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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: if fact for 0 < r < 1,

${\displaystyle |f(z)|\leq M(r)|z|}$

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 Schroeder's equation is satisfied:

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

The function h is the uniform limit on compacta of the normalized iterates ${\displaystyle g_{n}(z)=\lambda ^{-n}f^{n}(z)}$. 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 ${\displaystyle H=k\circ h^{-1}(z)}$ near 0. Thus H(0) =0, H'(0)=1 and for |z| small

${\displaystyle \lambda H(z)=\lambda h(k^{-1}(z))=h(f(k^{-1}(z))=h(k^{-1}(\lambda z)=H(\lambda z).}$

Substituting into the power series for H, it follows that H(z) = z near 0. Hence h = k near 0.

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

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

On the other hand

${\displaystyle g_{n}(z)=z\prod _{j=0}^{n-1}F(f^{j}(z)).}$

Hence g_n converges uniformly for |z| ≤ r by the Weierstrass M-test since

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

• Univalence. By Hurwitz's theorem, since each g 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 ${\displaystyle f_{t}(z)}$ be a semigroup of holomorphic univalent mappings of D into itself fixing 0 defined for ${\displaystyle t\in [0,\infty )}$ such that

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

Each ${\displaystyle f_{s}}$ with s > 0 has the same Koenigs function. In fact if h is the Koenigs function of f =f₁ then ${\displaystyle h(f_{s}(z))}$ satisfies Schroeder's equation and hence is proportion to h. Taking derivatives gives

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

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

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

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

${\displaystyle \partial _{t}(f_{t}(z))h^{\prime }(f_{t}(z))=\mu e^{\mu t}h(z)=\mu h(f_{t}(z)),}$

so that

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

and

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

the flow equation for a vector field.

Restricting to the case with 0 < λ < 1, the h(D) must be starlike so that

${\displaystyle \Re {zh^{\prime }(z) \over h(z)}\geq 0}$

Since the same result holds for the reciprocal,

${\displaystyle \Re {v(z) \over z}\leq 0.}$

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

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

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

${\displaystyle h(z)=z\exp \int _{0}^{z}{v^{\prime }(0) \over v(w)}-{1 \over w}\,dw.}$

Notes

1. Carleson & Gamelin 1993, p. 28-32
2. Shapiro 1993, p. 90-93

References

• Berkson, E.; Porta, H. (1978), "Semigroups of analytic functions and composition operators", Michigan Math. J., 25: 101–115
• 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 303460508
• Koenigs, G.P.X. (1884), "Recherches sur les intégrales de certaines équations fonctionnelles", Ann. Sci. Ecole Norm. Sup., 1: 2–41
• 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

• Complex analysis
• Dynamical systems