 Liouville volume of a surface geodesic flow

The positive Liouville volume for the base-orientation/positive-fibre orientation is $\mu ={\pi }^{\ast }{\Omega }_a\wedge \omega$. Its fibre integral is $2\pi {\Omega }_a$, so its total volume is $2\pi \mathrm{Area}(M)$. With the stated coframe convention, $\alpha \wedge d\alpha =-\mu$; the signed contact volume and the positive measure should not be confused. The fields $X,H,V$ preserve this volume by their structure equations.