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

Patterson–Sullivan Distributions and Quantum Ergodicity

Abstract.

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.