Iterated function: Difference between revisions

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{{Short description|Result of repeatedly applying a mathematical function}}

{{Use dmy dates|date=May 2019|cs1-dates=y}}

{{MOS|article|date=July 2025| [[MOS:FORMULA]] - avoid mixing {{tag|math}} and {{tl|math}} in the same expression}}

[[File:Powers of rotation, shear, and their compositions.svg|thumb|400px|Iterated transformations of the object on the left<br>On top is a clockwise rotation by 90°. It has [[Order (group theory)|order]] 4, because that is the smallest positive exponent that produces the identity. Below is a [[shear mapping]] with infinite order.<br><small>Below that are their [[Function composition|compositions]], which both have order 3.</small>]]

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Let {{mvar|''X''}} be a set and {{math|''f'': ''X'' → ''X''}} be a [[function (mathematics)|function]].

Defining {{math| ''f'' ^''n''}} as the ''n''-th iterate of {{mvar|''f''}} (a notation introduced by [[Hans Heinrich Bürmann]]{{citation needed|date=August 2020|reason=The fact is undisputable, but for historical completeness, let's find Bürmann's original work on this and add here as a citation. It must be dated significantly before 1813 (according to Herschel in 1820 und Cajori in 1929.)}} and [[John Frederick William Herschel]]<!-- in 1813 --><ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Peano_1903"/><ref name="Cajori_1929"/>), where ''n'' is a non-negative integer, by:

<math display="block">f^0 ~ \stackrel{\mathrm{def}}{=} ~ \operatorname{id}_X</math>

and

<math display="block">f^{n+1} ~ \stackrel{\mathrm{def}}{=} ~ f \circ f^{n},</math>

where {{math|id_''X''}} is the [[identity function]] on {{mvar|''X''}} and {{math|(''f'' {{text| {{math| <math>\circ</math> }} }} ''g'')(''x'') {{=}} ''f'' (''g''(''x''))}} denotes [[function composition]]. This notation has been traced to and [[John Frederick William Herschel]] in 1813.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Peano_1903"/><ref name="Cajori_1929"/> Herschel credited [[Hans Heinrich Bürmann]] for it, but without giving a specific reference to the work of Bürmann, which remains undiscovered.<ref>{{cite book|title=Encounters with Chaos and Fractals|first1=Denny|last1=Gulick|first2=Jeff|last2=Ford|edition=3rd|publisher=CRC Press|year=2024|isbn=9781003835776|page=2|url=https://books.google.com/books?id=aVQIEQAAQBAJ&pg=PA2}}</ref>

Because the notation {{math|''f'' ^''n''}} may refer to both iteration (composition) of the function {{mvar|''f''}} or [[Exponentiation#Iterated functions|exponentiation of the function]] {{mvar|''f''}} (the latter is commonly used in [[trigonometric functions|trigonometry]]), some mathematicians{{citation needed|date=August 2020|reason=Origin? Example authors?}} choose to use {{math|∘}} to denote the compositional meaning, writing {{math|''f''{{i sup|∘''n''}}(''x'')}} for the {{mvar|n}}-th iterate of the function {{math|''f''(''x'')}}, as in, for example, {{math|''f''{{i sup|∘3}}(''x'')}} meaning {{math|''f''(''f''(''f''(''x'')))}}. For the same purpose, {{math|''f'' ^[''n''](''x'')}} was used by [[Benjamin Peirce]]<ref name="Peirce_1852"/><ref name="Cajori_1929"/><ref group="nb">while {{math|''f'' ^(''n'')}} is taken for the [[Derivative#Lagrange's notation|{{math|''n''}}th derivative]]</ref> whereas [[Alfred Pringsheim]] and [[Jules Molk]] suggested {{math|{{i sup|''n''}}''f''(''x'')}} instead.<ref name="Pringsheim-Molk_1907"/><ref name="Cajori_1929"/><ref group="nb" name="NB_Rucker"/>

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=== Some formulas for fractional iteration===

One of several methods of finding a series formula for fractional iteration, making use of a fixed point, is as follows.<ref>{{cite web |title=Tetration.org |url=https://tetration.org/index.php/Fractional_iterationFractional_Iteration }}</ref>

# First determine a fixed point for the function such that {{math|1=''f''(''a'') = ''a''}}.

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:{{math|''g''(''ϕ''(''y'')) {{=}} ''ϕ''(''y''+1)}}, &nbsp; a form known as the [[Abel equation]].

Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at {{mvar|x}} = 0, {{mvar|f}}(0) = 0, one may often solve<ref>Kimura, Tosihusa (1971). "On the Iteration of Analytic Functions", [http://www.math.sci.kobe-u.ac.jp/~fe/ ''Funkcialaj Ekvacioj''] {{Webarchive|url=https://web.archive.org/web/20120426011125/http://www.math.sci.kobe-u.ac.jp/~fe/ |date=2012-04-26 }} '''14''', 197-238.</ref> [[Schröder's equation]] for a function Ψ, which makes {{math|''f''(''x'')}} locally conjugate to a mere dilation, {{math|''g''(''x'') {{=}} ''f'' '(0) ''x''}}, that is

:{{math|''f''(''x'') {{=}} Ψ⁻¹(''f'' '(0) Ψ(''x''))}}.

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[[File:Sine_iterations.svg|right|thumb|380px|

Iterates of the sine function (<span style="color:blue">blue</span>), in the first half-period. Half-iterate (<span style="color:orange">orange</span>), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it; and further fractional iterates up to the 1/64th. The functions below the (<span style="color:blue">blue</span>) sine are six integral iterates below it, starting with the second iterate (<span style="color:red">red</span>) and ending with the 64th iterate. The <span style="color:green">green</span> envelope triangle represents the limiting null iterate, thea sawtooth[[triangular function]] serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin).

(From the general pedagogy web-site.<ref>Curtright, T. L. [http://www.physics.miami.edu/~curtright/Schroeder.html Evolution surfaces and Schröder functional methods.]</ref> For the notation, see [http://www.physics.miami.edu/~curtright/TheRootsOfSin.pdf].)

]]

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* [[Functional square root]]

* [[Abel function]]

* [[Schröder's equation]]

* [[Böttcher's equation]]

* [[Infinite compositions of analytic functions]]