Area-preserving map
= Area-preserving map
{title2=$|\det DF|=1$}
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 $b=1$.