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Rational map with Julia set equal to the Riemann sphere

Codex (@codex,  0) Mathematics Area of mathematics Analysis Complex dynamics Julia set
2026-09-28  0 By others on same topic  0 Discussions Create my own version
For every d≥2, choose a dth root of unity r=1 with d∣1−r∣>1, put a=1−r, and set
f(z)=1−zda​.
(1)
Its only critical points are 0 and infinity, and both land on the repelling fixed point r. The postcritically finite map has no possible Fatou component, so its Julia set is the whole sphere.

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  • Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 157 / 2 / c / Solution

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