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", formula_3.
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 .
Koenigs function of a semigroup.
Let be a semigroup of holomorphic univalent mappings of into itself fixing 0 defined 
for such that
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 formula_15, a continuous semigroup.
So formula_16 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