Vol. 8, No. 4, 2015

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ISSN: 1948-206X (e-only)
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Power spectrum of the geodesic flow on hyperbolic manifolds

Semyon Dyatlov, Frédéric Faure and Colin Guillarmou

Vol. 8 (2015), No. 4, 923–1000
Abstract

We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott–Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.

Keywords
Pollicott–Ruelle resonances, hyperbolic manifolds
Mathematical Subject Classification 2010
Primary: 37D40
Milestones
Received: 28 October 2014
Revised: 30 January 2015
Accepted: 6 March 2015
Published: 21 June 2015
Authors
Semyon Dyatlov
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
United States
Frédéric Faure
Institut Fourier
Universite Joseph Fourier
100, rue des Maths, BP74
38402 St Martin d’Heres
France
Colin Guillarmou
DMA, U.M.R. 8553 CNRS
École Normale Supérieure
45 rue d’Ulm, DMA
75230 Paris
France