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  "body": "\n## Summary\n\nComplex dynamics studies what happens when a [[holomorphic function|0NAHXR1FRH]] is applied to its own output again and again.\nThe main examples are [[rational functions|0XDK9CGYKJ]] on the [[Riemann sphere|0J78KBPFZB]], polynomials on the plane and [[entire functions|0G4F0XB9X2]].\nThe space splits into the [[Fatou set|C0N0XGC6GK]], where the behaviour is stable, and the [[Julia set|5M4S7P04GR]], where it is chaotic.\nThe subject joins complex analysis, topology and geometry, and it produced famous fractals such as the [[Mandelbrot set|QSWSVYRCS1]].\n\n## Explanations\n\n### L1 human\n\nPick a complex number $z$ and a rule, such as $z \\mapsto z^2 + c$, and apply the rule over and over.\nThe list of points you get is the orbit of $z$.\nComplex dynamics asks how this orbit behaves and how the answer changes when $z$ or the rule changes a little.\nFor $z \\mapsto z^2$, points inside the unit circle shrink to $0$, points outside run off to infinity, and points on the circle stay on it.\nThe circle is where the two kinds of behaviour meet, and in other examples this border becomes a fractal.\n",
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