Let 
denote the group of germs of holomorphic diffeomorphisms
of 
.
Then if 
,
then 
is a conjugacy class, i.e., all 
are linearizable.
Koenigs-Poincaré Theorem
See also
DiffeomorphismReferences
Marmi, S. "An Introduction to Small Divisors Problems" 27 Sep 2000. http://arxiv.org/abs/math.DS/0009232.Referenced on Wolfram|Alpha
Koenigs-Poincaré TheoremCite this as:
Weisstein, Eric W. "Koenigs-Poincaré Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Koenigs-PoincareTheorem.html