Patterson--Sullivan distributions in higher rank
Spectral Theory
2011-05-31 v1 Representation Theory
Abstract
For a compact locally symmetric space of non-positive curvature, we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators. Using an -\psdiff\ calculus on , we define and study lifted quantum limits as weak-limit points of Wigner distributions. The Helgason boundary values of the eigenfunctions allow us to construct Patterson--Sullivan distributions on the space of Weyl chambers. These distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties, which makes them useful in the context of quantum ergodicity. Our results generalize results for compact hyperbolic surfaces obtained by Anantharaman and Zelditch.
Cite
@article{arxiv.1105.5788,
title = {Patterson--Sullivan distributions in higher rank},
author = {Sönke Hansen and Joachim Hilgert and Michael Schröder},
journal= {arXiv preprint arXiv:1105.5788},
year = {2011}
}