, Volume 8, Issue 2, pp 361-426,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 05 Apr 2007

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 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.

Communicated by Jens Marklof.
Research partially supported by NSF grant #DMS-0302518 and NSF Focussed Research Grant # FRG 0354386.
Submitted: April 20, 2006. Revised: July 10, 2006. Accepted: July 31, 2006.