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 corresponding spaces of [volume forms][3]? In other words, when is a map $\psi : \mathrm{\Omega}^n(T^*M) \to \mathrm{\Omega}^n(T^*M)$ *symplectic*?

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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