Patterson-Sullivan distributions for symmetric spaces of the noncompact type

Abstract

We generalize parts of a special non-Euclidean calculus of pseudodifferential operators, which was invented by S. Zelditch for hyperbolic surfaces, to symmetric spaces $X=G/K$ of the noncompact type and their compact quotients $Y=Γ\backslash G/K$. We sometimes restrict our results to the case of rank one symmetric spcaes. The non-Euclidean setting extends the defintion of so-called Patterson-Sullivan distributions, which were first defined by N. Anantharaman and S. Zelditch for hyperbolic systems, in a natural way to arbitrary symmetric spaces of the noncompact type. We find an explicit intertwining operator mapping Patterson-Sullivan distributions into Wigner distributions. We study the important invariance and equivariance properties of these distributions. Finally, we describe asymptotic properties of these distributions.