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

Choose a $d$th root of unity $r\ne 1$ far enough from one that $d|1-r|>1$, put $a=1-r$, and defineThis map has degree $d$ and only two critical points, $0$ and infinity, each of multiplicity $d-1$. Their orbits areThe multiplier at $r$ iswhose modulus exceeds one. Thus every critical orbit lands at a repelling fixed point.

An attracting or parabolic periodic Fatou component would capture a critical orbit, contrary to the displayed dynamics. A Siegel disc or Herman ring would have boundary in the closure of the postcritical set, but that set is finite and contained in the repelling grand orbit, whereas a rotation-domain boundary is infinite. By the Sullivan no-wandering-domain theorem, every Fatou component is eventually periodic, so the classification leaves no Fatou component. Therefore this rational map with Julia set equal to the Riemann sphere satisfies