Sharkovskii’s theorem

Every natural numberMathworldPlanetmath can be written as 2rp, where p is odd, and r is the maximum exponentMathworldPlanetmath such that 2r divides ( the given number. We define the Sharkovskii ordering of the natural numbers in this way: given two odd numbersMathworldPlanetmathPlanetmath p and q, and two nonnegative integers r and s, then 2rp2sq if

  1. 1.

    r<s and p>1;

  2. 2.

    r=s and p<q;

  3. 3.

    r>s and p=q=1.

This defines a linear ordering of , in which we first have 3,5,7,, followed by 23, 25,, followed by 223, 225,, and so on, and finally 2n+1,2n,,2,1. So it looks like this:


Sharkovskii’s theorem. Let I be an interval, and let f:I be a continuous functionMathworldPlanetmathPlanetmath. If f has a periodic pointMathworldPlanetmath of least period n, then f has a periodic point of least period k, for each k such that nk.

Title Sharkovskii’s theorem
Canonical name SharkovskiisTheorem
Date of creation 2013-03-22 13:16:11
Last modified on 2013-03-22 13:16:11
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Definition
Classification msc 37E05
Synonym Sharkovsky’s theorem
Defines Sharkovskii’s ordering
Defines Sharkovsky’s theorem