---
tag: WDZFV2ACXR
kind: concept
name: Quadratic polynomial dynamics
msc: [37F10, 37F46]
specializes: [BQQZ0ZCGQF]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_quadratic_polynomial
---

## Summary

Quadratic polynomial dynamics is the study of the iteration of the maps $f_c(z) = z^2 + c$ on the complex plane.
Up to an affine change of variable, every quadratic polynomial is one of these maps.
It is the simplest nontrivial case of [[complex dynamics|BQQZ0ZCGQF]] and the testing ground for most of its ideas.
Its parameter space is organised by the [[Mandelbrot set|QSWSVYRCS1]].

## Explanations

### L1 human

Take the rule $z \mapsto z^2 + c$ for a fixed complex number $c$ and follow the orbit of each starting point.
Some orbits run off to infinity and others stay bounded.
The bounded ones form the filled Julia set, and its border is the [[Julia set|5M4S7P04GR]].
For $c = 0$ the filled Julia set is the closed unit disc.
For $c = -1$ it is a shape made of infinitely many touching blobs, called the basilica.
The single orbit of $0$ already decides whether the Julia set is one piece or a cloud of dust.