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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]].\nAssume $f(z) = a_n z^n + a_{n+1} z^{n+1} + \\cdots$ near $0$, with $n \\ge 2$ and $a_n \\ne 0$.\nThen there is a holomorphic map $\\varphi$ on a neighbourhood $V$ of $0$ with $\\varphi(0) = 0$ and $\\varphi'(0) \\ne 0$ that solves *Böttcher's equation*\n$$\\varphi(f(z)) = \\varphi(z)^n \\quad \\text{for all } z \\in V \\text{ with } f(z) \\in V.$$\nSuch a $\\varphi$ is unique up to multiplication by an $(n-1)$-st root of unity.\n\n## Remarks\n\nSo near a superattracting [[fixed point|R6QVS6P21N]] of local degree $n$, the map $f$ is [[conjugate|HNXXAPQKG3]] to $w \\mapsto w^n$.\nThe idea of the proof: after a linear change of variable one may take $a_n = 1$.\nThen one sets $\\varphi(z) = \\lim_{k \\to \\infty} \\big(f^k(z)\\big)^{1/n^k}$, choosing the branches of the roots that are tangent to the identity.\nThe limit exists because $f^k(z)$ tends to $0$ very fast, and the correction factors form a convergent infinite product.\nExample: $f(z) = z^2 / (1 - 2z + 2z^2)$ satisfies the equation with $n = 2$ and $\\varphi(z) = z/(1 - z)$.\nIndeed $\\varphi(f(z)) = f(z)/(1 - f(z)) = z^2 / (1 - 2z + z^2) = \\varphi(z)^2$.\nFor a monic [[polynomial|1G0B73WPX8]] $P$ of degree $d \\ge 2$, the same result at the superattracting point $\\infty$ gives a map $\\varphi$ near $\\infty$ with $\\varphi(z) \\sim z$ and $\\varphi(P(z)) = \\varphi(z)^d$.\nThis is the Böttcher coordinate used to define external rays.\n\n## Explanations\n\n### L1 human\n\nThe simplest map with a very strongly attracting fixed point is $w \\mapsto w^n$, which squares or cubes distances to $0$ at each step.\nThe theorem says that every holomorphic map with such a fixed point looks exactly like $w \\mapsto w^n$ near it, after a change of coordinates.\n\n### L2 human\n\nThe equation was studied by Lucjan Böttcher in the early twentieth century.\nIt is the superattracting counterpart of Koenigs linearization, which handles multipliers $\\lambda$ with $0 \u003c \\abs{\\lambda} \u003c 1$.\nFor polynomials the Böttcher coordinate at infinity is the main tool for describing the basin of infinity.\nWhen the filled Julia set is connected, it extends to a conformal map from the basin of infinity onto the outside of the closed unit disc.\n",
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