Julia sets are Cantor circles and Sierpinski carpets for rational maps

Document Type : Research Paper


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


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 $.