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 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,

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,

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

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 &nbsp; 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

:

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

:

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

• is not an automorphism for > 0
• 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

Hence is the Koenigs function of .

Structure of univalent semigroups

On the domain , the maps become multiplication by , a continuous semigroup. So where is a uniquely determined solution of with Re < 0. It follows that the semigroup is differentiable at 0. Let

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

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 satisfies the conditions of

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

Notes

References

• ASIN: B0006BTAC2