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.

==Existence and uniqueness of Koenigs function== Let ''D'' be the [[unit disk]] in the complex numbers. Let {{mvar|f}} be a [[holomorphic function]] mapping ''D'' into itself, fixing the point 0, with {{mvar|f}} not identically 0 and {{mvar|f}} not an automorphism of ''D'', i.e. a [[Möbius transformation]] defined by a matrix in SU(1,1).

By the [[Denjoy-Wolff theorem]], {{mvar|f}} leaves invariant each disk |''z'' | < ''r'' and the iterates of {{mvar|f}} converge uniformly on compacta to 0: in fact for 0 < {{mvar|r}} < 1, :<math> |f(z)|\le M(r) |z|</math> for |''z'' | ≤ ''r'' with ''M''(''r'' ) < 1. Moreover {{mvar|f}} '(0) = {{mvar|λ}} with 0 < |{{mvar|λ}}| < 1.

{{harvtxt|Koenigs|1884}} proved that there is a unique holomorphic function ''h'' defined on ''D'', called the '''Koenigs function''', such that {{mvar|h}}(0) = 0, {{mvar|h}} '(0) = 1 and [[Schröder's equation]] is satisfied, :<math> h(f(z))= f^\prime(0) h(z) ~.</math>

The function ''h'' is ''the [[uniform limit]] on [[Compact space|compacta]] of the normalized iterates'', <math>g_n(z)= \lambda^{-n} f^n(z)</math>.

Moreover, if {{mvar|f}} is univalent, so is {{mvar|h}}.<ref>{{harvnb|Carleson|Gamelin|1993|pp=28–32}}</ref><ref>{{harvnb|Shapiro|1993|pp=90–93}}</ref>

As a consequence, when {{mvar|f}} (and hence {{mvar|h}}) are univalent, {{mvar|D}} can be identified with the open domain {{math|''U'' {{=}} ''h''(''D'')}}. Under this conformal identification, the mapping &nbsp; {{mvar|f}} becomes multiplication by {{mvar|λ}}, a dilation on {{mvar|U}}.

===Proof=== *''Uniqueness''. If {{mvar|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, ::<math>\lambda H(z)=\lambda h(k^{-1} (z)) = h(f(k^{-1}(z))=h(k^{-1}(\lambda z)= H(\lambda z)~.</math>

:Substituting into the [[power series]] for {{mvar|H}}, it follows that {{math|''H''(''z'') {{=}} ''z''}} near 0. Hence {{math|''h'' {{=}} ''k''}} near 0.

*''Existence''. If <math> F(z)=f(z)/\lambda z,</math> then by the [[Schwarz lemma]]

::<math>|F(z) - 1|\le (1+|\lambda|^{-1})|z|~.</math>

:On the other hand, ::<math> g_n(z) = z\prod_{j=0}^{n-1} F(f^j(z))~.</math>

:Hence ''g''<sub>''n''</sub> converges uniformly for |''z''| ≤ ''r'' by the [[Weierstrass M-test]] since

::<math> \sum \sup_{|z|\le r} |1 -F\circ f^j(z)| \le (1+|\lambda|^{-1}) \sum M(r)^j <\infty.</math>

*''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 {{mvar|h}} is also univalent.

==Koenigs function of a semigroup== Let {{math|''f''<sub>''t''</sub> (''z'')}} be a semigroup of holomorphic univalent mappings of {{mvar|D}} into itself fixing 0 defined for {{math| ''t'' ∈ [0, ∞)}} such that

*<math>f_s</math> is not an automorphism for {{mvar|s}} > 0 *<math> f_s(f_t(z))=f_{t+s}(z)</math> *<math> f_0(z)=z</math> *<math> f_t(z)</math> is jointly continuous in {{mvar|t}} and {{mvar|z}}

Each {{math|''f''<sub>''s''</sub>}} with {{mvar|s}} > 0 has the same Koenigs function, cf. [[iterated function]]. In fact, if ''h'' is the Koenigs function of {{math|''f'' {{=}} ''f''<sub>1</sub>}}, then {{math|''h''(''f''<sub>''s''</sub>(''z''))}} satisfies Schroeder's equation and hence is proportion to ''h''.

