---
tag: 5M4S7P04GR
kind: concept
name: Julia set
msc: [37F10]
sources:
  - url: https://en.wikipedia.org/wiki/Julia_set
same_as:
  wikidata: Q848427
  wikipedia: Julia set
---

## Summary

The Julia set of a [[rational function|0XDK9CGYKJ]] is the set of points near which the iterates of the function behave chaotically.
It is the complement of the [[Fatou set|C0N0XGC6GK]], where the iterates form a [[normal family|JTVWRXXAKM]].
For a map of degree at least $2$ it is a nonempty compact set.
The map sends it onto itself, and it is often a fractal.
Julia sets are the central objects of [[complex dynamics|BQQZ0ZCGQF]].

## Explanations

### L1 human

Iterate a rule such as $z \mapsto z^2 + c$ and colour each starting point by what its orbit does.
In some regions nearby starting points keep behaving alike; this is the Fatou set.
The Julia set is the border where tiny changes of the starting point can change the long-run behaviour completely.
For $z \mapsto z^2$ the Julia set is the unit circle: inside it orbits go to $0$, outside they go to infinity.
For most values of $c$ the border is no longer a smooth curve but an intricate fractal.