Rational map with Julia set equal to the Riemann sphere
= Rational map with Julia set equal to the Riemann sphere
For every $d\geq2$, choose a $d$th root of unity $r\ne1$ with $d|1-r|>1$, put $a=1-r$, and set
$$
f(z)=1-\frac a{z^d}.
$$
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.