# Circle Inversion Fractals

## Restricted Limit Sets

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 x_{i} lies in the
disc bounded by C_{j}, then the next orbit point, x_{i+1}, cannot be
I_{j}(x_{i}).

If the circles C_{i} 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 C_{i}.

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.