The limit set
That is,
To see this, note the outside of the disc bounded by A (on the left) inverts to the half of the plane above A' (on the right).
Next, following the orbit of any point outside the discs bounded by 1, 2, 3, 4, or 5, as soon as inversion in 5' is applied, the orbit drops below A'.
No subsequent inversion in 1' or 2' gives a point above A'.
No subsequent inversion in 3' or 4' gives a point above A'.
No subsequent inversion in 5' gives a point above A'.
Because any finite set of orbit points has empty limit set, we can ignore all finite collections of orbit points before inversion in 5' is applied.
The only remaining case is infinite orbits with no applications of 5'.
These generate limit points in
Consequently, no limit points lie above A'.
Return to Mandelbrot's method.