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,

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,

The function h is the uniform limit on compacta of the normalized iterates, .

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

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

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

• Existence. If then by the Schwarz lemma

On the other hand,

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

• 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.

Let f_t (z) be a semigroup of holomorphic univalent mappings of D into itself fixing 0 defined for t ∈ [0, ∞) such that

• is not an automorphism for s > 0
• 
• 
• 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

Hence h is the Koenigs function of f_s.

On the domain U = h(D), the maps f_s become multiplication by , a continuous semigroup. So where μ is a uniquely determined solution of e ^μ = λ with Reμ < 0. It follows that the semigroup is differentiable at 0. Let

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

Then

so that

and

the flow equation for a vector field.

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

Since the same result holds for the reciprocal,

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

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