Sierpinski Carpets as Julia sets for Imaginary 3-Circle Inversions

sends the point z ∈ C to its inversion image about the circle of radius r ∈ R centered at the point a ∈ C. Dynamically, this map is not interesting since iterating twice yields the identity mapping. Using three circles we invert a point z about each circle and form a new map by sending z to the arithmetic mean of the three inversion images. We refer to this as inverting z about the three circles and refer to the three circles as the generating circles. We define symmetric 3-circle inversion as the particular three circle inversion where all generating circles have the same radius, r, and each is centered on a cube root of unity. It is easy to verify this yields the degree-three antiholomorphic rational map