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,

[math]\displaystyle{ |f(z)|\le M(r) |z| }[/math]

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,

[math]\displaystyle{ h(f(z))= f^\prime(0) h(z) ~. }[/math]

The function h is the uniform limit on compacta of the normalized iterates, [math]\displaystyle{ g_n(z)= \lambda^{-n} f^n(z) }[/math].

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

[math]\displaystyle{ H=k\circ h^{-1} (z) }[/math]

near 0. Thus H(0) =0, H'(0)=1 and, for |z | small,

[math]\displaystyle{ \lambda H(z)=\lambda h(k^{-1} (z)) = h(f(k^{-1}(z))=h(k^{-1}(\lambda z)= H(\lambda z)~. }[/math]

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

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

[math]\displaystyle{ |F(z) - 1|\le (1+|\lambda|^{-1})|z|~. }[/math]

On the other hand,

[math]\displaystyle{ g_n(z) = z\prod_{j=0}^{n-1} F(f^j(z))~. }[/math]

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

[math]\displaystyle{ \sum \sup_{|z|\le r} |1 -F\circ f^j(z)| \le (1+|\lambda|^{-1}) \sum M(r)^j \lt \infty. }[/math]

• 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

• [math]\displaystyle{ f_s }[/math] is not an automorphism for s > 0
• [math]\displaystyle{ f_s(f_t(z))=f_{t+s}(z) }[/math]
• [math]\displaystyle{ f_0(z)=z }[/math]
• [math]\displaystyle{ f_t(z) }[/math] 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

[math]\displaystyle{ h(f_s(z)) =f_s^\prime(0) h(z). }[/math]

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 [math]\displaystyle{ \lambda(s)=f_s^\prime(0) }[/math], a continuous semigroup. So [math]\displaystyle{ \lambda(s)= e^{\mu s} }[/math] where μ is a uniquely determined solution of e ^μ = λ with Reμ < 0. It follows that the semigroup is differentiable at 0. Let

[math]\displaystyle{ v(z)=\partial_t f_t(z)|_{t=0}, }[/math]

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

Then

[math]\displaystyle{ \partial_t (f_t(z)) h^\prime(f_t(z))= \mu e^{\mu t} h(z)=\mu h(f_t(z)), }[/math]

so that

[math]\displaystyle{ v=v^\prime(0) {h\over h^\prime} }[/math]

and

[math]\displaystyle{ \partial_t f_t(z) = v(f_t(z)),\,\,\, f_t(z)=0 ~, }[/math]

the flow equation for a vector field.

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

[math]\displaystyle{ \Re {zh^\prime(z)\over h(z)} \ge 0 ~. }[/math]

Since the same result holds for the reciprocal,

[math]\displaystyle{ \Re {v(z)\over z}\le 0 ~, }[/math]

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

[math]\displaystyle{ v(z)= z p(z),\,\,\, \Re p(z) \le 0, \,\,\, p^\prime(0) \lt 0. }[/math]

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

[math]\displaystyle{ h(z)= z \exp \int_0^z {v^\prime(0) \over v(w)} -{1\over w} \, dw. }[/math]

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