 Lozi map

A piecewise affine planar map whose derivative changes across $x=0$. Away from that line its Jacobian is $\left(\begin{array}{cc}-a\mathrm{sgn}x& 1\\ b& 0\end{array}\right)$, with determinant $-b$. For $b\ne 0$ it is invertible; at $b=0$ an iteration collapses the image onto a line. The switching line requires piecewise analysis rather than applying a smooth-map theorem across it without checking its hypotheses.