Patterson-Sullivan distributions and quantum ergodicity

Abstract

We relate two types of phase space distributions associated to eigenfunctions $ϕ_{ir_j}$ of the Laplacian on a compact hyperbolic surface $X_Γ$: (1) Wigner distributions $\int_{S^*\X} a dW_{ir_j}=< Op(a)ϕ_{ir_j}, ϕ_{ir_j}>_{L^2(\X)}$, 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 $\lcal(s; a): = \sum_γ\frac{e^{-sL_γ}}{1-e^{-L_γ}} \int_{γ_0} a$ (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 $r_j \to \infty$. 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.