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  "body": "\n## Example\n\nFor parameters $a, b$, the *Hénon map* is\n$$H_{a,b}(x, y) = (1 - a x^2 + y, \\; b x).$$\nHénon studied it for real $a, b$ as a model of a dissipative system; for $a = 1.4$ and $b = 0.3$ its orbits accumulate numerically on a [[strange attractor|9WKCZGG906]].\n\nFor complex $a, b$ with $b \\ne 0$, $H_{a,b}$ is a holomorphic automorphism of $\\C^2$.\nCheck: if $(u, v) = H_{a,b}(x, y)$, then $x = v/b$ and $y = u - 1 + a v^2/b^2$, so $H_{a,b}$ is a bijection with polynomial inverse\n$$H_{a,b}^{-1}(u, v) = \\left(\\frac{v}{b}, \\; u - 1 + \\frac{a v^2}{b^2}\\right).$$\nIts Jacobian matrix is $\\begin{pmatrix} -2ax \u0026 1 \\\\ b \u0026 0 \\end{pmatrix}$, with [[Jacobian determinant|0ZCSMYHTBP]] $-b$, a nonzero constant.\nFor real $a, b$ it multiplies area in $\\R^2$ by $\\abs{b}$, so for $\\abs{b} \u003c 1$ it contracts area.\nAs a map of $\\C^2$ it multiplies volume by $\\abs{b}^2$.\n\n## Remarks\n\nIn complex dynamics one often writes generalized Hénon maps as $(x, y) \\mapsto (p(x) - a y, x)$ with $p$ a polynomial of degree $d \\ge 2$ and $a \\ne 0$.\nFriedland and Milnor showed that every polynomial automorphism of $\\C^2$ is conjugate either to a map of a simple triangular form or to a composition of generalized Hénon maps.\nSo Hénon maps are the main objects of [[holomorphic dynamics in several variables|6WCRMYEYF9]] on $\\C^2$.\nFor them one studies the sets $K^\\pm$ of points whose forward or backward orbits stay bounded, and the Julia sets $J^\\pm = \\partial K^\\pm$.\n\n## Explanations\n\n### L1 human\n\nThe Hénon map folds the plane and squeezes it, then applies the same rule again.\nUnlike the logistic map, it can be undone: from the image point one recovers the starting point by a formula.\nEven so, repeated application mixes points in a chaotic way, and for some parameters the orbits pile up on a fractal set.\nTaken over the complex numbers, it is the simplest interesting invertible holomorphic map of two-dimensional complex space.\n",
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