automorphism groups of newton’s method fractals

This is just a stub. When I see things like this or this, I keep wondering about the automorphism groups of Newton’s method fractals.
Just as Kleinian group limit sets are the result of applying a transformation over and over again, Newton’s method limit sets are the result of applying Newton’s method over and over again on the complex plane. In some ways, Newton’s method also involves a curious sort of Indra’s Pearls effect. For instance, in these youtube videos:

It is remarkable how the bulbs seem to have ‘reflection’ to all other bulbs contained within. Moreover, each node where a bulb joins another bulb has the same arrangement of colors (roots) around it. I suppose that there is some group ཞ for a particular function f(z) for which f(z) = f(z)

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One response to “automorphism groups of newton’s method fractals

  1. With the indras pearls type of fractal, the self-symmetry is perfect, for the newton’s method can we name a transformation that maps these forms to themselves after shrinking and distorting the plane?

    I’d like to view just the distortion function required… if such a thing exists… for newton’s method.

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