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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 ) | ≤ M ( r ) | z | {\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,

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

The function h is the uniform limit on compacta of the normalized iterates, g n ( z ) = λ − n f n ( z ) {\displaystyle g_{n}(z)=\lambda ^{-n}f^{n}(z)} .

Moreover, if f is univalent, so is h.¹²

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 ∘ h − 1 ( z ) {\displaystyle H=k\circ h^{-1}(z)} near 0. Thus H(0) =0, H'(0)=1 and, for |z | small, λ H ( z ) = λ h ( k − 1 ( z ) ) = h ( f ( k − 1 ( z ) ) = h ( k − 1 ( λ z ) = H ( λ z )   . {\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 F ( z ) = f ( z ) / λ z , {\displaystyle F(z)=f(z)/\lambda z,} then by the Schwarz lemma

| F ( z ) − 1 | ≤ ( 1 + | λ | − 1 ) | z |   . {\displaystyle |F(z)-1|\leq (1+|\lambda |^{-1})|z|~.} On the other hand, g n ( z ) = z ∏ j = 0 n − 1 F ( f j ( z ) )   . {\displaystyle g_{n}(z)=z\prod _{j=0}^{n-1}F(f^{j}(z))~.} Hence gn converges uniformly for |z| ≤ r by the Weierstrass M-test since ∑ sup | z | ≤ r | 1 − F ∘ f j ( z ) | ≤ ( 1 + | λ | − 1 ) ∑ M ( r ) j < ∞ . {\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 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 ft (z) be a semigroup of holomorphic univalent mappings of D into itself fixing 0 defined for t ∈ [0, ∞) such that

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

Each fs with s > 0 has the same Koenigs function, cf. iterated function. In fact, if h is the Koenigs function of f = f1, then h(fs(z)) satisfies Schroeder's equation and hence is proportion to h.

Taking derivatives gives

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

Hence h is the Koenigs function of fs.

Structure of univalent semigroups

On the domain U = h(D), the maps fs become multiplication by λ ( s ) = f s ′ ( 0 ) {\displaystyle \lambda (s)=f_{s}^{\prime }(0)} , a continuous semigroup. So λ ( s ) = e μ s {\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

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

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

Then

∂ t ( f t ( z ) ) h ′ ( f t ( z ) ) = μ e μ t h ( z ) = μ h ( f t ( z ) ) , {\displaystyle \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 ′ ( 0 ) h h ′ {\displaystyle v=v^{\prime }(0){h \over h^{\prime }}}

and

∂ t f t ( z ) = v ( f t ( z ) ) , f t ( z ) = 0   , {\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

ℜ z h ′ ( z ) h ( z ) ≥ 0   . {\displaystyle \Re {zh^{\prime }(z) \over h(z)}\geq 0~.}

Since the same result holds for the reciprocal,

ℜ v ( z ) z ≤ 0   , {\displaystyle \Re {v(z) \over z}\leq 0~,}

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

v ( z ) = z p ( z ) , ℜ p ( z ) ≤ 0 , p ′ ( 0 ) < 0. {\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 ft, with

h ( z ) = z exp ⁡ ∫ 0 z v ′ ( 0 ) v ( w ) − 1 w d w . {\displaystyle h(z)=z\exp \int _{0}^{z}{v^{\prime }(0) \over v(w)}-{1 \over w}\,dw.}

Notes

• 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

References

1. Carleson & Gamelin 1993, pp. 28–32 - 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 ↩

2. Shapiro 1993, pp. 90–93 - Shapiro, J. H. (1993), Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-94067-7 ↩