True statements:

• Given a Liouville domain, the space of Liouville structure is convex, hence contractible.
• Given any two Liouville structures on the same domain, we can find a canonical path connecting them.
• Given any two Liouville structures on a Liouville manifold (completion from a domain), we can find a diffeotopy, such that $h^* \lambda_1 - \lambda_0 = dh$ is exact and $dh$ vanishes outside a compact set. (note, $h$ may not be compactly supported, do we care?)
• All the GPS construction are done with choices made in contractible set.