= Fatou component
{c}
{title2=$U\subseteq F(R)$}
A Fatou component is a connected component of the <Fatou set> of a rational map. It is open. A rational map sends each Fatou component onto another one: complete invariance of the Fatou set puts it inside the inverse image of the target component; its maximal connectedness makes it one full inverse-image component. Restricting the globally proper sphere map to such a component is proper, since the inverse image of a compact subset intersected with the component is closed in a compact set. Its image is both open and closed in the connected target component, hence the whole target. A component is periodic if some iterate returns it to itself.
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