Julia sets are Cantor circles and Sierpinski carpets for rational maps

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 work, we study the  family of complex rational  maps which is given by
$$Q_{\beta }\left(z\right)=2{\beta }^{1-d}z^d-\frac{z^d(z^{2d}-{\beta }^{d+1})}{z^{2d}-{\beta }^{3d-1}},$$
where $d$ greater than or equal to 2 and  $\beta{\in }\mathbb{C}{\backslash }\{0\}$ such that $\beta^{1-d}\ne 1$ and $\beta^{2d-2}\ne 1$. We show that ${J(Q}_\beta$) is a  Cantor circle or a Sierpinski carpet or a degenerate Sierpinski carpet, whenever the image of one of the free critical points for $Q_\beta$ is not converge to $0$ or $\infty $.    

Keywords

[1] H.Q. Al-Salami, I. Al-shara, Rational maps whose Julia sets are quasi circles, Int. J. Nonlinear Anal. Appl. 12(2)
(2021) 2041–2048.
[2] A. Beardon, Iteration of Rational Functions, Grad. Texts in Math., vol. 132, Springer-Verlag, New York, 1991.
[3] M. Bonk, M. Lyubich and S. Merenkov, Quasisymmetries of Sierpi´nski carpet Julia sets, Adv. Math. 301 (2016)
383–422.
[4] R.L. Devaney, D. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana.
Univ. Math. J. 54 (2005) 1621–1634.
[5] R.L. Devaney, Singular perturbations of complex polynomials, Bull. Amer. Math. Soc. 50 (2013) 391–429.
[6] J. Fu and F. Yang, On the dynamics of a family of singularly perturbed rational maps, J. Math. Anal. Appl. 424
(2015) 104–121.
[7] A. Garijo and S. Godillon, On McMullen-like mappings, J. Fractal Geom. 2 (2015) 249–279.
[8] A. Garijo, S.M. Marotta and E.D. Russell, Singular perturbations in the quadratic family with multiple poles, J.
Difference Equ. Appl. 19 (2013) 124–145.
[9] D.M. Look, Singular Perturbations of Complex Polynomials and Circle Inversion Maps, Boston University, Thesis,
2005.[10] C. McMullen, Automorphisms of rational maps, in: Holomorphic functions and moduli I, Math. Sci. Res. Inst.
Publ. 10, Springer, 1988.
[11] J. Milnor, Dynamics in One Complex Variable, third edition, Princeton Univ. Press, Princeton, 2006.
[12] J. Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993) 37–83.
[13] K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets, Asterisque 261 (2000) 349–383.
[14] W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math. 229 (2012) 2525–2577.
[15] W. Qiu, F. Yang and J. Zeng, Quasisymmetric geometry of the Carpet Julia sets, Fund. Math. 244 (2019) 73–107.
[16] 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, Discrete Contin. Dyn. Syst. 39(9) (2019) 5185–5206.
[17] Y. Xiao, W. Qiu and Y. Yin, On the dynamics of generalized McMullen maps, Ergod. Th. Dynam. Syst. 34 (2014)
2093-2112.
Volume 13, Issue 1
March 2022
Pages 3937-3948
  • Receive Date: 01 August 2021
  • Revise Date: 04 September 2021
  • Accept Date: 15 October 2021