---
tag: 529D1C4W9H
kind: concept
name: Koenigs function
msc: [30D05, 37F10, 39B32]
sources:
  - url: https://en.wikipedia.org/wiki/Koenigs_function
same_as:
  wikidata: Q4493449
  wikipedia: Koenigs function
---

## Summary

The Koenigs function of a holomorphic map $f$ with an attracting, not superattracting, [[fixed point|R6QVS6P21N]] is the normalised solution of [[Schröder's equation|VTJV7JVJEA]] 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|G02NBNN6CJ]].
It is a basic tool in complex dynamics and in the study of composition operators.

## Explanations

### L1 human

If a map pulls points towards $0$ and shrinks distances near $0$ by the factor $\lambda$, then $n$ steps shrink by about $\lambda^n$.
Dividing the $n$-th iterate by $\lambda^n$ undoes this shrinking, and in the limit one gets a new coordinate.
In that coordinate the map is exactly multiplication by $\lambda$.
For $f(z) = z / (2 - z)$ the factor is $\lambda = 1/2$, and the Koenigs function is $z / (1 - z)$.