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.

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

$\sum \underset{|z|\le r}{\sup }|1-F\circ f^j(z)|\le (1+|\lambda {|}^{-1})\sum M(r{)}^j<\mathrm{\infty }.$

• 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 ∈ [0, ∞) 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 = f₁, 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 Reμ < 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

$v=v^{\prime }(0)\frac{h}{h^{\prime }}$

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

$\mathrm{\Re }\frac{z{h}^{\prime }(z)}{h(z)}\ge 0.$

Since the same result holds for the reciprocal,

$\mathrm{\Re }\frac{v(z)}{z}\le 0,$

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

$v(z)=zp(z),\mathrm{\Re }p(z)\le 0,p^{\prime }(0)<0.$

Conversely, reversing the above steps, any holomorphic vector field v(z) satisfying these conditions is associated to a semigroup f_t, with

$h(z)=z\exp \underset{0}{\overset{z}{\int }}\frac{v^{\prime }(0)}{v(w)}-\frac{1}{w}dw.$

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, https://archive.org/details/complexdynamics0000carl 
• 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. 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