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−d zd − 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 ∞.