Area-preserving map ($|\det DF|=1$)

A planar diffeomorphism is area-preserving if the phase-space area of every measurable set is unchanged. The change-of-variables formula proves this property when $|\det DF|=1$ everywhere, and conversely gives that determinant condition for a smooth map preserving all sufficiently small areas. Such a map cannot have an asymptotically attracting fixed point with a positive-area local basin: iterates of a small invariant neighborhood would have to enter arbitrarily small disks while retaining its phase-space area. For a map with constant positive determinant $b$, invariance of a bounded region of positive phase-space area instead forces $b=1$.

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.