How to recreate this complex iteration

I came across this image in a book Im reading:

enter image description here

(taken from Rational Iteration Complex Analytic Dynamical Systems from Steinmetz. I hope it is okay to share it here)

and I try to recreate it in Mathematica. I tried

JuliaSetPlot[(9 + 2 z + 9 z^2)/20, z]

which gives a complete different picture.

enter image description here

I also tried to use

ColorFunction -> With[{cf = ColorData["M10DefaultFractalGradient"]}, cf@Sqrt[#3] &]

which doesn’t improve anything. So JuliaSetPlot might not be the right method. Do you have any suggestions?