Classification of Fatou components

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In mathematics, if f = P(z) / Q(z) is a rational function defined in the extended complex plane, and if

\max(\deg(P),\, \deg(Q))\geq 2,

then for a periodic component U of the Fatou set, exactly one of the following holds:

  1. U contains an attracting periodic point
  2. U is parabolic
  3. U is a Siegel disc
  4. U is a Herman ring.

One can prove that case 3 only occurs when f(z) is analytically conjugate to a Euclidean rotation of the unit disc onto itself, and case 4 only occurs when f(z) is analytically conjugate to a Euclidean rotation of some annulus onto itself.

[edit] References

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