  We prove that, under some natural conditions, Hamiltonian systems on a
contact manifold $C$ can be split into a Reeb dynamics on an open subset of $C$
and a Liouville dynamics on a submanifold of $C$ of codimension 1. For the Reeb
dynamics we find an invariant measure. Moreover, we show that, under certain
completeness conditions, the existence of an invariant measure for the
Liouville dynamics can be characterized using the notion of a symplectic
sandwich with contact bread.
