The complex dynamics of rational maps having no Herman ring

Al-Shara`a, Iftichar Mudhar; Al-Salami, Hassanein Q.

doi:10.47974/JIM-2308

Open Access Research Article

The complex dynamics of rational maps having no Herman ring

^*Iftichar Mudhar Al-Shara`aCorresponding authorpure.iftichar.talb@uobabylon.edu.iqDepartment of MathematicsCollege of Education for Pure SciencesUniversity of BabylonBabylon, IraqView full profile → , Hassanein Q. Al-Salamihaszno732@gmail.comDepartment of MathematicsCollege of SciencesUniversity of BabylonBabylon, IraqView full profile →

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pp. 1255–1262Vol. 29Issue 5-AMay 2026DOI: 10.47974/JIM-2308XML

Received:: 01 May 2025
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2308
Pages:: 1255–1262

Abstract

This work aims to study the complex dynamics of rational maps, which, according to the Sullivan component, include sets of forward cyclic iterations, which include both equivalent and doubly connected components, where according to study the family of complex rational maps given by Qa (z) = 2an+1 z–n + z2n – a3n+1/(zn (z2n – an–1) where n ≥ 2 and a ∈ Λ ≔ C\{0} – {a : an+1 = 1 or a2n+2 = 1} So, the family would be considered a perturbation of a polynomial map. Therefore, we get the results. So Qa has neither infinitely connected attracting components nor equivalent (equivalent) components if the orbits of all free critical points are not attracted to the origin or infinity. Also, Qa does not contain Herman’s ring. So Qa has no Siegel disc.

Keywords

Julia sets Fatou components Herman ring Doubly connected

Subject Classifications

Primary 93A30Secondary 49K15

References

[1] A. F. Beardon, Iteration of Rational Functions: Complex Analytic Dynamical Systems, vol. 132 of Graduate Texts in Mathematics. New York, USA: Springer-Verlag, pp. 280 (1991).
[2] F. Yang, “Rational maps without Herman rings,” Proceedings of the American Mathematical Society, vol. 145, no. 4, pp. 1649–1659 (2017).
[3] M. Shishikura, “The connectivity of the Julia set and fixed points,” in Complex Dynamics: Families and Friends, D. Schleicher, Ed., Wellesley, MA, USA: A. K. Peters/CRC Press, pp. 257–276 (2009).
[4] Y. Yongcheng, “On the Julia sets of quadratic rational maps,” Complex Variables and Elliptic Equations, vol. 18, no. 3–4, pp. 141–147 (1992).
[5] J. Qiao and Y. Li, “On Connectivity of Julia Sets of Yang–Lee Zeros,” Commun. Math. Phys., vol. 222, no. 2, pp. 319–326 (2001).
[6] F. Yang and J. Zeng, “On the dynamics of a family of generated renormalization transformations,” J. Math. Anal. Appl., vol. 413, no. 1, pp. 361–377 (2014).
[7] R. L. Devaney and E. D. Russell, “Connectivity of Julia sets for singularly perturbed rational maps,” in Chaos, CNN, Memristors and Beyond: A Festschrift for Leon Chua With DVD-ROM, E. Bilotta, Ed., pp. 239–245 (2013).
[8] H. Q. Al-Salami and I. Al-Shara, “Julia sets are Cantor circles and Sierpinski carpets for rational maps,” International Journal of Nonlinear Analysis and Applications, vol. 13, no. 1, pp. 3937–3948 (2022).
[9] H. Q. Al-Salami and I. Al-Shara, “Q β has no doubly connected attracting Fatou or parabolic Fatou components,” AIP Conference Proceedings, vol. 3079, no. 1, p. 060035 (Apr. 2024).
[10] H. Q. Al-Salami, “Julia sets are quasicircles for rational maps of degree N,” Journal of Interdisciplinary Mathematics, vol. 26, no. 7, pp. 1421–1429 (2023).

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