= Noninjective map with constant Jacobian determinant
{title2=$a(s,r)=e^s(\cos(re^{-2s}),\sin(re^{-2s}))$}
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 $re^{-2s}$, 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.
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