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| In [[mathematics]], the '''Koenigs function''' is a function arising in [[complex analysis]] and [[dynamical systems]]. Introduced in 1884 by the French mathematician [[Gabriel Koenigs]], it gives a canonical representation as dilations of a [[univalent function|univalent holomorphic mapping]], or a [[semigroup]] of mappings, of the [[unit disk]] in the [[complex numbers]] into itself.
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| ==Existence and uniqueness of Koenigs function==
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| Let ''D'' be the [[unit disk]] in the complex numbers. Let ''f'' be a [[holomorphic function]] mapping ''D'' into itself, fixing the point 0. with ''f'' not identically ''0'' and ''f'' not an automorphism of ''D'', i.e. a [[Möbius transformation]] defined by a matrix in SU(1,1). By the [[Denjoy-Wolff theorem]], ''f'' leaves invariant each disk ''|z|'' < ''r'' and the iterates of ''f'' converge uniformly on compacta to 0: if fact for 0 < ''r'' < 1,
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| :<math> |f(z)|\le M(r) |z|</math>
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| for |''z''| ≤ ''r'' with ''M''(''r'') < 1. Moreover ''f'' '(0) = λ with 0 < |λ| < 1.
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| {{harvtxt|Koenigs|1884}} proved that there is a unique holomorphic function ''h'' defined on ''D'', called the
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| '''Koenigs function'''
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| such that ''h''(0) = 0, ''h'''(0) = 1 and [[Schroeder's equation]] is satisfied:
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| :<math> h(f(z))= f^\prime(0) h(z).</math>
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| The function ''h'' is the uniform limit on compacta of the normalized iterates <math>g_n(z)= \lambda^{-n} f^n(z)</math>. Moreover if ''f'' is univalent so is ''h''. <ref>{{harvnb|Carleson|Gamelin|1993|pp=28–32}}</ref><ref>{{harvnb|Shapiro|1993|pp=90–93}}</ref>
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| As a consequence, when ''f'' (and hence ''h'') are univalent, ''D'' ca be identified with the open domain ''U'' = ''h''(''D''). Under this conformal identification, the mapping ''f'' becomes multiplication by λ, a dilation on ''U''.
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| ===Proof===
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| *''Uniqueness''. If ''k'' is another solution then, by analyticity, it suffices to show that ''k'' = ''h'' near 0. Let <math> H=k\circ h^{-1} (z) </math> near 0. Thus ''H''(0) =0, ''H'''(0)=1 and for ''|z|'' small
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| ::<math>\lambda H(z)=\lambda h(k^{-1} (z)) = h(f(k^{-1}(z))=h(k^{-1}(\lambda z)= H(\lambda z).</math>
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| :Substituting into the power series for ''H'', it follows that ''H''(''z'') = ''z'' near 0. Hence ''h'' = ''k'' near 0.
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| *''Existence''. If <math> F(z)=f(z)/\lambda z,</math> then by the [[Schwarz lemma]]
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| ::<math>|F(z) - 1|\le (1+|\lambda|^{-1})|z|</math>
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| :On the other hand
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| ::<math> g_n(z) = z\prod_{j=0}^{n-1} F(f^j(z)).</math>
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| :Hence ''g''<sub>''n''</sub> converges uniformly for |''z''| ≤ ''r'' by the [[Weierstrass M-test]] since
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| ::<math> \sum \sup_{|z|\le r} |1 -F\circ f^j(z)| \le (1+|\lambda|^{-1}) \sum M(r)^j <\infty.</math>
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| *''Univalence''. By [[Hurwitz's theorem (complex analysis)|Hurwitz's theorem]], since each ''g''<sup>''n''</sup> is univalent and normalized, i.e. fixes 0 and has derivative 1 there , their limit ''h'' is also univalent.
