# Koenigs function

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In [mathematics](/source/Mathematics), the **Koenigs function** is a function arising in [complex analysis](/source/Complex_analysis) and [dynamical systems](/source/Dynamical_systems). Introduced in 1884 by the French mathematician [Gabriel Koenigs](/source/Gabriel_Koenigs), it gives a canonical representation as dilations of a [univalent holomorphic mapping](/source/Univalent_function), or a [semigroup](/source/Semigroup) of mappings, of the [unit disk](/source/Unit_disk) in the [complex numbers](/source/Complex_numbers) into itself.

## Existence and uniqueness of Koenigs function

Let *D* be the [unit disk](/source/Unit_disk) in the complex numbers. Let f be a [holomorphic function](/source/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](/source/M%C3%B6bius_transformation) defined by a matrix in SU(1,1).

By the [Denjoy-Wolff theorem](/source/Denjoy-Wolff_theorem), f leaves invariant each disk |*z* | < *r* and the iterates of f converge uniformly on compacta to 0: in fact for 0 < r < 1,

- | f ( z ) | ≤ M ( r ) | z | {\displaystyle |f(z)|\leq M(r)|z|}

for |*z* | ≤ *r* with *M*(*r* ) < 1. Moreover f '(0) = λ with 0 < |λ| < 1.

[Koenigs (1884)](#CITEREFKoenigs1884) proved that there is a unique holomorphic function *h* defined on *D*, called the **Koenigs function**, such that h(0) = 0, h '(0) = 1 and [Schröder's equation](/source/Schr%C3%B6der's_equation) is satisfied,

- h ( f ( z ) ) = f ′ ( 0 ) h ( z ) . {\displaystyle h(f(z))=f^{\prime }(0)h(z)~.}

The function *h* is *the [uniform limit](/source/Uniform_limit) on [compacta](/source/Compact_space) of the normalized iterates*, g n ( z ) = λ − n f n ( z ) {\displaystyle g_{n}(z)=\lambda ^{-n}f^{n}(z)} .

Moreover, if f is univalent, so is h.[1][2]

As a consequence, when f (and hence h) are univalent, D can be identified with the open domain *U* = *h*(*D*). Under this conformal identification, the mapping f becomes multiplication by λ, a dilation on U.

### Proof

- *Uniqueness*. If k is another solution then, by analyticity, it suffices to show that *k* = *h* near 0. Let

- - H = k ∘ h − 1 ( z ) {\displaystyle H=k\circ h^{-1}(z)}

- near 0. Thus *H*(0) =0, *H'*(0)=1 and, for |*z* | small, - λ H ( z ) = λ h ( k − 1 ( z ) ) = h ( f ( k − 1 ( z ) ) = h ( k − 1 ( λ z ) = H ( λ z ) . {\displaystyle \lambda H(z)=\lambda h(k^{-1}(z))=h(f(k^{-1}(z))=h(k^{-1}(\lambda z)=H(\lambda z)~.}

- Substituting into the [power series](/source/Power_series) for H, it follows that *H*(*z*) = *z* near 0. Hence *h* = *k* near 0.

- *Existence*. If F ( z ) = f ( z ) / λ z , {\displaystyle F(z)=f(z)/\lambda z,} then by the [Schwarz lemma](/source/Schwarz_lemma)

- - | F ( z ) − 1 | ≤ ( 1 + | λ | − 1 ) | z | . {\displaystyle |F(z)-1|\leq (1+|\lambda |^{-1})|z|~.}

- On the other hand, - g n ( z ) = z ∏ j = 0 n − 1 F ( f j ( z ) ) . {\displaystyle g_{n}(z)=z\prod _{j=0}^{n-1}F(f^{j}(z))~.}

- Hence *g**n* converges uniformly for |*z*| ≤ *r* by the [Weierstrass M-test](/source/Weierstrass_M-test) since

- - ∑ sup | z | ≤ r | 1 − F ∘ f j ( z ) | ≤ ( 1 + | λ | − 1 ) ∑ M ( r ) j < ∞ . {\displaystyle \sum \sup _{|z|\leq r}|1-F\circ f^{j}(z)|\leq (1+|\lambda |^{-1})\sum M(r)^{j}<\infty .}

- *Univalence*. By [Hurwitz's theorem](/source/Hurwitz's_theorem_(complex_analysis)), since each *g**n* is univalent and normalized, i.e. fixes 0 and has derivative 1 there, their limit h is also univalent.

## Koenigs function of a semigroup

Let *f**t* (*z*) be a semigroup of holomorphic univalent mappings of D into itself fixing 0 defined for *t* ∈ [0, ∞) such that

- f s {\displaystyle f_{s}} is not an automorphism for s > 0

- f s ( f t ( z ) ) = f t + s ( z ) {\displaystyle f_{s}(f_{t}(z))=f_{t+s}(z)}

- f 0 ( z ) = z {\displaystyle f_{0}(z)=z}

- f t ( z ) {\displaystyle f_{t}(z)} is jointly continuous in t and z

Each *f**s* with s > 0 has the same Koenigs function, cf. [iterated function](/source/Iterated_function). In fact, if *h* is the Koenigs function of *f* = *f*1, then *h*(*f**s*(*z*)) satisfies Schroeder's equation and hence is proportion to *h*.

