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: <math>f^m \circ f^n = f^n \circ f^m = f^{m+n}~.</math>
This is structurally identical to the property of [[exponentiation]] that {{math|1=''a''<sup>''m''</sup>''a''<sup>''n''</sup> = ''a''<sup>''m'' + ''n''</sup>}}.
In general, for arbitrary general (negative, non-integer, etc.) indices {{mvar|m}} and {{mvar|n}}, this relation is called the '''translation functional equation''', cf. [[Schröder's equation]] and [[Abel equation]]. On a logarithmic scale, this reduces to the '''nesting property''' of [[Chebyshev polynomials]], {{math|1=''T''<sub>''m''</sub>(''T''<sub>''n''</sub>(''x'')) = ''T''<sub>''m n''</sub>(''x'')}}, since {{math|1=''T''<sub>''n''</sub>(''x'') = cos(''n'' arccos(''x''))}}.
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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'''.
The ideas of attraction and repulsion generalize similarly; one may categorize iterates into [[stable manifold|stable set]]s and [[unstable set]]s, according to the behavior of small [[Neighbourhood (mathematics)|neighborhood]]s under iteration.
Other limiting behaviors are possible; for example, [[wandering point]]s are points that move away, and never come back even close to where they started.
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\sqrt{2}^{ \sqrt{2}^{\sqrt{2}^{\cdots}} } = f^n(1) = 2 - (\ln 2)^n + \frac{(\ln 2)^{n+1}((\ln 2)^n-1)}{4(\ln 2-1)} - \cdots
</math>
which, taking just the first three terms, is correct to the first decimal place when ''n'' is
For {{math|1= ''n'' = −1}}, the series computes the inverse function {{sfrac|2|ln ''x''|ln 2}}.
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==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>}}.
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==Lie's data transport equation==
{{see also|Shift operator#Functions of a real variable}}▼
Iterated functions crop up in the series expansion of combined functions, such as {{math|''g''(''f''(''x''))}}.
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The initial flow velocity {{mvar|v}} suffices to determine the entire flow, given this exponential realization which automatically provides the general solution to the ''translation functional equation'',<ref name="acz">Aczel, J. (2006), ''Lectures on Functional Equations and Their Applications'' (Dover Books on Mathematics, 2006), Ch. 6, {{ISBN|978-0486445236}}.</ref>
:<math>f_t(f_\tau (x))=f_{t+\tau} (x) ~.</math>
▲{{see also|Shift operator#Functions of a real variable}}
==See also==
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==External Links==
* {{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 }}
[[Category:Dynamical systems]]
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