Similar Fractals

While studying the periods of a superattracting varient of the tetration fractal using the map z → a ^z -z-1 instead of the typical map z → a ^z I noticed that I was looking at a fractal similar to the projective fractal of periods showing the location of the fixed points for the convergence of the tower function. Mathematica is used to compare the two period fractals and do the image processing necessary to display that which is common between the fractals and that which is different. I would be very interested to hear of other cases of similar fractals.