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  "body": "\n## Summary\n\nSchröder's equation asks for a function $\\Psi$ with $\\Psi(f(x)) = \\lambda \\Psi(x)$, where the map $f$ and the constant $\\lambda$ are given.\nA solution is a change of coordinates in which $f$ becomes multiplication by $\\lambda$.\nIt is one of the basic [[iterative functional equations|BCGBQ3HK26]]: once it is solved, every [[iterate|AYVTR2ES5B]] of $f$ has the closed form $f^n = \\Psi^{-1} \\circ (\\lambda^n \\Psi)$.\nNear a [[fixed point|R6QVS6P21N]] of a holomorphic map, the natural choice of $\\lambda$ is the derivative of $f$ at that point.\n\n## Explanations\n\n### L1 human\n\nIterating a map is often hard, but iterating multiplication by a number is easy: $n$ steps multiply by $\\lambda^n$.\nSchröder's equation looks for a new coordinate in which the given map is just such a multiplication.\nFor example, take $f(x) = x^2 + 2x$ on $x \u003e -1$ and $\\Psi(x) = \\log(1 + x)$.\nThen $\\Psi(f(x)) = \\log\\big((1 + x)^2\\big) = 2 \\Psi(x)$, so in the coordinate $\\Psi$ the map $f$ doubles.\nThis gives at once $f^n(x) = (1 + x)^{2^n} - 1$.\n",
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