Moser's celebrated theorem guarantees that every diffeomorphism of a closed manifold can be isotoped to a volume-preserving one. We show that this statement cannot be extended into the contact category: some connected components of contactomorphism groups contain no volume-preserving maps. Thus, the dissipation of volume appears for purely topological reasons. This phenomenon can be considered from different viewpoints: geometric (isometric action of the contact mapping class group on the moduli space of contact forms), topological (action in symplectic homology) and dynamical (propagation of trajectories for symplectic maps). We define a numerical invariant—a contact Lyapunov exponent—which leads to a quantitive version of the above-mentioned result.