Editing Iterated function (section)

==Conjugacy==

If {{mvar|f}} and {{mvar|g}} are two iterated functions, and there exists a [[homeomorphism]] {{mvar|h}} such that  {{math| ''g'' {{=}}  ''h''<sup>−1</sup> ○ ''f'' ○ ''h'' }},  then {{mvar|f}} and {{mvar|g}} are said to be [[Topological conjugacy|topologically conjugate]].

Clearly, topological conjugacy is preserved under iteration, as {{math|''g''<sup>''n''</sup>&nbsp;{{=}}&nbsp;''h''<sup>−1</sup> &nbsp;○&nbsp;''f'' <sup>''n''</sup> ○ ''h''}}.  Thus, if one can solve for one iterated function system, one also  has solutions for all topologically conjugate systems. For example, the [[tent map]] is topologically conjugate to the [[logistic map]]. As a special case, taking {{math|''f''(''x'') {{=}} ''x''&nbsp;+&nbsp;1}}, one has the iteration of {{math|''g''(''x'') {{=}} ''h''<sup>&minus;1</sup>(''h''(''x'')&nbsp;+&nbsp;1)}} as 
:{{math|''g''<sup>''n''</sup>(''x'') {{=}} ''h''<sup>&minus;1</sup>(''h''(''x'')&nbsp;+&nbsp;''n'')}},  &nbsp;  for any function {{mvar|h}}.

Making the substitution {{math|''x'' {{=}} ''h''<sup>&minus;1</sup>(''y'') {{=}} ''ϕ''(''y'')}} yields 
:{{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<sup>−1</sup>(''f'' '(0) Ψ(''x''))}}.

Thus, its iteration orbit, or flow, under suitable provisions (e.g., {{math|''f'' '(0) ≠ 1}}), amounts to the conjugate of the orbit of the monomial,
:{{math|Ψ<sup>−1</sup>(''f'' '(0)<sup>''n''</sup> Ψ(''x''))}},  
where {{mvar|n}} in this expression serves as a plain exponent: ''functional iteration has been reduced to multiplication!''  Here, however, the exponent {{mvar|n}} no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit:<ref>{{cite journal |author-last1=Curtright |author-first1=T. L. |author-link1=Thomas Curtright|author-last2=Zachos |author-first2=C. K. |author-link2=Cosmas Zachos | year=2009|title=Evolution Profiles and Functional Equations |journal=Journal of Physics A |volume=42|issue=48 |pages=485208|doi=10.1088/1751-8113/42/48/485208|arxiv=0909.2424|bibcode=2009JPhA...42V5208C|s2cid=115173476 }}</ref> the [[monoid]] of the Picard sequence (cf. [[transformation semigroup]]) has generalized to a full [[continuous group]].<ref>For explicit instance, example 2 above amounts to just {{math|''f'' <sup>''n''</sup>(''x'') {{=}} Ψ<sup>−1</sup>((ln&nbsp;2)<sup>''n''</sup> Ψ(''x''))}}, for ''any n'', not necessarily integer, where Ψ is the solution of the relevant [[Schröder's equation]], {{math|Ψ({{sqrt|2}}<sup>''x''</sup>) {{=}}&nbsp;ln&nbsp;2&nbsp;Ψ(''x'')}}. This solution is also the infinite ''m'' limit of {{math|(''f'' <sup>''m''</sup>(''x'')&nbsp;−&nbsp;2)/(ln&nbsp;2)<sup>''m''</sup>}}.</ref>

[[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, a [[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].)
]]

This method (perturbative determination of the principal [[eigenfunction]] Ψ, cf. [[Carleman matrix]]) is equivalent to the algorithm of the preceding section, albeit, in practice,  more powerful and systematic.