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  "body": "\n## Summary\n\nAn iterated function is a map of a set to itself composed with itself a given number of times.\nThe $n$-th iterate $f^n$ applies $f$ exactly $n$ times.\nIterates are the basic objects of discrete-time dynamics: the state after $n$ steps is $f^n$ of the initial state.\nThey also appear in [[fixed-point iteration|K6ZF91051B]], fractals and the study of functional equations.\n\n## Explanations\n\n### L1 human\n\nApply a function, then apply it again to the result, and keep going.\nThe $n$-th iterate is what one gets after $n$ rounds.\nFor $f(x) = x + 3$, the iterate $f^n$ is $x \\mapsto x + 3n$.\nFor $f(x) = 2x$, the iterate $f^n$ is $x \\mapsto 2^n x$.\nPressing the cosine key of a calculator many times, in radian mode, computes iterates of $\\cos$, and the display settles near $0.739$.\n",
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