This article gives relations between two types of phase space distributions associated to eigenfunctions ${\phi_{{{ir}}_{j}}}$ of the Laplacian on a compact hyperbolic surface X _Γ:

• Wigner distributions ${\int_{S^{*}X_{\Gamma}} a\quad dW_{{{ir}}_{j}} =\langle Op(a)\phi_{{{ir}}_{j}},\phi_{{{ir}}_{j}}\rangle_{{{L}^2}(X_\Gamma)}}$ , which arise in quantum chaos. They are invariant under the wave group.

• Patterson-Sullivan distributions ${PS_{{{ir}}_{j}}}$ , which are the residues of the dynamical zeta-functions ${\mathcal{Z}(s; a) := \sum_\gamma \frac{e^{-sL_\gamma}}{1-e^{-L_\gamma}}\int_{\gamma 0}\quad a}$ (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 ${r_j \rightarrow \infty}$ . 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.