SiegelDisc

A Siegel disc, also known as a Siegel domain, is a concept in complex dynamics and the study of holomorphic functions. It is named after the German mathematician Carl Ludwig Siegel. A Siegel disc is a special type of invariant domain that appears in the complex plane near a fixed point of a holomorphic function, specifically when the derivative of the function at the fixed point has a complex argument with bounded denominators.

Consider a holomorphic function $f(z)$ with a fixed point $z_0$ such that $f(z_0) = z_0$. If the derivative of the function at the fixed point, $f'(z_0)$, has an argument that is an irrational multiple of $\pi$, and the irrational number has bounded denominators (i.e., it is poorly approximable by rational numbers), then there exists a Siegel disc around the fixed point $z_0$. A Siegel disc has the following properties:

1. It is an open, connected, and simply connected domain in the complex plane containing the fixed point $z_0$.
2. The boundary of the Siegel disc, also called the Siegel boundary, is a closed curve that is invariant under the action of the holomorphic function $f(z)$.
3. The function $f(z)$ acts on the Siegel disc by rotation, meaning that points inside the disc are rotated around the fixed point $z_0$ under iteration.

Siegel discs are important in the study of complex dynamics because they are one of the possible local behaviors of a holomorphic function near a fixed point with a derivative of modulus 1 (i.e., $|f'(z_0)| = 1$). The other possible behavior in this case is the existence of a Herman ring, which is another type of invariant domain with a different structure. Siegel discs and Herman rings are related to the so-called small divisors problem, which arises in the study of dynamical systems and Diophantine approximation.