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)|\le 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|\le (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|\le r} |1 -F\circ f^j(z)| \le (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)} \ge 0 ~.

Since the same result holds for the reciprocal,

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

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

v(z)= z p(z),\,\,\, \Re p(z) \le 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 208, Springer, ISBN 978-3034605083
Koenigs, G.P.X. (1884), "Recherches sur les intégrales de certaines équations fonctionnelles", Ann. Sci. Ecole 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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