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  "body": "\n## Definition\n\nLet $X$ be a set, let $f \\colon X \\to X$ be a map, and let $f^n$ denote the $n$-th [[iterate|AYVTR2ES5B]] of $f$.\nA point $x \\in X$ is a *periodic point* of $f$ if there is an integer $n \\ge 1$ with $f^n(x) = x$.\nSuch an $n$ is a *period* of $x$.\nEquivalently, $x$ is a [[fixed point|R6QVS6P21N]] of $f^n$ for some $n \\ge 1$.\n\n## Remarks\n\nIf $n$ is a period of $x$, so is every positive multiple of $n$, since $f^{kn}(x) = (f^n)^k(x) = x$.\nThe orbit $\\set{x, f(x), \\dots, f^{n-1}(x)}$ of a periodic point is a finite set, called a periodic orbit or cycle.\nA point $x$ is *eventually periodic*, or preperiodic, if some $f^m(x)$ with $m \\ge 0$ is periodic.\n\nExample: for $f(x) = x^2 - 1$ on $\\R$, the point $0$ is periodic with period $2$.\nIndeed $f(0) = -1$ and $f(-1) = 0$, so $f^2(0) = 0$.\n\nNon-example: for the same $f$, the point $1$ is not periodic.\nOne has $f(1) = 0$, after which the orbit alternates between $-1$ and $0$ and never returns to $1$.\nSo $1$ is eventually periodic but not periodic.\n",
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    "name": "Periodic point",
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        "url": "https://en.wikipedia.org/wiki/Periodic_point"
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  "msc": [
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  "name": "Periodic point",
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