An Interesting Application of the Intermediate Value Theorem: A Simple Proof of Sharkovsky's Theorem and the Towers of Periodic Points

TL;DR

This paper provides a straightforward proof of Sharkovsky's theorem using the Intermediate Value Theorem, aimed at calculus students, and explores the structure of periodic points, including towers, in continuous interval maps.

Contribution

It offers a simple, accessible proof of Sharkovsky's theorem and introduces the concept of towers of periodic points without requiring prior dynamical systems knowledge.

Findings

Proof of Sharkovsky's theorem using the Intermediate Value Theorem

Identification of towers of periodic points with infinitely many layers

Nested structure of periodic orbits in unimodal maps

Abstract

This note is intended primarily for college calculus students right after the introduction of the Intermediate Value Theorem, to show them how the Intermediate Value Theorem is used repeatedly and straightforwardly to prove the celebrated Sharkovsky's theorem on the periods of coexistent periodic orbits of continuous maps on an interval. Furthermore, if the maps have a periodic orbit P of odd period > 1, then we find more periodic points, in the appendix, which constitute what we call the towers of periodic points associated with P. These towers of periodic points have infinitely many layers. In this note, no knowledge of Dynamical Systems Theory is required. In this revision, we add the result (Proposition 1 on page 9) that the periodic orbits of continuous unimodal maps on the interval $[0, 1]$ of least periods $\ge 2$ are nested in the sense that if $P$ and $Q$ are periodic orbits of a continuous unimodal map on $[0, 1]$ of least periods $\ge 2$ and if $\max P < \max Q$, then $[\min P, \max P] \subset [\min Q, \max Q]$.