2013/11/03 by Benjamin Hoffman, Hoffman, Benjamin
Mathematics · #37F99 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DS #msc:37F99
paper · pdf · doi:10.48550/arxiv.1311.0535
arxiv created 2013/11/03 · openalex publication_date 2013/11/03 · arxiv updated 2013/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→\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.