On the spectral theory and dynamics of asymptotically hyperbolic manifolds

Rowlett, Julie

Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2461-2492.

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.

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.

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
Mot clés : variété asymptotiquement hyperbolique, conformement compact, courbures negatives, spectre des longeurs géodesiques, flot géodesique, dynamique, formule de trace dynamique, entropie topologique, mécanique quantique et classique

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