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# 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 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. $r<s$ and $p>1$;

2. $r=s$ and $p<q$;

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

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

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

Sharkovskii’s theorem. Let $I\subset\mathbb{R}$ be an interval, and let $f:I\rightarrow\mathbb{R}$ 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 $n\succ k$.

## Mathematics Subject Classification

37E05*no label found*

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