Liouville's Theorem ( the continuity equation in phase space ) shows that the comoving volume in phase space is unchanged by Hamilton's equations of motion. Symplectic integrators preserve not only this property but are also said to preserve "Poincaré invariants" which are lower dimensional geometrical constructs of some kind. I would appreciate an example which is interesting ( not a harmonic chain or ideal gas ) and which has Poincaré invariants which can be followed numerically in time. Help appreciated.
Please see: Maruskin, Jared M., Daniel J. Scheeres, and Anthony M. Bloch. "Dynamics of symplectic subvolumes." SIAM Journal on Applied Dynamical Systems 8.1 (2009): 180-201.
Scheeres, D. J., et al. "Fundamental limits on spacecraft orbit uncertainty and distribution propagation." The Journal of the Astronautical Sciences 54.3-4 (2006): 505-523.
These papers discuss some cases.