Koenigs function explained

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 be a holomorphic function mapping D into itself, fixing the point 0, with not identically 0 and not an automorphism of D, i.e. a Möbius transformation defined by a matrix in SU(1,1).

By the Denjoy-Wolff theorem, leaves invariant each disk |z | < r and the iterates of converge uniformly on compacta to 0: in fact for 0 < < 1,

|f(z)|\le M(r) |z| for |z | ≤ r with M(r) < 1. Moreover '(0) = with 0 < || < 1.

proved that there is a unique holomorphic function h defined on D, called the Koenigs function, such that (0) = 0, '(0) = 1 and Schröder's equation is satisfied,

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)} .

Moreover, if is univalent, so is .

As a consequence, when (and hence) are univalent, can be identified with the open domain . Under this conformal identification, the mapping   becomes multiplication by, a dilation on .

Proof

• Uniqueness. If is another solution then, by analyticity, it suffices to show that k = h near 0. Let

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)~.

Substituting into the power series for, it follows that near 0. Hence near 0.

• Existence. If

F(z) = f(z)/ λ z, then by the Schwarz lemma|F(z) - 1 |\le ( 1+ | λ | ^{-1} )|z|~.

On the other hand,

g _{n(z)} = z\prod ^{n -1} _{j =0} F(f ^{j(z))~.}

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

\sum \sup _{|z|\le} | 1 - F\circ f ^{j(z)|} \le ( 1+ | λ | ^{-1} ) \sum M(r) ^{j} < inf ty.

• Univalence. By Hurwitz's theorem, since each gⁿ is univalent and normalized, i.e. fixes 0 and has derivative 1 there, their limit is also univalent.

Koenigs function of a semigroup

Let be a semigroup of holomorphic univalent mappings of into itself fixing 0 defined for such that

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

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

Taking derivatives gives

h(f _{s(z))} = f ^{\prime( 0 )} _{s} h(z). Hence is the Koenigs function of .

Structure of univalent semigroups

On the domain, the maps become multiplication by

λ (s) = f ^{\prime( 0 )} _{s} , a continuous semigroup.So λ (s) = e ^{\mu} where is a uniquely determined solution of with Re < 0. It follows that the semigroup is differentiable at 0. Letv(z) = \partial _{t} f _{t(z)|} _{t=0} , a holomorphic function on with v(0) = 0 and = .

Then

\partial _{t} (f _{t(z))} h ^{\prime(f} _{t(z)) =} \mu e ^{\mu} h(z) = \mu h(f _{t(z)),} so thatv = v ^{\prime(0)} {h\over h ^{\prime}} and\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

\Re {zh ^{\prime(z)\over} h(z)} \ge 0 ~.

Since the same result holds for the reciprocal,

\Re {v(z)\over z}\le 0 ~, so that satisfies the conditions of v(z) = z p(z), \Re p(z) \le 0 , p ^{\prime(0)} < 0 .

Conversely, reversing the above steps, any holomorphic vector field satisfying these conditions is associated to a semigroup, with

h(z) = z \exp \int ^{z} _{0} {v ^{\prime(0)} \over v(w)} - { 1 \over w} dw.

References

• • • • • Book: Kuczma, Marek. Marek Kuczma

. Functional equations in a single variable . Marek Kuczma. Monografie Matematyczne . 1968 . PWN – Polish Scientific Publishers . Warszawa. ASIN: B0006BTAC2