Julia sets are Cantor circles and Sierpinski carpets for rational maps

In this work, we study the family of complex rational maps which is given by Qβ(z)=2β1−dzd−zd(z2d−βd+1)z2d−β3d−1, where d greater than or equal to 2 and β∈C∖{0} such that β1−d≠1 and β2d−2≠1. We show that J(Qβ) is a Cantor circle or a Sierpinski carpet or a degenerate Sierpinski carpet, whenever the image of one of the free critical points for Qβ is not converge to 0 or ∞.