---
tag: C0N0XGC6GK
kind: concept
name: Fatou set
msc: [37F10]
sources:
  - url: https://en.wikipedia.org/wiki/Julia_set
---

## Summary

The Fatou set of a [[rational function|0XDK9CGYKJ]] is the open set of points near which its iterates behave in a stable, predictable way.
Precisely, it is where the iterates form a [[normal family|JTVWRXXAKM]].
Its complement is the [[Julia set|5M4S7P04GR]].
The map permutes the connected components of the Fatou set.
The possible types of periodic components are completely classified.

## Explanations

### L1 human

Iterate a rational map and look at a small disc of starting points.
If all points of the disc move together and do the same thing in the long run, the disc is in the Fatou set.
For $z \mapsto z^2$ the Fatou set has two pieces.
One is the open unit disc, whose points are pulled to $0$.
The other is the outside of the unit circle, whose points run off to infinity.
The Fatou set is the calm region, and the Julia set is its chaotic border.