If is a map, then, for ,

In other words, the map induces a map , where denotes the set of all subsets of .

The standard notation for the iterates will be used, namely,

The notion of convergence in the "Hausdorff metric" will be used implicitly throughout the paper. However, since the concept will not be used in its full generality, the following notation is introduced for the special case of interest here.

Let , , be a sequence of subsets of , and let . The notation

means that

(1) ,
(2) , and
(3) neighborhood of , , such that .

Recall that, if is compact for every , then so is . If, in addition, each is nonempty, then so is .

For , denote