Iterated function: Difference between revisions

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Limit is a triangular function
References: fixed name in reference to Ikuo Kimura
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:{{math|''g''(''ϕ''(''y'')) {{=}} ''ϕ''(''y''+1)}},   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, TosihusaIkuo (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''))}}.