  In this paper, the chaotic ray dynamics inside dielectric cavities is
described by the properties of an invariant chaotic saddle. I show that the
localization of the far field emission in specific directions is related to the
filamentary pattern of the saddle's unstable manifold, along which the energy
inside the cavity is distributed. For cavities with mixed phase space, the
chaotic saddle is divided in hyperbolic and non-hyperbolic components, related,
respectively, to the intermediate exponential (t<t_c) and the asymptotic
power-law (t>t_c) decay of the energy inside the cavity. The alignment of the
manifolds of the two components of the saddle explains why even if the energy
concentration inside the cavity dramatically changes from t<t_c to t>t_c, the
far field emission changes only slightly. Simulations in the annular billiard
confirm and illustrate the predictions.
