Koenigs function

Summary

The Koenigs function of a holomorphic map $f$ with an attracting, not superattracting, fixed point is the normalised solution of Schröder's equation at that point. It is the limit of the rescaled iterates $f^n(z) / \lambda^n$, where $\lambda = f'(0)$. In its coordinate the map $f$ becomes multiplication by $\lambda$; this is the content of the Koenigs linearization theorem. It is a basic tool in complex dynamics and in the study of composition operators.