 Contact conformal factor along an isotopy

If a time-dependent vector field satisfies ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, its flow maps satisfy ${\rho }_t^{\ast }{\alpha }_t=u_t{\alpha }_0$, where $u_t(x)=\exp ({\int }_0^t{h}_s({\rho }_s(x))ds)$. Thus the conformal factor is smooth and positive even when $h_t$ changes sign.