Dispersion with a nonlinear velocity map ($\Vert f(t){\Vert }_{L_x^{\infty }{L}_v^1}\le |t{|}^{-d}\Vert m_a{\Vert }_{\infty }\Vert f_0{\Vert }_{L_x^1{L}_v^{\infty }}$)

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 $|\det Da|\ge \alpha >0$, 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.