Nonlinear shear map ($(u,v)\mapsto (u+f(v),v)$)

This triangular planar map has diagonal Jacobian matrix entries one and determinant one, for differentiable $f$. Its inverse subtracts $f(v)$, so it is an area-preserving map. The analogous vertical map $(u,v)\mapsto (u,v+g(u))$ has the same property. Their composition preserves area even when the shifts are nonlinear, as in the logarithmic area-preserving seasonal predator-prey map.