---
tag: KNDYA9T8ZK
kind: concept
name: Siegel disc
msc: [37F50, 37F10]
sources:
  - url: https://en.wikipedia.org/wiki/Siegel_disc
same_as:
  wikidata: Q4162128
  wikipedia: Siegel disc
---

## Summary

A Siegel disc is a component of the [[Fatou set|C0N0XGC6GK]] of a rational map on which an iterate acts as an irrational rotation.
The rotation is seen after a conformal change of coordinates.
It surrounds a neutral periodic point whose multiplier is $e^{2\pi i \theta}$ with $\theta$ irrational.
Whether such a disc exists depends on the arithmetic of $\theta$, through small divisor problems.
Siegel discs are one of the four types of periodic Fatou components.

## Explanations

### L1 human

Imagine a disc that the map turns by a fixed angle at each step, like a wheel.
The angle is an irrational fraction of a full turn, so no point ever returns exactly to where it started.
Each point travels forever along a closed curve around a centre point.
For $z \mapsto e^{2\pi i \theta} z + z^2$ with $\theta$ the golden mean, such a turning disc exists around $0$.
Its boundary is a complicated fractal curve.