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 $f$ 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,

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

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,

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 $f$ 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\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 $D$ into itself fixing 0 defined for $t \in [0, \infty)$ such that

$f_{s}$ is not an automorphism for $s$ > 0
$f_{s}(f_{t}(z))=f_{t+s}(z)$
$f_{0}(z)=z$
$f_{t}(z)$ is jointly continuous in $t$ and $z$

Each $f s$ with $s$ > 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 $h$ is the Koenigs function of $f s$ .

Structure of univalent semigroups

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

v(z)=\partial _{t}f_{t}(z)|_{t=0},

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

Then

\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^{\prime }(0){h \over h^{\prime }}

and

\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 so that

\Re {zh^{\prime }(z) \over h(z)}\geq 0~.

Since the same result holds for the reciprocal,

\Re {v(z) \over z}\leq 0~,

so that $v (z)$ satisfies the conditions of Berkson & Porta (1978)

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 \int _{0}^{z}{v^{\prime }(0) \over v(w)}-{1 \over w}\,dw.

Notes

Carleson & Gamelin 1993, pp. 28–32
Shapiro 1993, pp. 90–93

References

Berkson, E.; Porta, H. (1978), "Semigroups of analytic functions and composition operators", Michigan Math. J., 25: 101–115, doi:10.1307/mmj/1029002009
Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 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 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 (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 0-387-94067-7
Shoikhet, D. (2001), Semigroups in geometrical function theory, Kluwer Academic Publishers, ISBN 0-7923-7111-9

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[1] Carleson & Gamelin 1993, pp. 28–32

[2] Shapiro 1993, pp. 90–93