Patterson–Sullivan distributions in higher rank
Identifieur interne : 000272 ( Main/Exploration ); précédent : 000271; suivant : 000273Patterson–Sullivan distributions in higher rank
Auteurs : Sönke Hansen [Allemagne] ; Joachim Hilgert [Allemagne] ; Michael Schröder [Allemagne]Source :
- Mathematische Zeitschrift [ 0025-5874 ] ; 2012-10-01.
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Abstract
Abstract: For a compact locally symmetric space X Γ 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 h-pseudo-differential calculus on X Γ , 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.
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DOI: 10.1007/s00209-011-0952-1
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Z.</title><idno type="ISSN">0025-5874</idno><idno type="eISSN">1432-1823</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>Berlin/Heidelberg</pubPlace><date type="published" when="2012-10-01">2012-10-01</date><biblScope unit="volume">272</biblScope><biblScope unit="issue">1-2</biblScope><biblScope unit="page" from="607">607</biblScope><biblScope unit="page" to="643">643</biblScope></imprint><idno type="ISSN">0025-5874</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0025-5874</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Geometric pseudo-differential analysis</term><term>Lifted quantum limits</term><term>Locally symmetric spaces</term><term>Patterson–Sullivan distributions</term><term>Quantum ergodicity</term><term>Weyl chamber flow</term><term>Wigner distributions</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: For a compact locally symmetric space X Γ 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 h-pseudo-differential calculus on X Γ , 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. 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