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 ${\int }_{S^{\ast }{X}_{\Gamma }}ad{W}_{i{r}_j}=<\mathrm{Op}(a){\varphi }_{i{r}_j},{\varphi }_{i{r}_j}{>}_{L^2(X_{\Gamma })}$ and Patterson--Sullivan distributions $P{S}_{i{r}_j}$ 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 $P{S}_{{\lambda }_j,{\lambda }_k}$. Further, we give explicit relations between off-diagonal Patterson--Sullivan distributions and off-diagonal Wigner distributions and describe the asymptotic relation between these distributions.

25 pages

References in corpus (1)

• Patterson-Sullivan distributions and quantum ergodicity

Cited by in corpus (1)

• Patterson--Sullivan distributions in higher rank