Koenigs function

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 |

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 )   .

The function h is the uniform limit on compacta of the normalized iterates, g n ( z ) = λ − 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

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 )   . 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 , 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.

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

f s is not an automorphism for s > 0

f s ( f t ( z ) ) = f t + s ( z )

f 0 ( z ) = z

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

h ( f s ( z ) ) = f s ′ ( 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 λ ( s ) = f s ′ ( 0 ) , a continuous semigroup. So λ ( s ) = e μ 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 ,

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 ) ) ,

so that

v = v ′ ( 0 ) h h ′

and

∂ 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   .

Since the same result holds for the reciprocal,

ℜ v ( z ) z ≤ 0   ,

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

v ( z ) = z p ( z ) , ℜ p ( z ) ≤ 0 , p ′ ( 0 ) < 0.

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

h ( z ) = z exp ⁡ ∫ 0 z v ′ ( 0 ) v ( w ) − 1 w d w .