---
tag: DP4WWQBJJF
kind: concept
name: Rotation number
msc: ["37E45", "37E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Rotation_number
same_as:
  wikidata: Q643156
  wikipedia: Rotation number
---

## Summary

The rotation number of an orientation-preserving [[homeomorphism|0QT00PB5G2]] of the [[circle|060W5CBZP6]] is the average amount by which the map turns points around the circle, per step.
For the rotation by an angle $\alpha$ it is $\alpha$.
It is a topological invariant: homeomorphisms conjugate by an orientation-preserving homeomorphism have the same rotation number.
Poincaré introduced it, and showed that it decides much of the dynamics: rational rotation number means periodic orbits, irrational rotation number means none.
It is the basic invariant of circle dynamics and of quasi-periodic motion.

## Explanations

### L1 human

Follow a point as a map moves it around a circle many times, and count the total number of turns made.
Divide by the number of steps; in the long run this ratio settles to a number, the rotation number.
For a clock hand that moves a quarter turn each tick, the rotation number is $1/4$.
A map can move different points by different amounts, but the long-run average turn is the same for every starting point.