Julia sets are Cantor circles and Sierpinski carpets for rational maps

	Julia sets are Cantor circles and Sierpinski carpets for rational maps
International Journal of Nonlinear Analysis and Applications
دوره 13، شماره 1، خرداد 2022، صفحه 3937-3948 اصل مقاله (668.05 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.6193
نویسندگان
Hassanein Q. Al-Salami^* ¹؛ Iftichar Al-Shara²
¹Department of Biology, College of Sciences, University of Babylon, Iraq
²Department of Mathematics, College of Education of Pure Sciences, University of Babylon, Iraq
چکیده
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 $.
کلیدواژه‌ها
Julia sets؛ Cantor circle؛ Sierpinski carpet؛ degenerate Sierpinski carpet
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