# Patterson–Sullivan Distributions and Quantum Ergodicity

Open AccessArticle

- First Online:

DOI: 10.1007/s00023-006-0311-7

- Cite this article as:
- Anantharaman, N. & Zelditch, S. Ann. Henri Poincaré (2007) 8: 361. doi:10.1007/s00023-006-0311-7

- 6 Citations
- 178 Downloads

## 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
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.

*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.

Download to read the full article text

## Copyright information

© Birkhäuser Verlag, Basel/Switzerland 2007