Siegel disc

In complex dynamics, one often analyse the behaviour of a set when it is iterated under a function, for example, let $C$ be the unit circle in the complex plane, and let the function be $f(z)\; =\; z(cos(sqrt\{2\}pi)\; +\; isin(sqrt\{2\}pi))$. The effect is that the points in the set are rotated $360/sqrt\{2\}$ degrees around the origin. The set $C$ is an example of a Siegel disc, (with respect to $f$). "Note that $z\; =\; 0$ is a fixed point, i.e. $f(z)=z$, and that $f\text{'}(z)\; =\; e^\{isqrt\{2\}pi\}$."

The definition of siegel discs is rather technical, and requires some knowledge in Complex analysis.

Formal definition

Let $F\_0$ be a component of the Fatou set $F(R)$, where $R$ is a rational function.

If$F\_0$ contains a fixed point $zeta$ such that $f\text{'}(zeta)=e^\{2pi\; heta\; i\}$ where $heta$ is irrational, then it is called a Siegel disc.

It can be proved [Alan F. Beardon "Iteration of Rational Functions", Springer 1991, Theorem 6.3.3] that $R:F\_0\; ightarrow\; F\_0$ is analytically conjugate to a rotation of infinite order of the unit disc.

This is part of the result from the Classification of Fatou components.

ee also

* Herman ring

References

* Lennart Carleson and Theodore W. Gamelin, "Complex Dynamics", Springer 1993.
* Alan F. Beardon "Iteration of Rational Functions", Springer 1991.