---
tag: F666F5W806
kind: definition
name: Siegel disc
msc: [37F50, 37F10]
defines: KNDYA9T8ZK
uses: [C0N0XGC6GK, 0XDK9CGYKJ, 8G5ABXRK85, 0YYT36R4YD, 0DZXAXEJQ9, T75CZ5H47N]
sources:
  - url: https://en.wikipedia.org/wiki/Siegel_disc
---

## Definition

Let $f \colon \hat{\C} \to \hat{\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$.
A *Siegel disc* of $f$ is a component $U$ of the [[Fatou set|C0N0XGC6GK]] of $f$ with the following properties.

1. There is $p \ge 1$ with $f^p(U) = U$.
2. There is a [[conformal map|0YYT36R4YD]] $\varphi \colon U \to \mathbb{D}$ onto the [[unit disc|8G5ABXRK85]] $\mathbb{D}$.
3. There is an [[irrational number|0DZXAXEJQ9]] $\theta$ with $\varphi(f^p(z)) = e^{2\pi i \theta} \varphi(z)$ for all $z \in U$.

## Remarks

The point $z_0 = \varphi^{-1}(0)$ is a [[periodic point|T75CZ5H47N]] of $f$, and its multiplier is $(f^p)'(z_0) = e^{2\pi i \theta}$.
So the periodic point is neutral and irrationally indifferent, and the map is linearizable near it.
Example: Siegel proved in 1942 that $f(z) = e^{2\pi i \theta} z + z^2$ is linearizable at $0$ when $\theta$ is Diophantine, for instance the golden mean $(\sqrt{5} - 1)/2$.
The component of the Fatou set containing $0$ is then a Siegel disc.
Non-example: the unit disc for $f(z) = z^2$ fails clause 3, since $f$ is not injective there, while a rotation is.
The analogous domain with an annulus in place of the disc is a Herman ring.