For simplicity, here we consider only circles C1, ... , CN bounding discs D1, ... , DN having disjoint interiors.
If the discs intersect, then depending on the angle of the intersections, quite complicated relations can arise between the inversions.
This case we consider in overlapping circles.
Starting with a point z0 outside the discs, we pick a circle at random and invert the starting point, obtaining a new point z1.
Then we pick another circle at random, with the restriction that it cannot be the same circle across which we just inverted (property (v)), and invert z1, obtaining a new point z2.
Keep repeating the process.
In limit set convergence, we shall see the sequence of points
ziconverges to
the limit set
Return to inversion limit sets.