Iterated functions and the Cantor set in one dimension

Abstract

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 ${\Lambda }^{\ast }$ embedded in $\mathbb{R}$, there exists a continuous function $F^{\ast }:\mathbb{R}\to \mathbb{R}$ such that the points that are bounded under iterations of $F^{\ast }$ are just those points in ${\Lambda }^{\ast }$. 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.