Selberg Trace Formulae and Equidistribution Theorems for Closed Geodesics and Laplace Eigenfunctions
Finite Area Surfaces
Seiten
1992
American Mathematical Society (Verlag)
978-0-8218-2526-6 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-2526-6 (ISBN)
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This work is concerned with a pair of dual asymptotics problems on a finite-area hyperbolic surface. The first problem is to determine the distribution of closed geodesics in the unit tangent bundle. The second problem is to determine the distribution of eigenfunctions (in microlocal sense) in the unit tangent bundle.
This work is concerned with a pair of dual asymptotics problems on a finite-area hyperbolic surface. The first problem is to determine the distribution of closed geodesics in the unit tangent bundle. The author's results give a quantitative form to Bowen's equidistribution theory: they refine Bowen's theorem much as the prime geodesic theorem on hyperbolic quotients refines the asymptotic formula for the number of closed geodesics of length less than T. In particular, the author gives a rate of equidistribution in terms of low eigenvalues of the Laplacian. The second problem is to determine the distribution of eigenfunctions (in microlocal sense) in the unit tangent bundle. The main result here (which is needed for the equidistribution theory of closed geodesics) is a proof of a signed and averaged version of the mean Lindelof hypothesis for Rankin-Selberg zeta functions. The main tool used here is a generalization of Selberg's trace formula.
This work is concerned with a pair of dual asymptotics problems on a finite-area hyperbolic surface. The first problem is to determine the distribution of closed geodesics in the unit tangent bundle. The author's results give a quantitative form to Bowen's equidistribution theory: they refine Bowen's theorem much as the prime geodesic theorem on hyperbolic quotients refines the asymptotic formula for the number of closed geodesics of length less than T. In particular, the author gives a rate of equidistribution in terms of low eigenvalues of the Laplacian. The second problem is to determine the distribution of eigenfunctions (in microlocal sense) in the unit tangent bundle. The main result here (which is needed for the equidistribution theory of closed geodesics) is a proof of a signed and averaged version of the mean Lindelof hypothesis for Rankin-Selberg zeta functions. The main tool used here is a generalization of Selberg's trace formula.
Review of (Z.2) on the compact case; Trace formula for Tr *sRrj (*s cuspidal); Trace formula for Tr*sRrj (*s Einstein); Spectral estimates (Weyl laws); Asymptotic expansions (Prime geodesic theorems); Equidistribution of closed goedesics.
| Erscheint lt. Verlag | 30.7.1992 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 0-8218-2526-7 / 0821825267 |
| ISBN-13 | 978-0-8218-2526-6 / 9780821825266 |
| Zustand | Neuware |
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