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  "body": "\n## Summary\n\nA point is chain recurrent when it can return to itself along a path that follows the dynamics up to arbitrarily small jumps.\nSuch paths are called $\\varepsilon$-chains or $\\varepsilon$-pseudo-orbits.\nThe chain recurrent set is the weakest of the usual notions of recurrence: it contains the [[non-wandering set|EY2W34JZA4]].\nBy [[Conley's fundamental theorem|D07MBM0S3V]], outside the chain recurrent set the dynamics is gradient-like.\n\n## Explanations\n\n### L1 human\n\nImagine following the orbit of a point, but allowing yourself a tiny error at each step, as a computer with rounding would.\nThe point is chain recurrent if, for every allowed error size, you can get back to the starting point this way.\nA recurrent point is chain recurrent: its orbit comes back close to it, and one small final jump closes the loop.\nFor the map $x \\mapsto x/2$ on $[0, 1]$, a point $x \u003e 0$ is not chain recurrent: with small errors you still cannot climb back from $x/2$ to $x$.\n",
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