---
tag: B763QH5521
kind: concept
name: Topological mixing
msc: [37B02, 37B05]
specializes: [FR49Y9GQHR]
sources:
  - url: https://en.wikipedia.org/wiki/Mixing_(mathematics)
---

## Summary

A continuous map is topologically mixing when every nonempty open set, after some time, meets every other nonempty open set at all later times.
It strengthens [[topological transitivity|FR49Y9GQHR]], which only asks for one such time.
It is the topological counterpart of strong [[mixing|S3RJQ9CX2P]] in ergodic theory.
Expanding maps such as the doubling map of the circle are mixing, while rotations are never mixing.

## Explanations

### L1 human

Put a drop of ink in a glass of water and stir with a fixed rule.
Mixing means that after enough stirring the ink reaches every small region of the glass, and stays present there at every later moment.
The doubling map $x \mapsto 2x \bmod 1$ of the circle is mixing: it stretches every small arc until it covers the whole circle.
An irrational rotation is transitive but not mixing: a small arc just moves around rigidly and keeps missing a given region at many times.