  We study the time-dependent Schr\"odinger equation $ \imath\frac{\partial
u}{\partial t}=-1/2\Delta u,$ on a compact riemannian manifold on which the
geodesic flow has the Anosov property. Using the notion of semiclassical
measures, we prove various results related to the dispersive properties of the
Schr\"odinger propagator, and to controllability.
