A Simple Proof of Sharkovsky's Theorem Rerevisited

TL;DR

This paper presents simplified proofs of Sharkovsky's cycle coexistence theorem using various strategies and introduces a new doubling operator, including a graph-based proof suitable for calculus courses.

Contribution

It offers new, simplified proofs of Sharkovsky's theorem and introduces a novel doubling operator, making the theorem more accessible for educational purposes.

Findings

Multiple simple proofs of Sharkovsky's theorem

Introduction of a new general doubling operator

Graph-based proof suitable for calculus courses

Abstract

Based on various strategies and a new general doubling operator, we obtain several simple proofs of the celebrated Sharkovsky's cycle coexistence theorem. A simple non-directed graph proof which is especially suitable for a calculus course right after the introduction of Intermediate Value Theorem is also given (in section 3).