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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $k \\ge 2$, and let $f \\colon X \\to X$ be a [[holomorphic map|3MQTWKBBBT]].\nFor $m \\ge 0$, the [[iterate|AYVTR2ES5B]] $f^m$ is the $m$-fold composition of $f$, with $f^0 = \\id$.\nThe *orbit* of $x \\in X$ is the sequence $x, f(x), f^2(x), \\ldots$.\n*Holomorphic dynamics in several variables* is the study of the orbits of such a [[discrete-time dynamical system|5AA9AJEARJ]] $(X, f)$.\n\nLet $X$ be compact.\nThe *Fatou set* of $f$ is the set of points $x \\in X$ with a neighbourhood $U$ on which the iterates $(f^m|_U)_{m \\ge 0}$ form a [[normal family|JTVWRXXAKM]].\nThis means that every sequence of these iterates has a subsequence converging uniformly on compact subsets of $U$.\nThe *Julia set* of $f$ is the complement of the Fatou set.\n\n## Remarks\n\nThe main cases are $X = \\mathbb{P}^k$ with $f$ a holomorphic endomorphism of algebraic degree $d \\ge 2$, and $X = \\C^2$ with $f$ a polynomial automorphism.\n\nExample: $f[z_0 : \\cdots : z_k] = [z_0^d : \\cdots : z_k^d]$ with $d \\ge 2$ is a holomorphic self-map of $\\mathbb{P}^k$.\nIt is well defined because the $z_j^d$ vanish together only when all $z_j$ vanish.\nIf one coordinate $z_i$ has strictly larger modulus than all the others, then the ratios $(z_j/z_i)^{d^m}$ tend to $0$ uniformly near $[z]$.\nSo the orbit tends to the coordinate point with $1$ in place $i$, the iterates converge near $[z]$, and $[z]$ lies in the Fatou set.\n\nExample: a [[Hénon map|KS4FJ6H5JH]] $(x, y) \\mapsto (p(x) - a y, x)$ with $a \\ne 0$ and $\\deg p \\ge 2$ is a holomorphic automorphism of $\\C^2$.\n\nNon-example: the map $(z, w) \\mapsto (\\bar z^2, w)$ of $\\C^2$ is smooth but not holomorphic, so its iteration is not part of holomorphic dynamics.\n\nFor $k = 1$ the same definitions give one-variable [[complex dynamics|BQQZ0ZCGQF]].\n",
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