Iterated function: Difference between revisions

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

[[File:An iterated direct similarity yields spirals.svg|thumb|upright=1.8|[[Function composition|Composed]]  with  itself  '''repeatedly''',  [[Similarity (geometry)|similarity]]  {{math|''F''}} of [[Similarity (geometry)#Similarity_with_a_center|center]] ''S ''enlarges the smallest [[Regular polygon|'''regular''']] [[pentagon]] into successive [[Concentric objects|concentric]] pentagons,  in manner that the outline of each one<br

/>passes through all vertices of the previous pentagon, of which it is the [[Image_(mathematics)#Image_of_an_element|image]] under {{math|''F''}}.  If  [[Transformation (function)|transformation]]  {{math|''F''}}<br

/>of [[Similarity (geometry)#Similarity_with_a_center|center]] ''S '' enlarges the smallest [[Regular polygon|'''regular''']] [[pentagon]] into successive [[Concentric objects|concentric]] pentagons,  in manner that the outline of each one<br

/>is  '''iterated'''  indefinitely,   then  ''A ''  and  ''K''<br

/>passes through all vertices of the previous pentagon,<br

/>are  the  starting  points  of  two infinite infinite  [[spiral]]s.]]

/>of which it is the [[Image_(mathematics)#Image_of_an_element|image]] under {{math|''F''}}.  If  [[Transformation (function)|transformation]]  {{math|''F''}}<br

/>is  '''iterated'''  indefinitely,   then  ''A ''  and  ''K''<br

/>are  the  starting  points  of  two  infinite  [[spiral]]s.]]

In [[mathematics]], an '''iterated function''' is a function {{math|''X&nbsp;→&nbsp;X''}} (that is, a function from some [[Set (mathematics)|set]] {{mvar|X}} to itself) which is obtained by [[function composition|composing]] another function {{math|''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''X''}} with itself a certain number of 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 in the function as input, and this process is repeated. For example on the image on the right: