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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 $\psi : \mathrm{\Omega}^{2n}(T^*M) \to \mathrm{\Omega}^{2n}(T^*M)$ compatible with the symplectic structure of $T^*M$?

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?

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?

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 $\psi : \mathrm{\Omega}^n(T^*M) \to \mathrm{\Omega}^n(T^*M)$ compatible with the symplectic structure of $T^*M$?

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?

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 $\psi : \mathrm{\Omega}^{2n}(T^*M) \to \mathrm{\Omega}^{2n}(T^*M)$ compatible with the symplectic structure of $T^*M$?

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?

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?

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 $\psi : \mathrm{\Omega}^n(T^*M) \to \mathrm{\Omega}^n(T^*M)$ compatible with the symplectic structure of $T^*M$?

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?