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.
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, for |z | ≤ r with M(r ) < 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, The function h is the uniform limit on compacta of the normalized iterates,
Under this conformal identification, the mapping f becomes multiplication by λ, a dilation on U.
Let ft (z) be a semigroup of holomorphic univalent mappings of D into itself fixing 0 defined for t ∈ [0, ∞) such that Each fs with s > 0 has the same Koenigs function, cf.
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 h is the Koenigs function of fs.
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 Berkson & Porta (1978) Conversely, reversing the above steps, any holomorphic vector field v(z) satisfying these conditions is associated to a semigroup ft, with