# Koenigs function

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 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 Template:Mvar be a holomorphic function mapping *D* into itself, fixing the point 0, with Template:Mvar not identically 0 and Template:Mvar not an automorphism of *D*, i.e. a Möbius transformation defined by a matrix in SU(1,1).

By the Denjoy-Wolff theorem, Template:Mvar leaves invariant each disk |*z* | < *r* and the iterates of Template:Mvar converge uniformly on compacta to 0: in fact for 0 < Template:Mvar < 1,

for |*z* | ≤ *r* with *M*(*r* ) < 1. Moreover Template:Mvar '(0) = Template:Mvar with 0 < |Template:Mvar| < 1.

Template:Harvtxt proved that there is a unique holomorphic function *h* defined on *D*, called the **Koenigs function**,
such that Template:Mvar(0) = 0, Template:Mvar '(0) = 1 and Schröder's equation is satisfied,

The function *h* is the uniform limit on compacta of the normalized iterates . Moreover, if Template:Mvar is univalent, so is *h*.^{[1]}^{[2]}

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

### Proof

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

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

*Existence*. If then by the Schwarz lemma

- Hence
*g*_{n}converges uniformly for |*z*| ≤*r*by the Weierstrass M-test since

*Univalence*. By Hurwitz's theorem, 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 Template:Mvar into itself fixing 0 defined
for *t* ∈ [0, ∞) such that

- is not an automorphism for Template:Mvar > 0
- is jointly continuous in Template:Mvar and Template:Mvar

Each *f*_{s} with Template:Mvar > 0 has the same Koenigs function, cf. 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

Hence Template:Mvar is the Koenigs function of *f*_{s}.

## Structure of univalent semigroups

On the domain *U* = *h*(*D*), the maps *f*_{s} become multiplication by , a continuous semigroup.
So where μ is a uniquely determined solution of with Re μ < 0. It follows that the semigroup is differentiable at 0. Let

a holomorphic function on *D* with *v*(0) = 0 and *v'*(0) = Template:Mvar. Then

so that

and

the flow equation for a vector field.

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

Since the same result holds for the reciprocal,

so that *v*(*z*) satisfies the conditions of Template:Harvtxt

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

## Notes

## References

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