Patterson--Sullivan distributions for rank one symmetric spaces of the noncompact type

Abstract

There is a remarkable relation between two kinds of phase space distributions associated to eigenfunctions of the Laplacian of a compact hyperbolic manifold: It was observed in \cite{AZ} that for compact hyperbolic surfaces $X_{\Gamma}=\Gamma\backslash\mathbb{H}$ Wigner distributions $\int_{S^* X_{\Gamma}} a dW_{ir_j} = < \mathrm{Op}(a)\phi_{ir_j},\phi_{ir_j} >_{L^2(X_{\Gamma})}$ and Patterson--Sullivan distributions $PS_{ir_j}$ are asymptotically equivalent as $r_j\to\infty$. We generalize the definitions of these distributions to all rank one symmetric spaces of noncompact type and introduce off-diagonal elements $PS_{\lambda_j,\lambda_k}$. Further, we give explicit relations between off-diagonal Patterson--Sullivan distributions and off-diagonal Wigner distributions and describe the asymptotic relation between these distributions.