---
tag: HZX6Q514X8
kind: concept
name: Herman ring
msc: [37F50, 37F10]
sources:
  - url: https://en.wikipedia.org/wiki/Herman_ring
same_as:
  wikidata: Q4228993
  wikipedia: Herman ring
---

## Summary

A Herman ring is a component of the [[Fatou set|C0N0XGC6GK]] of a rational map that is a topological annulus.
On it some iterate of the map acts as an irrational rotation.
It is one of the four types of periodic Fatou components, next to attracting basins, parabolic basins and [[Siegel discs|KNDYA9T8ZK]].
Unlike a Siegel disc, it has no fixed point at its centre.
Polynomials never have Herman rings, so they appear only for rational maps that are not polynomials.

## Explanations

### L1 human

Picture a ring, like a washer, that the map turns around and around by an angle that is an irrational fraction of a full turn.
Points travel forever on closed loops that circle the hole of the ring.
The hole and the outside of the ring both contain points of the Julia set.
Michael Herman showed that such rings really occur for rational maps.