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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=== 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]]