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,

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

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 Schröder'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.^[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.

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

• ${\displaystyle f_{s}}$ 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 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

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

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 e ^μ = λ with Reμ < 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

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