= Dispersion with a nonlinear velocity map
{title2=$\|f(t)\|_{L_x^\infty L_v^1}\leq|t|^{-d}\|m_a\|_\infty\|f_0\|_{L_x^1L_v^\infty}$}
For $f(t,x,v)=f_0(x-ta(v),v)$, apply the <area formula> to the velocity map and bound the initial data by their velocity <essential supremum>. This gives the displayed <mixed Lebesgue norm> estimate whenever the <weighted inverse multiplicity> is essentially bounded. If $a$ is injective and $|\det Da|\geq\alpha>0$, the constant is at most $1/\alpha$. In dimension one a derivative bounded away from zero gives a global <diffeomorphism>; in higher dimensions determinant bounds alone do not.
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