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  "body": "\n## Example\n\nLet $n \\ge 2$.\nA *Fatou–Bieberbach domain* is an open set $\\Omega \\subseteq \\C^n$ with $\\Omega \\ne \\C^n$ that is [[biholomorphic|FX92PP3HZR]] to $\\C^n$.\n\nA concrete one comes from an [[automorphism of $\\C^2$|2J7JB3H3FD]].\nLet $F(z, w) = (w/2, \\, z/2 + w^2)$.\nIt is a polynomial map with polynomial inverse $(u, v) \\mapsto (2v - 8u^2, \\, 2u)$, so it is a holomorphic automorphism of $\\C^2$.\nThe origin is a fixed point, and the Jacobian matrix of $F$ there has eigenvalues $1/2$ and $-1/2$.\nSo the origin is an attracting fixed point.\nLet $\\Omega$ be its basin of attraction, the set of points $q$ with $F^m(q) \\to 0$ as $m \\to \\infty$.\nIt is a known theorem that the basin of attraction of an attracting fixed point of a holomorphic automorphism of $\\C^n$ is biholomorphic to $\\C^n$.\nThe point $(3/8, 3/4)$ is a second fixed point of $F$, so it is not in $\\Omega$.\nHence $\\Omega$ is a proper open subset of $\\C^2$ biholomorphic to $\\C^2$.\n\n## Remarks\n\nNo such domain exists in dimension one.\nIf $\\Omega \\subsetneq \\C$ were biholomorphic to $\\C$, then $\\Omega$ would be simply connected, so its complement has at least two points.\nA biholomorphism $\\C \\to \\Omega$ would then be an entire function omitting two values, hence constant by Picard's little theorem, which is absurd.\n\n## Explanations\n\n### L1 human\n\nIn the plane, the whole complex line cannot be squeezed holomorphically and bijectively onto a smaller region.\nIn two or more complex dimensions this can happen: all of $\\C^n$ fits, holomorphically and without distortion of its complex structure, inside a proper part of itself.\nThe pieces left out can even contain open sets.\nThis shows how much more flexible holomorphic maps become in several variables.\n",
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