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[[File:An iterated direct similarity yields spirals.svg|thumb|upright=1.8|[[Function composition|Composed]] with itself '''repeatedly''', [[Similarity (geometry)|similarity]] {{math|''F''}}<br
/>of [[Similarity (geometry)#
/>into successive [[Concentric objects|concentric]] pentagons, in manner that the outline<br
/>of each one passes through all vertices of the previous pentagon,<br
/>of which it is the [[
/>is '''iterated''' indefinitely, then ''A '' and ''K'' are<br
/>the starting points of two infinite [[spiral]]s.]]
In [[mathematics]], an '''iterated function''' is a function that is obtained by [[function composition|composing]] another function with itself two or several times. The process of repeatedly applying the same function is called [[iteration]]. In this process, starting from some initial object, the result of applying a given function is fed again into the function as input, and this process is repeated.
For example on the image on the right:
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<math display="block">f^{n+1} ~ \stackrel{\mathrm{def}}{=} ~ f \circ f^{n},</math>
where {{math|id<sub>''X''</sub>}} is the [[identity function]] on {{mvar|''X''}} and {{math|(''f'' {{text| {{math| <math>\circ</math> }} }} ''g'')(''x'') {{=}} ''f'' (''g''(''x''))}} denotes [[function composition]].
Because the notation {{math|''f'' <sup>''n''</sup>}} 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'' <sup>[''n'']</sup>(''x'')}} was used by [[Benjamin Peirce]]<ref name="Peirce_1852"/><ref name="Cajori_1929"/><ref group="nb">while {{math|''f'' <sup>(''n'')</sup>}} 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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==Limiting behaviour==
Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an [[attractive fixed point]]. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an [[unstable fixed point]].<ref>Istratescu, Vasile (1981). ''Fixed Point Theory, An Introduction'', D. Reidel, Holland. {{ISBN|90-277-1224-7}}.</ref>
When the points of the orbit converge to one or more limits, the set of [[accumulation point]]s of the orbit is known as the '''[[limit set]]''' or the '''ω-limit set'''.
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}}
==External
* {{cite web |url=https://www.researchgate.net/publication/362010262 |author-link=John Gill (climber) |first=John |last=Gill |title=A Primer on the Elementary Theory of Infinite Compositions of Complex Functions |publisher=Colorado State University |date=January 2017 }}
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