Doi proved that the $L^2_t H^{ 1/2}_x$ local smoothing effect for Schr\"odinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and $L^1\to L^\infty$ dispersive estimates still hold without loss for $e^{ it\Delta}$ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.
Categories: math.AP