Patterson-Sullivan distributions and quantum ergodicity

This article gives relations between two types of phase space distributions associated to eigenfunctions φirj of the Laplacian on a compact hyperbolic surface X _Γ: Wigner distributions ∫S*XΓ a dWirj = Op(a)φirj,φ 〈 irj L²Γ, which arise in quantum chaos. They are invariant under the wave group. Patterson-Sullivan distributions PSirj, which are the residues of the dynamical zeta-functions Z(s; a) := ∑γ -sL_γ1-e-L_γ∫γ 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 → ∞ . 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.