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 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: in fact for 0 < r < 1,
![{\displaystyle |f(z)|\leq M(r)|z|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d77aada5bc2eb9e1436b3c6ed280e56f825efd4)
for |z | ≤ r with M(r ) < 1. Moreover f '(0) = λ with 0 < |λ| < 1.
Koenigs (1884) 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 is satisfied,
![{\displaystyle h(f(z))=f^{\prime }(0)h(z)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a931af5ba489d747dec46915928991c0bd5b375)
The function h is the uniform limit on compacta of the normalized iterates,
.
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
![{\displaystyle H=k\circ h^{-1}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d55eb4a3d620dd034b082470221e724e49f18dc)
- near 0. Thus H(0) =0, H'(0)=1 and, for |z | small,
![{\displaystyle \lambda H(z)=\lambda h(k^{-1}(z))=h(f(k^{-1}(z))=h(k^{-1}(\lambda z)=H(\lambda z)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08ec7feaaf780650d6f2f72576562b46a69423b0)
- 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
![{\displaystyle |F(z)-1|\leq (1+|\lambda |^{-1})|z|~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae411f86f7209c0184c3f806e98f22e690407429)
- On the other hand,
![{\displaystyle g_{n}(z)=z\prod _{j=0}^{n-1}F(f^{j}(z))~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b97e3c353206a70fb6ab0f4cb0ba4646339b5c6)
- Hence gn converges uniformly for |z| ≤ r by the Weierstrass M-test since
![{\displaystyle \sum \sup _{|z|\leq r}|1-F\circ f^{j}(z)|\leq (1+|\lambda |^{-1})\sum M(r)^{j}<\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6f98680e217c38e7de9672441c7398bbf66b7a)
- Univalence. By Hurwitz's theorem, since each gn 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 ft (z) be a semigroup of holomorphic univalent mappings of D into itself fixing 0 defined
for t ∈ [0, ∞) such that
is not an automorphism for s > 0
![{\displaystyle f_{s}(f_{t}(z))=f_{t+s}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3d565e8fd47b76c7cf5df188e1ed568ba8c475)
![{\displaystyle f_{0}(z)=z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a65c8214e75f756bffb536feaa3f7977bc8ce0fa)
is jointly continuous in t and z
Each fs with s > 0 has the same Koenigs function, cf. iterated function. In fact, if h is the Koenigs function of
f = f1, then h(fs(z)) satisfies Schroeder's equation and hence is proportion to h.
Taking derivatives gives
![{\displaystyle h(f_{s}(z))=f_{s}^{\prime }(0)h(z).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/809bb869d5cc5871fdf53fd1f388af75fccf293e)
Hence h is the Koenigs function of fs.
Structure of univalent semigroups
On the domain U = h(D), the maps fs become multiplication by
, a continuous semigroup.
So
where μ is a uniquely determined solution of e μ = λ with Reμ < 0. It follows that the semigroup is differentiable at 0. Let
![{\displaystyle v(z)=\partial _{t}f_{t}(z)|_{t=0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/243c6f8a4132ff067cd0ad29d58ab6608088808c)
a holomorphic function on D with v(0) = 0 and v'(0) = μ.
Then
![{\displaystyle \partial _{t}(f_{t}(z))h^{\prime }(f_{t}(z))=\mu e^{\mu t}h(z)=\mu h(f_{t}(z)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d802de5fb0f70b60440d0f94418c9199a9731bf)
so that
![{\displaystyle v=v^{\prime }(0){h \over h^{\prime ))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a923fd7a88ddb3fb94d4b3c6ec133215becdd7d7)
and
![{\displaystyle \partial _{t}f_{t}(z)=v(f_{t}(z)),\,\,\,f_{t}(z)=0~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0a06992b67b7fb879bb7d0170493ef31db7827)
the flow equation for a vector field.
Restricting to the case with 0 < λ < 1, the h(D) must be starlike so that
![{\displaystyle \Re {zh^{\prime }(z) \over h(z)}\geq 0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6557152f9b0e46b275549e94d0ae143b304238ee)
Since the same result holds for the reciprocal,
![{\displaystyle \Re {v(z) \over z}\leq 0~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7903a6936bedf6fe218c72c574a04c61ecc3d486)
so that v(z) satisfies the conditions of Berkson & Porta (1978)
![{\displaystyle v(z)=zp(z),\,\,\,\Re p(z)\leq 0,\,\,\,p^{\prime }(0)<0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3340a6e66a1f32f85f571c8e6f2a1547abe968)
Conversely, reversing the above steps, any holomorphic vector field v(z) satisfying these conditions is associated to a semigroup ft, with
![{\displaystyle h(z)=z\exp \int _{0}^{z}{v^{\prime }(0) \over v(w)}-{1 \over w}\,dw.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c051a1a11a1ffd65c9135f2b26553f86150d5f)