For a $n$ dimensional smooth manifold $M$, I consider the cotangent bundle $T^*M$ with the canonical symplectic form $\omega$. A symplectic map $\phi : T^*M \to T^* M$ is a map which leaves the symplectic form invariant, i.e. $\phi^* \omega = \omega$.

Question: Is there a notion of symplectic maps between the corresponding spaces of volume forms? In other words, when is a map $\psi : \mathrm{\Omega}^n(T^*M) \to \mathrm{\Omega}^n(T^*M)$ symplectic?

Background: For numerical simulation of Hamiltonian equations, it is good to use symplectic integrators, such as symplectic Euler. However, I am interested in solving Liouville's equations and this raised the question what a corresponding symplectic integrator would be in that case?