2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59998A theorem of Moser 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 contact category: some connected components of contactomorphism groups of certain contact manifolds contain no volume-preserving diffeomorphisms. 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 (diffusion). We define a numerical invariant - a kind of contact Lyapunov exponent - which leads to a quantitive version of the abovementioned result.Latex, 14 pages, preliminary versionSymplectic GeometryDynamical Systems(2000) 53Dxx (Primary) 37Jxx (Secondary)text