Taking derivatives gives :<math>h(f_s(z)) =f_s^\prime(0) h(z).</math> Hence {{mvar|h}} is the Koenigs function of {{math|''f''<sub>''s''</sub>}}.

==Structure of univalent semigroups== On the domain {{math|''U'' {{=}} ''h''(''D'')}}, the maps {{math|''f''<sub>''s''</sub>}} become multiplication by <math>\lambda(s)=f_s^\prime(0)</math>, a continuous semigroup. So <math>\lambda(s)= e^{\mu s}</math> where {{mvar|μ}} is a uniquely determined solution of {{math|''e <sup>μ</sup> {{=}} λ''}} with Re{{mvar|μ}} < 0. It follows that the semigroup is differentiable at 0. Let :<math> v(z)=\partial_t f_t(z)|_{t=0},</math> a holomorphic function on {{mvar|D}} with ''v''(0) = 0 and {{math|''v'''(0)}} = {{mvar|μ}}.

Then :<math>\partial_t (f_t(z)) h^\prime(f_t(z))= \mu e^{\mu t} h(z)=\mu h(f_t(z)),</math> so that :<math> v=v^\prime(0) {h\over h^\prime}</math> and :<math>\partial_t f_t(z) = v(f_t(z)),\,\,\, f_t(z)=0 ~,</math> the flow equation for a vector field.

Restricting to the case with 0 < λ < 1, the ''h''(''D'') must be [[star domain|starlike]] so that :<math>\Re {zh^\prime(z)\over h(z)} \ge 0 ~.</math>

Since the same result holds for the reciprocal, :<math> \Re {v(z)\over z}\le 0 ~,</math> so that {{math|''v''(''z'')}} satisfies the conditions of {{harvtxt|Berkson|Porta|1978}} :<math> v(z)= z p(z),\,\,\, \Re p(z) \le 0, \,\,\, p^\prime(0) < 0.</math>

Conversely, reversing the above steps, any holomorphic vector field {{math|''v''(''z'')}} satisfying these conditions is associated to a semigroup {{math|''f''<sub>''t''</sub>}}, with :<math> h(z)= z \exp \int_0^z {v^\prime(0) \over v(w)} -{1\over w} \, dw.</math>

==Notes== {{reflist}}

==References== *{{citation|last=Berkson|first=E.|last2= Porta|first2= H.|title=Semigroups of analytic functions and composition operators|journal=Michigan Math. J.|volume= 25|year= 1978|pages= 101–115|doi=10.1307/mmj/1029002009|doi-access=free}} *{{citation|last=Carleson|first=L.|author-link=Lennart Carleson|last2=Gamelin|first2=T. D. W.|author-link2=Theodore Gamelin|title=Complex dynamics|series=Universitext: Tracts in Mathematics|publisher=Springer-Verlag|year=1993|isbn=0-387-97942-5|url-access=registration|url=https://archive.org/details/complexdynamics0000carl}} *{{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= 978-3034605083}} *{{citation|first=G.P.X.|last=Koenigs|title=Recherches sur les intégrales de certaines équations fonctionnelles|journal=Ann. Sci. École Norm. Sup.|volume= 1|year=1884|pages= 2–41}} *{{cite book |title=Functional equations in a single variable |last=Kuczma |first=Marek|authorlink=Marek Kuczma|series=Monografie Matematyczne |year=1968 |publisher=PWN – Polish Scientific Publishers |location=Warszawa}} ASIN: B0006BTAC2 *{{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}} *{{citation|last=Shoikhet|first=D.|title=Semigroups in geometrical function theory|publisher= Kluwer Academic Publishers|year= 2001|isbn=0-7923-7111-9 }}

[[Category:Complex analysis]] [[Category:Dynamical systems]] [[Category:Types of functions]]