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  "body": "\n## Summary\n\nA recurrent point of a continuous map is a point whose orbit returns arbitrarily close to it again and again.\nFormally, the point lies in its own $\\omega$-[[limit set|YF4XH2BEG5]].\nRecurrence is the topological form of the idea that a system comes back near its initial state.\nEvery point of a [[minimal set|CRGZZKQ7Z9]] is recurrent, so on a nonempty compact metric space every continuous map has recurrent points.\n\n## Explanations\n\n### L1 human\n\nFollow a point under repeated application of a map.\nThe point is recurrent if, however small a window you draw around it, the orbit enters that window at infinitely many later times.\nA periodic point is recurrent, since it comes back exactly.\nFor a rotation of the circle by an irrational angle, every point is recurrent without ever coming back exactly.\nFor the map $x \\mapsto x/2$ on the real line, the point $1$ is not recurrent: its orbit $1, 1/2, 1/4, \\ldots$ runs off to $0$.\n",
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