Editing Iterated function (section)

==Abelian property and iteration sequences==
In general, the following identity holds for all non-negative integers {{mvar|m}} and {{mvar|n}},

: <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''))}}.

The relation  {{math|1=(''f''<sup> ''m''</sup>)<sup>''n''</sup>(''x'') = (''f''<sup> ''n''</sup>)<sup>''m''</sup>(''x'') = ''f''<sup> ''mn''</sup>(''x'')}} also holds, analogous to the property of exponentiation that  {{math|1=(''a''<sup>''m''</sup>)<sup>''n''</sup> = (''a''<sup>''n''</sup>)<sup>''m''</sup> = ''a''<sup>''mn''</sup>}}.

The sequence of functions  {{math|''f'' <sup>''n''</sup>}} is called a '''Picard sequence''',<ref>{{cite book |title=Functional equations in a single variable |last=Kuczma |first=Marek| author-link=Marek Kuczma|series=Monografie Matematyczne |year=1968 |publisher=PWN – Polish Scientific Publishers |location=Warszawa}}</ref><ref>{{cite book|title=Iterative Functional Equations| last=Kuczma| first=M., Choczewski B., and Ger, R. |year=1990|publisher=Cambridge University Press|isbn= 0-521-35561-3|url=https://archive.org/details/iterativefunctio0000kucz|url-access=registration}}</ref> named after [[Charles Émile Picard]].

For a given {{mvar|x}} in {{mvar|X}}, the [[sequence]] of values  {{math|''f''<sup>''n''</sup>(''x'')}} is called the '''[[orbit (dynamics)|orbit]]''' of {{mvar|x}}.

If  {{math|1=''f'' <sup>''n''</sup> (''x'') =  ''f'' <sup>''n''+''m''</sup> (''x'')}} for some integer {{math|m > 0}}, the orbit is called a '''periodic orbit'''. The smallest such value of {{mvar|m}} for a given {{mvar|x}} is called the '''period of the orbit'''. The point {{mvar|x}} itself is called a [[periodic point]]. The [[cycle detection]] problem in computer science is the [[algorithm]]ic problem of finding the first periodic point in an orbit, and the period of the orbit.