# ANNALES DE L'INSTITUT FOURIER

On the spectral theory and dynamics of asymptotically hyperbolic manifolds
Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2461-2492.

We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.

Cet article est une présentation rapide de la théorie spectrale et de la dynamique des variétés asymptotiquement hyperboliques à volume infini. Nous commençons par leur géométrie et quelques exemples, nous poursuivons en rappelant leur théorie spectrale, puis continuons sur des développements récents de leur dynamique. Nous concluons par une discussion des résultats qui démontrent un rapport entre leurs mécaniques quantiques et classiques et enfin, nous offrons quelques idées et conjectures.

DOI: 10.5802/aif.2615
Classification: 37D40, 58J50, 53C22
Keywords: Asymptotically hyperbolic, conformally compact, wave trace, negative curvature, resonances, length spectrum, topological entropy, dynamics, geodesic flow, prime orbit theorem, quantum and classical mechanics
Rowlett, Julie 1

1 Hausdorff Center for Mathematics Villa Maria Endenicher Allee 62 53115 Bonn (Deutschland)
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Rowlett, Julie. On the spectral theory and dynamics of asymptotically hyperbolic manifolds. Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2461-2492. doi : 10.5802/aif.2615. https://aif.centre-mersenne.org/articles/10.5802/aif.2615/

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