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.
Prof. Elizabeth Stanhope, Department of Mathematics, Lewis & Clark College
"Iterated functions and the Cantor set in one dimension,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 14
, Article 5.
Available at: http://scholar.rose-hulman.edu/rhumj/vol14/iss2/5