  We relate two types of phase space distributions associated to eigenfunctions
$\phi_{ir_j}$ of the Laplacian on a compact hyperbolic surface $X_{\Gamma}$:
  (1) Wigner distributions $\int_{S^*\X} a dW_{ir_j}=< Op(a)\phi_{ir_j},
\phi_{ir_j}>_{L^2(\X)}$, which arise in quantum chaos. They are invariant under
the wave group.
  (2) Patterson-Sullivan distributions $PS_{ir_j}$, which are the residues of
the dynamical zeta-functions $\lcal(s; a): = \sum_\gamma
\frac{e^{-sL_\gamma}}{1-e^{-L_\gamma}} \int_{\gamma_0} a$ (where the sum runs
over closed geodesics) at the poles $s = {1/2} + ir_j$. They are invariant
under the geodesic flow.
  We prove that these distributions (when suitably normalized) are
asymptotically equal as $r_j \to \infty$. We also give exact relations between
them. This correspondence gives a new relation between classical and quantum
dynamics on a hyperbolic surface, and consequently a formulation of quantum
ergodicity in terms of classical ergodic theory.
