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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nA point $x \\in X$ is *non-wandering* for $f$ if for every open neighbourhood $U$ of $x$ there is an integer $n \\ge 1$ with $f^n(U) \\cap U \\neq \\emptyset$.\nThe *non-wandering set* $\\Omega(f)$ is the set of non-wandering points.\n\n## Remarks\n\nSo $x$ is non-wandering exactly when it is not a wandering point in the sense of [[wandering sets|VNW5FHCQ2M]].\nThe set $\\Omega(f)$ is closed, and $f(\\Omega(f)) \\subseteq \\Omega(f)$.\nIf $X$ is compact and nonempty, then $\\Omega(f)$ is nonempty.\nEvery [[recurrent point|4SQHHRPKMV]], and so every point on a [[periodic orbit|0AA97XA70K]], is non-wandering.\nA non-wandering point need not be recurrent.\nExample: for $f(x) = x/2$ on $\\R$, the point $0$ is fixed, so $0 \\in f^n(U) \\cap U$ for every neighbourhood $U$ of $0$.\nNon-example: for the same map, a point $x \u003e 0$ has the neighbourhood $U = (2x/3, 4x/3)$, and $f^n(U) \\subseteq (0, 2x/3]$ for $n \\ge 1$, so $f^n(U) \\cap U = \\emptyset$.\nIn fact $\\Omega(f) = \\set{0}$ for this map.\n\n## Explanations\n\n### L1 human\n\nA point is non-wandering when every small region around it comes back, at some later time, to overlap with itself.\nThe non-wandering set collects these points.\n",
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