Iterated functions and the Cantor set in one dimension

In this paper we consider the long-term behavior of points in ${\mathbb R}$ under iterations of continuous functions. We show that, given any Cantor set $Λ^*$ embedded in ${\mathbb R}$, there exists a continuous function $F^*:{\mathbb R}\to{\mathbb R}$ such that the points that are bounded under iterations of $F^*$ are just those points in $Λ^*$. In the course of this, we find a striking similarity between the way in which we construct the Cantor middle-thirds set, and the way in which we find the points bounded under iterations of certain continuous functions.

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