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

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 [1] that for compact hyperbolic surfaces XΓ = Γ\H Wigner distributions R S∗XΓ a dWirj = 〈Op(a)φirj , φirj 〉L2(XΓ) and Patterson–Sullivan distributions PSirj are asymptotically equivalent as rj → ∞. We generalize the definitions of these distributions to all rank one symmetric spaces of noncompact type and introduce offdiagonal elements PSλj,λk . Further, we give explicit relations between offdiagonal Patterson–Sullivan distributions and off-diagonal Wigner distributions and describe the asymptotic relation between these distributions. 2000 Mathematics Subject Classification: Primary 53C35; Secondary 58C40, 58J50.