Sharkovskii’s theorem

Every natural number can be written as $2^{r}p$, where $p$ is odd, and $r$ is the maximum exponent such that $2^r$ divides (http://planetmath.org/Divisibility) the given number. We define the Sharkovskii ordering of the natural numbers in this way: given two odd numbers $p$ and $q$, and two nonnegative integers $r$ and $s$,then $2^{r}p\succ 2^{s}q$ if

1. and $p>1$;

2. $r=s$ and ;

3. $r>s$ and $p=q=1$.

This defines a linear ordering of $\mathbb{N}$, in which we first have $3,5,7,\mathrm{\dots }$, followed by $2\cdot 3$, $2\cdot 5,\mathrm{\dots }$,followed by $2^2\cdot 3$, $2^2\cdot 5,\mathrm{\dots }$,and so on, and finally $2^{n+1},2^n,\mathrm{\dots },2,1$. So it looks like this:

$$3\succ 5\succ \mathrm{\cdots }\succ 3\cdot 2\succ 5\cdot 2\succ \mathrm{\cdots }\succ 3\cdot 2^n\succ 5\cdot 2^n\succ \mathrm{\cdots }\succ 2^2\succ 2\succ 1.$$

Sharkovskii’s theorem. Let $I\subset ℝ$ be an interval, and let $f:I\to ℝ$ be a continuous function. If $f$ has a periodic point of least period $n$, then $f$ has a periodic point of least period $k$, for each $k$ such that $n\succ k$.