One way to avoid the problems that can result when points inside a circle are inverted outside the circle (as can happen when the inverting circles overlap) is to forbid all combinations involving inverting from inside to outside.
Introduced in Clancy and Frame, the restricted limit set is the limit set of the orbit of a point, with the restriction that if some orbit point xi lies in the disc bounded by Cj, then the next orbit point, xi+1, cannot be Ij(xi).
If the circles Ci bound disjoint discs, then this condition is just the familiar requirement that we never invert successively in the same circle.
If the circles intersect, this condition can be more interesting.
With this restriction, inversions will never be expanding maps and the restricted limit set is contained in the discs bounded by the Ci.
|Example 1: the restricted limit set for inversions in five circles|
|Example 2: the restricted limit set for inversions in eight circles|
|Example 3: the restricted limit set for inversions in eight circles|
|Example 4: the restricted limit set for inversions in five circles|
Return to circle inversion fractals.