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We generalize parts of a special non-Euclidean calculus of pseudodifferentialoperators, which was invented by S. Zelditch for hyperbolic surfaces, tosymmetric spaces $X=G/K$ of the noncompact type and their compact quotients$Y=\Gamma\backslash G/K$. We sometimes restrict our results to the case of rankone symmetric spcaes. The non-Euclidean setting extends the defintion ofso-called Patterson-Sullivan distributions, which were first defined by N.Anantharaman and S. Zelditch for hyperbolic systems, in a natural way toarbitrary symmetric spaces of the noncompact type. We find an explicitintertwining operator mapping Patterson-Sullivan distributions into Wignerdistributions. We study the important invariance and equivariance properties ofthese distributions. Finally, we describe asymptotic properties of thesedistributions.

Patterson-Sullivan distributions for symmetric spaces of the noncompact type