  In this paper we give an overview over some aspects of the modern
mathematical theory of Ruelle resonances for chaotic, i.e. uniformly
hyperbolic, dynamical systems and their implications in physics. First we
recall recent developments in the mathematical theory of resonances, in
particular how invariant Ruelle distributions arise as residues of weighted
zeta functions. Then we derive a correspondence between weighted and
semiclassical zeta functions in the setting of negatively curved surfaces.
Combining this with results of Hilgert, Guillarmou and Weich yields a high
frequency interpretation of invariant Ruelle distributions as quantum
mechanical matrix coefficients in constant negative curvature. We finish by
presenting numerical calculations of phase space distributions in the more
physical setting of 3-disk scattering systems.
