The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem

TL;DR

This paper investigates the minimal number of periodic orbits guaranteed by Sharkovskii's theorem for continuous functions on intervals, providing a complete answer to whether the lower bound can be improved beyond one.

Contribution

The paper offers a comprehensive analysis that determines the exact minimal number of periodic orbits for periods related by Sharkovskii's ordering, extending the classical theorem.

Findings

Established the minimal number of periodic orbits for certain periods

Provided conditions under which the lower bound exceeds one

Enhanced understanding of periodic orbit structure in interval maps

Abstract

Let f(x) be a continuous function from a compact real interval into itself with a periodic orbit of minimal period m, where m is not an integral power of 2. Then, by Sharkovsky's theorem, for every positive integer n with m \prec n in the Sharkovsky's ordering defined below, a lower bound on the number of periodic orbits of f(x) with minimal period n is 1. Could we improve this lower bound from 1 to some larger number? In this paper, we give a complete answer to this question.