I'm having difficulty understanding why the algorithm presented in this paper works.
If I understand correctly, to construct a Schottky group start with $2n$ circles: $A_1...A_n$ and $B_1...B_n$, then the group is generated by all Mobius transformations $\alpha_i$, taking the exterior of $A_i$ to the interior of $B_i$, and $\beta_i$, taking the exterior of $B_i$ to the interior of $A_i$, where $i$ runs from $1$ to $n$. Based on the author's diagrams, a kissing Schottky group is one with 4 circles where each circle is tangent to the other pair of circles.
To render the limit set of such a group, I would think you would have to iterate the generators of the group on each pixel, and count how many times it takes for the pixel to fall outside any of the circles. But instead, the author renders them simply by iteratively inverting each circle, and counting how many inversions it takes to get outside any of the circles. Why does this work? Is it dependent upon the fact that some of the circles are tangent to each other?