Patterson-Sullivan distributions for symmetric spaces of the noncompact type

Details Patterson-Sullivan distributions for symmetric spaces of the noncompact type Patterson-Sullivan distributions for symmetric spaces of the noncompact type

2010-12-06

arxiv.org/abs/1012.1113v1

Mathematics

Group Theory

Scientific paper

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 $Y=\Gamma\backslash G/K$. 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.

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