---
tag: 75B8VHDGSA
kind: concept
name: Iterated function system
msc: [28A80, 37C70, 37H12]
sources:
  - url: https://en.wikipedia.org/wiki/Iterated_function_system
same_as:
  wikidata: Q1502167
  wikipedia: Iterated function system
---

## Summary

An iterated function system is a finite family of [[contractions|0QN69YA357]] of a complete metric space.
It has a unique nonempty compact set, its attractor, that is the union of its images under the maps of the family.
Many fractals, such as the [[Cantor set|0BXNDZFQME]] and the [[Sierpiński carpet|PCM3H6J3EE]], are attractors of iterated function systems.
The attractor can be drawn by applying the maps at random to a point, which is the chaos game.

## Explanations

### L1 human

Take a few rules, each of which shrinks the plane towards some point.
Apply all rules to a picture and put the results together, and repeat.
Whatever picture you start from, the results converge to one shape, which is made of shrunk copies of itself.
For the two rules $x \mapsto x/3$ and $x \mapsto x/3 + 2/3$ on the line, the shape is the Cantor set: two copies of itself, each one third as large.