Sullivan's no wandering domains theorem

Remarks

A Fatou component that is not eventually periodic is called a wandering domain, so the theorem says that rational maps have none. The proof, due to Dennis Sullivan, uses quasiconformal mappings. A wandering domain would support an infinite-dimensional family of quasiconformal deformations of $f$. Each deformation is again a rational map of the same degree, and these maps form a finite-dimensional space, a contradiction. The hypothesis that $f$ is rational is needed. The transcendental entire function $g(z) = z + \sin z + 2\pi$ has wandering domains. Indeed $h(z) = z + \sin z$ has superattracting fixed points at the odd multiples of $\pi$, and $g = h + 2\pi$ moves each of their basins to the next one.

Going deeper

For a student

Together with the classification of periodic Fatou components, the theorem gives a complete picture of the Fatou set of a rational map. Each component eventually lands on a cycle of attracting basins, parabolic basins, Siegel discs or Herman rings. The method of proof, quasiconformal deformation, became a basic tool of the subject. For transcendental entire functions wandering domains exist, and they remain an active topic of research.