Koenigs function#Structure of univalent semigroups

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,

:$|f(z)|\backslash le\; M(r)\; |z|$

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,

:$h(f(z))=\; f^\backslash prime(0)\; h(z)\; ~.$

The function h is the uniform limit on compacta of the normalized iterates, $g\_n(z)=\; \backslash lambda^\{-n\}\; f^n(z)$.

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

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   {{mvar|f}} becomes multiplication by {{mvar|λ}}, a dilation on {{mvar|U}}.

• Uniqueness. If {{mvar|k}} is another solution then, by analyticity, it suffices to show that k = h near 0. Let

::$H=k\backslash circ\; h^\{-1\}\; (z)$

:near 0. Thus H(0) =0, H'(0)=1 and, for |z | small,

::$\backslash lambda\; H(z)=\backslash lambda\; h(k^\{-1\}\; (z))\; =\; h(f(k^\{-1\}(z))=h(k^\{-1\}(\backslash lambda\; z)=\; H(\backslash lambda\; z)~.$

: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 $F(z)=f(z)/\backslash lambda\; z,$ then by the Schwarz lemma

::$|F(z)\; -\; 1|\backslash le\; (1+|\backslash lambda|^\{-1\})|z|~.$

:On the other hand,

::$g\_n(z)\; =\; z\backslash prod\_\{j=0\}^\{n-1\}\; F(f^j(z))~.$

:Hence g_n converges uniformly for |z| ≤ r by the Weierstrass M-test since

::

• Univalence. By Hurwitz's theorem, since each gⁿ is univalent and normalized, i.e. fixes 0 and has derivative 1 there , their limit {{mvar|h}} is also univalent.

Let {{math|f_t (z)}} be a semigroup of holomorphic univalent mappings of {{mvar|D}} into itself fixing 0 defined

for {{math| t ∈ [0, ∞)}} such that

• $f\_s$ is not an automorphism for {{mvar|s}} > 0
• $f\_s(f\_t(z))=f\_\{t+s\}(z)$
• $f\_0(z)=z$
• $f\_t(z)$ is jointly continuous in {{mvar|t}} and {{mvar|z}}

Each {{math|f_s}} with {{mvar|s}} > 0 has the same Koenigs function, cf. iterated function. In fact, if h is the Koenigs function of

{{math|f {{=}} f₁}}, then {{math|h(f_s(z))}} satisfies Schroeder's equation and hence is proportion to h.

Taking derivatives gives

:$h(f\_s(z))\; =f\_s^\backslash prime(0)\; h(z).$

Hence {{mvar|h}} is the Koenigs function of {{math|f_s}}.

On the domain {{math|U {{=}} h(D)}}, the maps {{math|f_s}} become multiplication by $\backslash lambda(s)=f\_s^\backslash prime(0)$, a continuous semigroup.

So $\backslash lambda(s)=\; e^\{\backslash mu\; s\}$ where {{mvar|μ}} is a uniquely determined solution of {{math|e ^μ {{=}} λ}} with Re{{mvar|μ}} < 0. It follows that the semigroup is differentiable at 0. Let

:$v(z)=\backslash partial\_t\; f\_t(z)|\_\{t=0\},$

a holomorphic function on {{mvar|D}} with v(0) = 0 and {{math|v'(0)}} = {{mvar|μ}}.

Then

:$\backslash partial\_t\; (f\_t(z))\; h^\backslash prime(f\_t(z))=\; \backslash mu\; e^\{\backslash mu\; t\}\; h(z)=\backslash mu\; h(f\_t(z)),$

so that

:$v=v^\backslash prime(0)\; \{h\backslash over\; h^\backslash prime\}$

and

:$\backslash partial\_t\; f\_t(z)\; =\; v(f\_t(z)),\backslash ,\backslash ,\backslash ,\; f\_t(z)=0\; ~,$

the flow equation for a vector field.

Restricting to the case with 0 < λ < 1, the h(D) must be starlike so that

:$\backslash Re\; \{zh^\backslash prime(z)\backslash over\; h(z)\}\; \backslash ge\; 0\; ~.$

Since the same result holds for the reciprocal,

:$\backslash Re\; \{v(z)\backslash over\; z\}\backslash le\; 0\; ~,$

so that {{math|v(z)}} satisfies the conditions of {{harvtxt|Berkson|Porta|1978}}

:

Conversely, reversing the above steps, any holomorphic vector field {{math|v(z)}} satisfying these conditions is associated to a semigroup {{math|f_t}}, with

:$h(z)=\; z\; \backslash exp\; \backslash int\_0^z\; \{v^\backslash prime(0)\; \backslash over\; v(w)\}\; -\{1\backslash over\; w\}\; \backslash ,\; dw.$

{{reflist}}

• {{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.|last2=Gamelin|first2=T. D. W.|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