In this paper we consider the long-term behavior of points in ℜ under iterations of continuous functions. We show that, given any Cantor set Λ* embedded in ℜ, there exists a continuous function F*:ℜ → 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.

Author Bio

Benjamin Hoffman graduated from Lewis & Clark College in May of 2013 with degrees in mathematics and philosophy. He is spending a year working at a wolf sanctuary in a remote part of southern Colorado. He will begin to pursue a PhD in mathematics in the Fall of 2014.