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 spaces of volume forms over the phase space $T^*M$?

*edit:*

In more detail: Let me denote the space of all volume forms over $T^*M$ as $\Omega^{2n}(T^*M) := \Gamma( \Lambda^{2n} (T^*M) )$ (the notation is from John M. Lee's book 'Intro to smooth manifolds'.) Then for a map $\psi$ which transforms volume forms, i.e. $$\psi : \mathrm{\Omega}^{2n}(T^*M) \to \mathrm{\Omega}^{2n}(T^*M),$$ I am looking for a condition which ensures that $\psi$ is compatible with the symplectic structure of $T^*M$?

*Example:*
A symplectic map $\phi : T^* M \to T^*M$ implies a map $$\psi : \mathrm{\Omega}^{2n}(T^*M) \to \mathrm{\Omega}^{2n}(T^*M) : \eta \mapsto \phi^* \eta.$$
Such a map should be compatible for example.

But I would expect that there are more maps which are compatible and that not all of them are derived from symplectic maps like in the example.

*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?

allvolume forms is not right. Perhaps you are interested in the diffeomorphisms that preserve $\omega^n$, with no further conditions, of which there are many. $\endgroup$6more comments