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Incoming links: Fixed Fatou component classification
Past exam of the mathematics course of the University of Cambridge 2021 iii Paper 157 2 b Solution 2026-09-28
The fixed Fatou component classification has four cases:
- An immediate attracting or superattracting basin contains a critical point whose forward orbit converges to the attracting cycle.
- An immediate parabolic basin contains a critical point whose forward orbit converges to the parabolic cycle along an attracting direction.
- A Siegel disc is conformally conjugate to an irrational rotation of a disc. It contains no critical point, and its boundary is contained in the closure of the postcritical set.
- A Herman ring is conformally conjugate to an irrational rotation of an annulus. It contains no critical point, and both boundary components are contained in the closure of the postcritical set.