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| ==Koenigs function of a semigroup==
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| Let <math>f_t(z)</math> be a semigroup of holomorphic univalent mappings of ''D'' into itself fixing 0 defined for <math> t\in [0,\infty)</math> such that
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| *<math>f_s</math> is not an automorphism for ''s'' > 0
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| *<math> f_s(f_t(z))=f_{t+s}(z)</math>
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| *<math> f_0(z)=z</math>
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| *<math> f_t(z)</math> is jointly continuous in ''t'' and ''z''
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| Each <math> f_s</math> with ''s'' > 0 has the same Koenigs function, cf. [[iterated function]]. In fact if ''h'' is the Koenigs function of ''f'' =''f''<sub>1</sub> then
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| <math> h(f_s(z))</math> satisfies Schroeder's equation and hence is proportion to ''h''. Taking derivatives gives
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| :<math>h(f_s(z)) =f_s^\prime(0) h(z).</math>
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| Hence ''h'' is the Koenigs function of ''f''<sub>''s''</sub>.
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| ==Structure of univalent semigroups==
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| On the domain ''U'' = ''h''(''D''), the maps ''f''<sub>''s''</sub> become multiplication by <math>\lambda(s)=f_s^\prime(0)</math>, a continuous semigroup.
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| So <math>\lambda(s)= e^{\mu s}</math> where μ is a uniquely determined solution of <math> e^\mu=\lambda</math> with Re μ < 0. It follows that the semigroup is differentiable at 0. Let
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| :<math> v(z)=\partial_t f_t(z)|_{t=0},</math>
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| a holomorphic function on ''D'' with ''v''(0) = 0 and ''v'''(0) = μ. Then | |
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| :<math>\partial_t (f_t(z)) h^\prime(f_t(z))= \mu e^{\mu t} h(z)=\mu h(f_t(z)),</math>
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| so that
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| :<math> v=v^\prime(0) {h\over h^\prime}</math>
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| and | |
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| :<math>\partial_t f_t(z) = v(f_t(z)),\,\,\, f_t(z)=0</math>
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| the flow equation for a vector field.
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| Restricting to the case with 0 < λ < 1, the ''h''(''D'') must be [[star domain|starlike]] so that
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| :<math>\Re {zh^\prime(z)\over h(z)} \ge 0</math> | |
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| Since the same result holds for the reciprocal,
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| :<math> \Re {v(z)\over z}\le 0.</math>
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| so that ''v''(''z'') satisfies the conditions of {{harvtxt|Berkson|Porta|1978}}
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| :<math> v(z)= z p(z),\,\,\, \Re p(z) \le 0, \,\,\, p^\prime(0) < 0.</math>
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| Conversely, reversing the above steps, any holomorphic vector field ''v''(''z'')
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| satisfying these conditions is associated to a semigroup ''f''<sub>''t''</sub>, with
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| :<math> h(z)= z \exp \int_0^z {v^\prime(0) \over v(w)} -{1\over w} \, dw.</math>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|last=Berkson|first=E.|last2= Porta|first2= H.|title=Semigroups of analytic functions and composition operators|journal=
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| Michigan Math. J.|volume= 25|year= 1978|pages= 101–115}}
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| *{{citation|last=Carleson|first=L.|last2= Gamelin|first2= T. D. W.|title=Complex dynamics|series=
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| Universitext: Tracts in Mathematics|publisher= Springer-Verlag|year=1993|isbn=0-387-97942-5}}
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| *{{citation|last2=Shoikhet|first2=D.|title=Linearization Models for Complex Dynamical Systems: Topics in Univalent Functions, Functional Equations and Semigroup Theory|volume=208|series= Operator Theory: Advances and Applications|first=M.|last= Elin|publisher=Springer|year= 2010|isbn=303460508}}
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| *{{citation|first=G.P.X.|last=Koenigs|title=Recherches sur les intégrales de certaines équations fonctionnelles|journal=Ann. Sci. Ecole Norm. Sup.|volume= 1|year=1884|pages= 2–41}}
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| *{{citation|last=Shapiro|first=J. H.|title=Composition operators and classical function theory|series=Universitext: Tracts in Mathematics|publisher= Springer-Verlag|year= 1993|isbn=0-387-94067-7}}
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| *{{citation|last=Shoikhet|first=D.|title=Semigroups in geometrical function theory|publisher= Kluwer Academic Publishers|year= 2001|isbn=0-7923-7111-9 }}
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| [[Category:Complex analysis]]
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| [[Category:Dynamical systems]]
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| [[Category:Types of functions]]
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