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  "body": "\n## Summary\n\nAn iterative functional equation is a [[functional equation|SRAN4EKWS0]] in one variable in which the unknown function is composed with a given map or with itself.\nA typical form is $\\varphi(f(x)) = g(x, \\varphi(x))$, where $f$ and $g$ are given and $\\varphi$ is unknown.\nSuch equations describe how a quantity changes under one step of an [[iterated map|AYVTR2ES5B]].\nThe Abel equation, the Schröder equation and the equation $\\varphi(\\varphi(x)) = f(x)$ for a functional square root are the main examples.\n\n## Explanations\n\n### L1 human\n\nIteration means applying the same map again and again.\nAn iterative functional equation asks for a function that interacts in a fixed way with one step of this process.\nFor example, a function $\\varphi$ with $\\varphi(2x) = \\varphi(x) + 1$ counts how many doublings one has made, and $\\varphi(x) = \\log_2 x$ works on the positive numbers.\nSuch a function lets one ask what half a step of the process should be.\n",
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