---
tag: BQQZ0ZCGQF
kind: concept
name: Complex dynamics
msc: [37F10, 30D05]
specializes: [BBZP1R872N]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_dynamics
same_as:
  wikidata: Q264224
  wikipedia: Complex dynamics
---

## Summary

Complex dynamics studies what happens when a [[holomorphic function|0NAHXR1FRH]] is applied to its own output again and again.
The main examples are [[rational functions|0XDK9CGYKJ]] on the [[Riemann sphere|0J78KBPFZB]], polynomials on the plane and [[entire functions|0G4F0XB9X2]].
The space splits into the [[Fatou set|C0N0XGC6GK]], where the behaviour is stable, and the [[Julia set|5M4S7P04GR]], where it is chaotic.
The subject joins complex analysis, topology and geometry, and it produced famous fractals such as the [[Mandelbrot set|QSWSVYRCS1]].

## Explanations

### L1 human

Pick a complex number $z$ and a rule, such as $z \mapsto z^2 + c$, and apply the rule over and over.
The list of points you get is the orbit of $z$.
Complex dynamics asks how this orbit behaves and how the answer changes when $z$ or the rule changes a little.
For $z \mapsto z^2$, points inside the unit circle shrink to $0$, points outside run off to infinity, and points on the circle stay on it.
The circle is where the two kinds of behaviour meet, and in other examples this border becomes a fractal.