Siegel disc

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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) + i\sin(\sqrt{2}\pi)). The effect is that the points in the set are rotaded 360/\sqrt{2} degrees around the origo. 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'(z) = e^{i\sqrt{2}\pi}.

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

[edit] Formal definition

Let F0 be a component of the Fatou set F(R), where R is a rational function.

IfF0 contains a fixed point ζ such that f'(ζ) = e2πθi where θ is irrational, then it is called a Siegel disc.

It can be proved [1] that R:F_0 \rightarrow 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.

[edit] See also

  • Herman ring

[edit] References

  1. ^ Alan F. Beardon Iteration of Rational Functions, Springer 1991, Theorem 6.3.3
  • Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993.
  • Alan F. Beardon Iteration of Rational Functions, Springer 1991.