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,

|f(z)|\leq M(r)|z|

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,

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

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

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

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

\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 for H, it follows that H(z) = z near 0. Hence h = k near 0.

Existence. If $F(z)=f(z)/\lambda z,$ then by the Schwarz lemma

|F(z)-1|\leq (1+|\lambda |^{-1})|z|

On the other hand,

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 since

\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, 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 $f t (z)$ be a semigroup of holomorphic univalent mappings of Template:Mvar into itself fixing 0 defined for $t \in [0, \infty)$ such that

$f_{s}$ is not an automorphism for Template:Mvar > 0
$f_{s}(f_{t}(z))=f_{t+s}(z)$
$f_{0}(z)=z$
$f_{t}(z)$ 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

h(f_{s}(z))=f_{s}^{\prime }(0)h(z).

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 $\lambda (s)=f_{s}^{\prime }(0)$ , a continuous semigroup. So $\lambda (s)=e^{\mu s}$ where μ is a uniquely determined solution of $e^{\mu }=\lambda$ with Re μ < 0. It follows that the semigroup is differentiable at 0. Let