Although we did not emphasize this point when describing the random IFS algorithm, the fractal generated is not the orbit of a point, but rather the limit set of the orbit.
Suppose X is a set of points in the plane.
The limit set L(X) of X is the set of all points that can be approximated arbitrarily closely by points of X.
That is, a point q belongs to L(X) if for every distance d > 0, there is a point of X closer to q than d.
More precisely, q is a limit point of X, if for every d > 0 there is a point w in X with 0 < dist(w, q) < d.
Note that 0 < d(w, q) implies that w is not equal to q.
Here dist(w, q) denotes the Euclidean distance between w and q.
The set of all limit points of X is the limit set L(X).
Here are three examples of limit sets.
| Example 1 If X is a finite set of points, then L(X) is empty. |
| Example 2 If X is the set of all points
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| Example 3 If X is the Cantor set, then L(X) = X. |
Return to inversion limit sets.