Residual Julia Sets for Meromorphic Functions with Countably Many Essential Singularities,” Journal of Difference Equations and Applications, Vol. 16, No. 5-6, 2010, pp. 519-522.

http://dx.doi.org/10.1080/10236190903203879