International Journal of Nonlinear Analysis and Applications

Rational maps whose Julia sets are quasi circles

Document Type : Research Paper

Authors

1 Department of Biology, College of Sciences, University of Babylon, Iraq

2 Department of Mathematics, College of Education of Pure Sciences, University of Babylon, Iraq

Abstract

In this paper, we give a family of rational maps whose Julia sets are quasicircles also we the boundaries of $I_0\ , I_\infty$ are quasicircles , we have the family of complex rational maps are given by
\begin{equation}\label{e1}
\mathcal{Q}_\alpha(Z)=2\alpha^{1-n}\ Z^n -\frac{z^n \left(z^{2n}-\alpha^{n+1}\right)}{z^{2n}-\alpha^{3n-1}},
\end{equation}
where $n\geq 2$ and $\alpha \in C\backslash \{0\},$ but $\alpha^{2n-2}\neq 1,\;\;\alpha^{1-n}\neq 1.$

Keywords

[1] A. Beardon, Iteration of Rational Functions, Grad. Texts in Math., 132, Springer-Verlag, New York, 1991.
[2] L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993.
[3] R. Devaney, D. Look, D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ.
Math. J. 54 (2005) 1621-1634.
[4] R.L. Devaney, Singular perturbations of complex polynomials, Bull. Amer. Math. Soc. 50 (2013) 391–429.[5] A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ec. Norm. Super. 18 (1985)
287–343.
[6] A. Garijo and S. Godillon, On McMullen-Like mappings, J. Fractal Geom. 2 (2015) 249–279.
[7] A. Garijo, S.M. Marotta and E.D. Russell, Singular perturbations in the quadratic family with multiple poles, J.
Diff. Equ. Appl. 19 (2013) 124–145.
[8] J. Fu and F. Yang, On the dynamics of a family of singularly perturbed rational maps, J. Math. Anal. Appl. 424
(2015) 104–121.
[9] J. Fu and Y. Zhang, Connectivity of the Julia sets of singularly perturbed rational maps, Proc. Indian Acad. Sci.
Math. Sci. 29(3) (2013) 239–245.
[10] D. M. Look, Singular perturbations of complex polynomials and circle inversion maps, Boston University, Ph.D.
Thesis, 2005.
[11] C. McMullen, Automorphisms of rational maps,in holomorphic functions and moduli I, Math. Sci. Res. Inst.
Publ., 10 Springer, 1988.
[12] W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math. 229 (2012) 2525–2577.
[13] P. Roesch, On local connectivity for the julia set of rational maps: newton’s famous example, Ann. Math. 168
(2008) 127–174.
[14] Y. Xiao, W. Qiu and Y. Yin, On the dynamics of generalized McMullen maps, Ergod. Th. Dyn. Syst. 34 (2014)
2093–2112.
[15] Y. Wang, F. Yang, S. Zhang and L. Liao, Escape quartered theorem and the connectivity of the Julia sets of a
family of rational maps, Disc. Contin. Dyn. Syst. 39 (2019) 5185–5206.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 2
November 2021
Pages 2041-2048
  • Receive Date: 11 March 2021
  • Revise Date: 27 June 2021
  • Accept Date: 09 July 2021