For a $n$ dimensional smooth manifold $M$, I consider the cotangent bundle $T^*M$ with [the canonical symplectic][1] form $\omega$.
A [symplectic map][2] $\phi : T^*M \to T^* M$ 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.

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*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][4] and this raised the question what a corresponding symplectic integrator would be in that case?



  [1]: https://en.wikipedia.org/wiki/Cotangent_bundle#The_cotangent_bundle_as_phase_space
  [2]: https://en.wikipedia.org/wiki/Symplectomorphism
  [3]: https://en.wikipedia.org/wiki/Volume_form
  [4]: https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)#Liouville_equations