Editing Sharkovskii's theorem (section)

==Statement==
For some [[interval (mathematics)|interval]] <math>I\subset \mathbb{R}</math>, suppose that
<math display=block>f : I \to I</math>
is a [[continuous function]]. The number <math>x</math> is called a ''periodic point of period <math>m</math>'' if <math>f^{(m)}(x)=x</math>, where <math>f^{(m)}</math> denotes the [[iterated function]] obtained by composition of <math>m</math> copies of <math>f</math>. The number <math>x</math> is said to have ''least period <math>m</math>'' if, in addition, <math>f^{(k)}(x)\ne x</math> for all <math>0<k<m</math>. Sharkovskii's theorem concerns the possible least periods of periodic points of <math>f</math>. Consider the following ordering of the positive [[integer]]s, sometimes called the Sharkovskii ordering:<ref>K. Burns, B. Hasselblatt, "[https://www.math.arizona.edu/~dwang/BurnsHasselblattRevised-1.pdf The Sharkovsky Theorem: A Natural Direct Proof]" (2008). Accessed 3 February 2023.</ref>
<math display=block>\begin{array}{cccccccc}
3 & 5 & 7 & 9 & 11 & \ldots & (2n+1)\cdot2^{0} & \ldots\\
3\cdot2 & 5\cdot2 & 7\cdot2 & 9\cdot2 & 11\cdot2 & \ldots & (2n+1)\cdot2^{1} & \ldots\\
3\cdot2^{2} & 5\cdot2^{2} & 7\cdot2^{2} & 9\cdot2^{2} & 11\cdot2^{2} & \ldots & (2n+1)\cdot2^{2} & \ldots\\
3\cdot2^{3} & 5\cdot2^{3} & 7\cdot2^{3} & 9\cdot2^{3} & 11\cdot2^{3} & \ldots & (2n+1)\cdot2^{3} & \ldots\\
 & \vdots\\
\ldots & 2^{n} & \ldots & 2^{4} & 2^{3} & 2^{2} & 2 & 1\end{array}</math>

It consists of:
* [[odd number|the odd numbers]] ''excluding'' <math> 1 </math> <math> = (2n+1)\cdot2^0</math> in ''increasing'' order,
* 2 times the odd numbers <math> = (2n+1)\cdot2^1</math> in ''increasing'' order,
* 4 times the odd numbers <math> = (2n+1)\cdot2^2</math> in ''increasing'' order,
* 8 times the odd numbers <math> = (2n+1)\cdot2^3</math>,
* etc. <math> = (2n+1)\cdot2^m</math>
* finally, the powers of two <math> = 2^n</math> in ''decreasing'' order.

This ordering is a [[total order]]: every positive integer appears exactly once somewhere on this list. However, it is not a [[well-order]]. In a well-order, every subset would have an earliest element, but in this order there is no earliest power of two.

Sharkovskii's theorem states that if <math>f</math> has a periodic point of least period <math>m</math>, and <math>m</math> precedes <math>n</math> in the above ordering, then <math>f</math> has also a periodic point of least period <math>n</math>.

One consequence is that if <math>f</math> has only finitely many periodic points, then they must all have periods that are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods.

Sharkovskii's theorem does not state that there are ''stable'' cycles of those periods, just that there are cycles of those periods. For systems such as the [[logistic map]], the [[bifurcation diagram]] shows a range of parameter values for which apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer-generated picture.

The assumption of continuity is important. Without this assumption, the discontinuous [[piecewise linear function]] <math>f:[0,3) \to [0,3)</math> defined as:
<math display=block>f: x\mapsto \begin{cases}x+1 &\mathrm{for\ } 0\le x<2 \\ x-2 &\mathrm{for\ } 2\le x< 3\end{cases}</math>
for which every value has period 3, would be a counterexample. Similarly essential is the assumption of <math>f</math> being defined on an interval. Otherwise <math>f : x \mapsto (1 - x)^{-1}</math>, which is defined on real numbers except the one: <math>\mathbb R\setminus\{1\},</math> and for which every non-zero value has period 3, would be a counterexample.