Koenigs function

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

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

for |z | ≤ r with M(r ) < 1. Moreover Template:Mvar '(0) = Template:Mvar with 0 < |Template:Mvar| < 1.

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

The function h is the uniform limit on compacta of the normalized iterates . Moreover, if Template:Mvar is univalent, so is h.[1][2]

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

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.
On the other hand,
Hence gn 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 ft (z) be a semigroup of holomorphic univalent mappings of Template:Mvar into itself fixing 0 defined for t ∈ [0, ∞) such that

Each fs with Template:Mvar > 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

Hence Template:Mvar is the Koenigs function of fs.

Structure of univalent semigroups

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

a holomorphic function on D with v(0) = 0 and v'(0) = Template:Mvar. 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 Template:Harvtxt

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

Notes

References

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