The Julia set is the complement of the Fatou set. It is nonempty, closed, completely invariant, and perfect for every rational map of degree at least two.
If a closed set satisfies for a rational map of degree at least two, then either has at most two points and lies in the Fatou set, or . The complement omits every point of , so Montel theorem proves normality when .
The accumulation-point set of is closed and completely invariant. Applying the classification of completely invariant closed sets shows that it equals ; hence the Julia set is perfect.
For every , choose a th root of unity with , put , and set
(1)
Its only critical points are and infinity, and both land on the repelling fixed point . The postcritically finite map has no possible Fatou component, so its Julia set is the whole sphere.
The degree- Chebyshev polynomial has Julia set the interval 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.

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A Julia set is a complex fractal that is associated with a particular complex quadratic polynomial, typically in the form \( f(z) = z^2 + c \), where \( z \) is a complex number and \( c \) is a complex constant. The behavior of the Julia set depends on the value of the constant \( c \).
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