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  "body": "\n## Statement\n\nLet $D \\subseteq \\C^n$ be a [[bounded|09QSZVVM26]] connected open set and let $p \\in D$.\nLet $f \\colon D \\to D$ be a [[holomorphic map|3MQTWKBBBT]] with $f(p) = p$.\nSuppose that the complex [[Jacobian matrix|047NDTEPCV]] of $f$ at $p$ is the identity matrix.\nThen $f(z) = z$ for all $z \\in D$.\n\n## Remarks\n\nThe proof uses iteration.\nTake $p = 0$ and suppose that $f(z) = z + P_k(z) + \\cdots$, where $P_k$ is the first nonzero homogeneous term of degree $k \\ge 2$ in the Taylor expansion of $f - \\id$.\nThe $m$-th iterate of $f$ then has the expansion $z + m P_k(z) + \\cdots$.\nAll iterates map $D$ into the bounded set $D$, so the Cauchy estimates bound their Taylor coefficients independently of $m$, which forces $P_k = 0$.\n\nA consequence: if $g, h$ are [[biholomorphisms|FX92PP3HZR]] of $D$ onto a domain $D'$ with $g(p) = h(p)$ and the same Jacobian matrix at $p$, then $g = h$.\nSo an automorphism of a bounded domain is determined by its value and its derivative at one point.\n\nBoundedness is necessary.\nOn $\\C^2$ the map $f(z_1, z_2) = (z_1, z_2 + z_1^2)$ fixes $0$ and has the identity as Jacobian matrix at $0$, but it is not the identity map.\n\n## Explanations\n\n### L1 human\n\nA holomorphic self-map of a bounded domain that fixes a point and looks like the identity to first order at that point must be the identity everywhere.\nIt is a rigidity result: first-order information pins down the whole map.\nIn one variable it is a cousin of the [[Schwarz lemma|0MB48XDBY3]].\n\n### L2 human\n\nThe theorem is the basic tool for studying the automorphism group of a bounded domain.\nIt shows that the map sending an automorphism fixing $p$ to its Jacobian matrix at $p$ is injective.\nWith it one proves that automorphisms of a bounded circular domain fixing the origin are linear maps.\nIt is also a step toward showing that the automorphism group of a bounded domain is a Lie group.\n",
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  "name": "Cartan's uniqueness theorem",
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