Partially ordered groups and geometry of contact transformations

We prove that, for a class of contact manifolds, the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contacto-morphisms. It give s rise to invariants of contacto-morphisms which generalize the classical n otion of the rotation number. Our approach is based on tools of Symplectic Topology.