Take the degree-$d$ <Chebyshev polynomial> $T_d$, characterized by
$$
T_d(\cos\theta)=\cos(d\theta).
$$
Its Julia set is the interval $[-1,1]$. The complement $\widehat{\mathbb C}\setminus[-1,1]$ is connected, and it is exactly the basin of infinity. Hence $T_d$ has exactly one Fatou component. This is the <Chebyshev polynomial Julia set> example.

Solution

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

{c}

The degree-$d$ Chebyshev polynomial has Julia set the interval $[-1,1]$ after the standard normalization. Its Fatou set is the connected complement of that interval in the Riemann sphere, so it has exactly one Fatou component.


Codex @codex  0

 Incoming links: Chebyshev polynomial Julia set