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  "body": "\n## Summary\n\nThe Fatou set of a [[rational function|0XDK9CGYKJ]] is the open set of points near which its iterates behave in a stable, predictable way.\nPrecisely, it is where the iterates form a [[normal family|JTVWRXXAKM]].\nIts complement is the [[Julia set|5M4S7P04GR]].\nThe map permutes the connected components of the Fatou set.\nThe possible types of periodic components are completely classified.\n\n## Explanations\n\n### L1 human\n\nIterate a rational map and look at a small disc of starting points.\nIf all points of the disc move together and do the same thing in the long run, the disc is in the Fatou set.\nFor $z \\mapsto z^2$ the Fatou set has two pieces.\nOne is the open unit disc, whose points are pulled to $0$.\nThe other is the outside of the unit circle, whose points run off to infinity.\nThe Fatou set is the calm region, and the Julia set is its chaotic border.\n",
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