The Koenigs-Poincaré theorem is a local linearization theorem for holomorphic maps near a fixed
point. Let 
be the group, under composition,
of germs at 0 of locally invertible holomorphic
functions fixing 0. Two functions define the same
germ if they agree on some neighborhood
of 0. For a nonzero complex number 
, set
 |
The theorem states that if 
,
then every 
admits a locally invertible holomorphic function 
defining a coordinate transformation,
with 
and 
, such that
 |
Thus 
is a conjugacy class in 
(Marmi 2000, pp. 3, 5, 7).
