Patterson-sullivan distributions and quantum ergodicity

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-Gamma: Wigner distributions integral(S*X Gamma) a dWir(j) = [Op(a)phi ir(j), phi ir(j))L-2 (x(Gamma)) which arise in quantum chaos. They are invariant under the wave group. Patterson-Sullivan distributions Psir(j), which are the residues of the dynamical zeta-functions Z(s;a):= Sigma(gamma) e(-sL gamma)/1-e-L gamma integral(gamma o) 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) -> infinity. 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.