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).
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,
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.
- Uniqueness. If k is another solution then, by analyticity, it suffices to show that k = h near 0. Let
- 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
- 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
- is not an automorphism for Template:Mvar > 0
- is jointly continuous in Template:Mvar and Template:Mvar
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
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