Hénon map

Example

For parameters $a, b$, the Hénon map is $$H_{a,b}(x, y) = (1 - a x^2 + y, \; b x).$$ Hénon 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.

For complex $a, b$ with $b \ne 0$, $H_{a,b}$ is a holomorphic automorphism of $\C^2$. Check: 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 $$H_{a,b}^{-1}(u, v) = \left(\frac{v}{b}, \; u - 1 + \frac{a v^2}{b^2}\right).$$ Its Jacobian matrix is $\begin{pmatrix} -2ax & 1 \\ b & 0 \end{pmatrix}$, with Jacobian determinant $-b$, a nonzero constant. For real $a, b$ it multiplies area in $\R^2$ by $\abs{b}$, so for $\abs{b} < 1$ it contracts area. As a map of $\C^2$ it multiplies volume by $\abs{b}^2$.