Lozi map
= Lozi map
{c}
{title2=$F(x,y)=(1+y-a|x|,bx)$}
A piecewise affine planar map whose <derivative> changes across $x=0$. Away from that line its Jacobian is $\bigl(\begin{smallmatrix}-a\operatorname{sgn}x&1\\b&0\end{smallmatrix}\bigr)$, with <determinant> $-b$. For $b\ne0$ 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.