 Noninjective map with constant Jacobian determinant

The displayed smooth map from ${\mathbb{R}}^2$ to the punctured plane has Jacobian determinant one: in polar coordinates its radius is $e^s$ and its angle is $r{e}^{-2s}$, so the determinant is $e^{2s}{e}^{-2s}=1$. Nevertheless $a(0,2\pi k)=(1,0)$ for every integer $k$. Thus a local volume-preserving map need not be injective or have bounded weighted inverse multiplicity. Compactly supported data distributed over many inverse branches disprove a determinant-only dispersion with a nonlinear velocity map estimate.