Patterson–Sullivan Distributions and Quantum Ergodicity
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DOI: 10.1007/s0002300603117
 Cite this article as:
 Anantharaman, N. & Zelditch, S. Ann. Henri Poincaré (2007) 8: 361. doi:10.1007/s0002300603117
Abstract.

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.

PattersonSullivan distributions \({PS_{{{ir}}_{j}}}\), which are the residues of the dynamical zetafunctions \({\mathcal{Z}(s; a) := \sum_\gamma \frac{e^{sL_\gamma}}{1e^{L_\gamma}}\int_{\gamma 0}\quad a}\) (where the sum runs over closed geodesics) at the poles s = 1/2 + ir _{ j }. They are invariant under the geodesic flow.