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  "body": "\n## Statement\n\nLet $U \\subseteq \\C$ be an open set containing $0$, and let $f \\colon U \\to \\C$ be [[holomorphic|0NAHXR1FRH]] with $f(0) = 0$.\nLet $\\lambda = f'(0)$ and assume $\\lambda \\ne 0$ and $\\abs{\\lambda} \\ne 1$.\nThen there is a holomorphic map $\\varphi$ on a neighbourhood $V$ of $0$ with $\\varphi(0) = 0$ and $\\varphi'(0) = 1$ such that\n$$\\varphi(f(z)) = \\lambda \\, \\varphi(z) \\quad \\text{for all } z \\in V \\text{ with } f(z) \\in V.$$\nThe map $\\varphi$ is unique.\n\n## Remarks\n\nIn the attracting case $0 \u003c \\abs{\\lambda} \u003c 1$, the map is given by $\\varphi(z) = \\lim_{n \\to \\infty} f^n(z) / \\lambda^n$.\nThe limit converges uniformly near $0$, because $f^n(z) = \\lambda^n z (1 + O(\\abs{z}))$ with errors that shrink geometrically.\nThe repelling case $\\abs{\\lambda} \u003e 1$ follows by applying the attracting case to the local inverse of $f$.\nSo near a [[fixed point|R6QVS6P21N]] that is attracting but not superattracting, or repelling, $f$ is locally [[conjugate|HNXXAPQKG3]] to its linear part $z \\mapsto \\lambda z$.\nExample: $f(z) = z / (2 - z)$ has $f'(0) = 1/2$, and $\\varphi(z) = z / (1 - z)$ satisfies $\\varphi(f(z)) = \\varphi(z) / 2$.\nThe hypotheses are needed.\nFor $\\lambda = 0$, the map $z \\mapsto z^2$ is not conjugate to $z \\mapsto 0$, since it is not constant.\nFor $\\lambda = 1$, the map $z \\mapsto z + z^2$ is not conjugate to the identity, since it is not the identity.\n\n## Explanations\n\n### L1 human\n\nNear a fixed point, a smooth map looks like multiplication by its derivative there.\nThe theorem says that for holomorphic maps this is exactly true after a change of coordinates, as long as the derivative has size neither $0$ nor $1$.\nIn the new coordinate, iterating the map just means multiplying by $\\lambda$ again and again.\n\n### L2 human\n\nThe function $\\varphi$ is called the Koenigs function of $f$ at the fixed point.\nIn the attracting case it extends to the whole basin of attraction, where it still satisfies $\\varphi \\circ f = \\lambda \\varphi$.\nThe cases left out lead to other theorems: Böttcher's theorem for $\\lambda = 0$, the Leau–Fatou flower theorem for roots of unity, and Siegel and Brjuno's small divisor results for other $\\abs{\\lambda} = 1$.\n",
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  "name": "Koenigs linearization theorem",
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