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  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to X$ be a [[map|0EPAT51DXQ]].\nThe *iterates* of $f$ are the maps $f^n \\colon X \\to X$ for $n \\in \\N$, defined recursively by\n$$f^0 = \\id_X, \\qquad f^{n+1} = f \\circ f^n,$$\nwhere $\\circ$ is [[function composition|0THBK9V825]].\nThe map $f^n$ is the *$n$-th iterate* of $f$.\n\n## Remarks\n\nBy induction, $f^m \\circ f^n = f^{m + n}$ and $(f^m)^n = f^{mn}$ for all $m, n \\in \\N$.\nIf $f$ is a bijection, one also sets $f^{-n} = (f^{-1})^n$, and the rule $f^m \\circ f^n = f^{m+n}$ then holds for all $m, n \\in \\Z$.\n\nExample: for $f(x) = x^2$ on $\\R$, one has $f^2(x) = (x^2)^2 = x^4$ and in general $f^n(x) = x^{2^n}$.\nThe check is the induction step $f^{n+1}(x) = f(x^{2^n}) = x^{2^{n+1}}$.\n\nNon-example: for $g(x) = x + 1$ on $\\R$, the pointwise square $x \\mapsto g(x)^2 = (x+1)^2$ is not the second iterate, which is $g^2(x) = x + 2$.\nThe notation $f^n$ can mean either object, so texts state which one is meant; for example $\\sin^2 x$ usually means $(\\sin x)^2$.\n",
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