Editing Iterated function (section)

==Examples==
There are [[List of chaotic maps|many chaotic maps]]. Well-known iterated functions include the [[Mandelbrot set]] and [[iterated function systems]].

[[Ernst Schröder (mathematician)|Ernst Schröder]],<ref name="schr">{{cite journal |last=Schröder |first=Ernst |author-link=Ernst Schröder (mathematician) |year=1870 |title=Ueber iterirte Functionen|journal=Math. Ann. |volume=3 |issue= 2|pages=296–322 | doi=10.1007/BF01443992 |s2cid=116998358 }}</ref> in 1870, worked out special cases of the [[logistic map]], such as the chaotic case {{math|1=''f''(''x'') = 4''x''(1 − ''x'')}}, so that {{math|1=Ψ(''x'') = arcsin({{radic|''x''}})<sup>2</sup>}}, hence {{math|1=''f'' <sup>''n''</sup>(''x'') = sin(2<sup>''n''</sup> arcsin({{radic|''x''}}))<sup>2</sup>}}.

A nonchaotic case Schröder also illustrated with his method, {{math|1=''f''(''x'') = 2''x''(1 − ''x'')}}, yielded {{math|1=Ψ(''x'') = −{{sfrac|1|2}} ln(1 − 2''x'')}}, and hence  {{math|1=''f''<sup>''n''</sup>(''x'') = −{{sfrac|1|2}}((1 − 2''x'')<sup>2<sup>''n''</sup></sup> − 1)}}.

If {{mvar|''f''}} is the [[Group action (mathematics)|action]] of a group element on a set, then the iterated function corresponds to a [[free group]].

Most functions do not have explicit general [[closed-form expression]]s for the ''n''-th iterate. The table below lists some<ref name="schr"/> that do. Note that all these expressions are valid even for non-integer and negative ''n'', as well as non-negative integer ''n''.

{| class=wikitable width=100%
!<math>f(x)</math>
!<math>f^n(x)</math>
|-
|<math>x+b</math>
|<math>x+nb</math>
|-
|<math>ax+b \ (a\ne 1)</math>
|<math>a^nx+\frac{a^n-1}{a-1}b</math>
|-
|<math>ax^b \ (b\ne 1)</math>
|<math>a^{\frac{b^n-1}{b-1}}x^{b^n}</math>
|-
|<math>ax^2 + bx + \frac{b^2 - 2b}{4a}</math> (see note)<br>
|<math>\frac{2\alpha^{2^n} - b}{2a}</math><br>
where:
*<math>\alpha = \frac{2ax + b}{2}</math>
|-
|<math>ax^2 + bx + \frac{b^2 - 2b - 8}{4a}</math> (see note)<br>
|<math>\frac{2\alpha^{2^n} + 2\alpha^{-2^n} - b}{2a}</math><br>
where:
*<math>\alpha = \frac{2ax + b \pm \sqrt{(2ax + b)^2 - 16}}{4}</math>
|-
|<math>\frac{ax + b}{cx + d}</math> &nbsp; ([[fractional linear transformation]])<ref>Brand, Louis, "A sequence defined by a difference equation," ''[[American Mathematical Monthly]]'' '''62''', September 1955, 489&ndash;492.   [https://www.jstor.org/discover/10.2307/2307362  online]</ref>
|<math>\frac{a}{c} + \frac{bc - ad}{c} \left [ \frac{(cx - a + \alpha)\alpha^{n - 1} - (cx - a + \beta)\beta^{n - 1}}{(cx - a + \alpha)\alpha^{n} - (cx - a + \beta)\beta^{n}} \right ]</math><br>
where:
*<math>\alpha = \frac{a + d + \sqrt{(a - d)^2 + 4bc}}{2}</math>
*<math>\beta = \frac{a + d - \sqrt{(a - d)^2 + 4bc}}{2}</math>
|-
|<math>g^{-1}\Big(h\bigl(g(x)\bigr)\Big)</math>
|<math>g^{-1}\Bigl(h^n\bigl(g(x)\bigr)\Bigr)</math>
|-
|<math>g^{-1}\bigl(g(x)+b\bigr)</math> &nbsp; (generic [[Abel equation]])
|<math>g^{-1}\bigl(g(x)+nb\bigr)</math>
|-
| <math>\sqrt{x^2 + b}</math>
| <math>\sqrt{x^2 + bn}</math>
|-
|<math>g^{-1}\Bigl(a\ g(x)+b\Bigr) \ (a\ne 1 \vee b=0)</math>
|<math>g^{-1}\Bigl(a^ng(x)+\frac{a^n-1}{a-1}b\Bigr)</math>
|-
| <math>\sqrt{ax^2 + b}</math>
| <math>\sqrt{a^nx^2 + \frac{a^n - 1}{a - 1}b}</math>
|-
| <math>T_m (x)=\cos (m \arccos x)</math> ([[Chebyshev polynomials#Trigonometric definition|Chebyshev polynomial]] for integer ''m'')
| <math>T_{mn}=\cos(m^n \arccos x)</math>
|}
Note: these two special cases of {{math|''ax''<sup>2</sup> + ''bx'' + ''c''}} are the only cases that have a closed-form solution. Choosing ''b'' = 2 = –''a'' and ''b'' = 4 = –''a'', respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table.

Some of these examples are related among themselves by simple conjugacies.