Taking derivatives gives

- h ( f s ( z ) ) = f s ′ ( 0 ) h ( z ) . {\displaystyle h(f_{s}(z))=f_{s}^{\prime }(0)h(z).}

Hence h is the Koenigs function of *f**s*.

## Structure of univalent semigroups

On the domain *U* = *h*(*D*), the maps *f**s* become multiplication by λ ( s ) = f s ′ ( 0 ) {\displaystyle \lambda (s)=f_{s}^{\prime }(0)} , a continuous semigroup. So λ ( s ) = e μ s {\displaystyle \lambda (s)=e^{\mu s}} where μ is a uniquely determined solution of *e μ = λ* with Reμ < 0. It follows that the semigroup is differentiable at 0. Let

- v ( z ) = ∂ t f t ( z ) | t = 0 , {\displaystyle v(z)=\partial _{t}f_{t}(z)|_{t=0},}

a holomorphic function on D with *v*(0) = 0 and *v'*(0) = μ.

Then

- ∂ t ( f t ( z ) ) h ′ ( f t ( z ) ) = μ e μ t h ( z ) = μ h ( f t ( z ) ) , {\displaystyle \partial _{t}(f_{t}(z))h^{\prime }(f_{t}(z))=\mu e^{\mu t}h(z)=\mu h(f_{t}(z)),}

so that

- v = v ′ ( 0 ) h h ′ {\displaystyle v=v^{\prime }(0){h \over h^{\prime }}}

and

- ∂ t f t ( z ) = v ( f t ( z ) ) , f t ( z ) = 0 , {\displaystyle \partial _{t}f_{t}(z)=v(f_{t}(z)),\,\,\,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](/source/Star_domain) so that

- ℜ z h ′ ( z ) h ( z ) ≥ 0 . {\displaystyle \Re {zh^{\prime }(z) \over h(z)}\geq 0~.}

Since the same result holds for the reciprocal,

- ℜ v ( z ) z ≤ 0 , {\displaystyle \Re {v(z) \over z}\leq 0~,}

so that *v*(*z*) satisfies the conditions of [Berkson & Porta (1978)](#CITEREFBerksonPorta1978)

- v ( z ) = z p ( z ) , ℜ p ( z ) ≤ 0 , p ′ ( 0 ) < 0. {\displaystyle v(z)=zp(z),\,\,\,\Re p(z)\leq 0,\,\,\,p^{\prime }(0)<0.}

Conversely, reversing the above steps, any holomorphic vector field *v*(*z*) satisfying these conditions is associated to a semigroup *f**t*, with

- h ( z ) = z exp ⁡ ∫ 0 z v ′ ( 0 ) v ( w ) − 1 w d w . {\displaystyle h(z)=z\exp \int _{0}^{z}{v^{\prime }(0) \over v(w)}-{1 \over w}\,dw.}

## Notes

1. **[^](#cite_ref-1)** [Carleson & Gamelin 1993](#CITEREFCarlesonGamelin1993), pp. 28–32

1. **[^](#cite_ref-2)** [Shapiro 1993](#CITEREFShapiro1993), pp. 90–93

## References

- Berkson, E.; Porta, H. (1978), "Semigroups of analytic functions and composition operators", *Michigan Math. J.*, **25**: 101–115, [doi](/source/Doi_(identifier)):[10.1307/mmj/1029002009](https://doi.org/10.1307%2Fmmj%2F1029002009)

- [Carleson, L.](/source/Lennart_Carleson); [Gamelin, T. D. W.](/source/Theodore_Gamelin) (1993), [*Complex dynamics*](https://archive.org/details/complexdynamics0000carl), Universitext: Tracts in Mathematics, Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [0-387-97942-5](https://en.wikipedia.org/wiki/Special:BookSources/0-387-97942-5)

- Elin, M.; Shoikhet, D. (2010), *Linearization Models for Complex Dynamical Systems: Topics in Univalent Functions, Functional Equations and Semigroup Theory*, Operator Theory: Advances and Applications, vol. 208, Springer, [ISBN](/source/ISBN_(identifier)) [978-3034605083](https://en.wikipedia.org/wiki/Special:BookSources/978-3034605083)

- Koenigs, G.P.X. (1884), "Recherches sur les intégrales de certaines équations fonctionnelles", *Ann. Sci. École Norm. Sup.*, **1**: 2–41

- [Kuczma, Marek](/source/Marek_Kuczma) (1968). *Functional equations in a single variable*. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers. ASIN: B0006BTAC2

- Shapiro, J. H. (1993), *Composition operators and classical function theory*, Universitext: Tracts in Mathematics, Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [0-387-94067-7](https://en.wikipedia.org/wiki/Special:BookSources/0-387-94067-7)

- Shoikhet, D. (2001), *Semigroups in geometrical function theory*, Kluwer Academic Publishers, [ISBN](/source/ISBN_(identifier)) [0-7923-7111-9](https://en.wikipedia.org/wiki/Special:BookSources/0-7923-7111-9)

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