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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nA point $x \\in X$ is *recurrent* for $f$ if there is a strictly increasing sequence of positive integers $n_1 \u003c n_2 \u003c \\cdots$ such that $f^{n_k}(x) \\to x$ as $k \\to \\infty$.\n\n## Remarks\n\nEquivalent form: $x$ is recurrent if and only if $x \\in \\omega(x)$, where $\\omega(x)$ is the $\\omega$-[[limit set|YF4XH2BEG5]] of $x$, the set of limits of sequences $f^{n_k}(x)$ with $n_k \\to \\infty$.\nThe same condition $x \\in \\omega(x)$ defines recurrence on a general topological space.\nEvery point on a [[periodic orbit|0AA97XA70K]] is recurrent: if $f^p(x) = x$, take $n_k = kp$.\nEvery point of a [[minimal set|CRGZZKQ7Z9]] is recurrent.\nExample: for $f(x) = x + \\alpha \\bmod 1$ on the circle with $\\alpha$ irrational, Dirichlet's approximation theorem gives infinitely many $n$ with $n\\alpha$ within $1/n$ of an integer, so $f^{n}(x) \\to x$ along these $n$.\nNon-example: for $f(x) = x/2$ on $\\R$, the point $x = 1$ has $f^n(1) = 2^{-n} \\to 0$, so no subsequence tends to $1$.\n\n## Explanations\n\n### L1 human\n\nA recurrent point is a point that its own orbit approaches again and again.\n",
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