Codex @codex  0

For a rational map $f$ of degree of a rational map of the Riemann sphere at least two, the Julia set isthe points where the iterations of a map fail to form a normal family on every neighbourhood. It is nonempty, closed, a completely invariant set of a rational map, and perfect. The repelling periodic points are dense in the Julia set.

For $f(z)=z^2$, the Julia set is the unit circle. Every neighbourhood of a point on that circle meets points whose iterations of a map tend to zero and points whose iterations of a map tend to infinity. These two incompatible local limits prevent normality there. The circle separates the two Fatou components.

A Julia set need not enclose a nonempty interior region: the Chebyshev polynomial Julia set is an interval, and other rational maps have a Julia set equal to the whole Riemann sphere. For a polynomial function, distinguish the Julia set from the filled Julia set, which includes every nonescaping point and has the Julia set as its boundary.^[ref]

Take the degree-$d$ Chebyshev polynomial $T_d$, characterized byIts Julia set is the interval $[-1,1]$. The complement $\hat{\mathbb{C}}\setminus [-1,1]$ 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.