Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 4 (2015), 923-1000.

Power spectrum of the geodesic flow on hyperbolic manifolds

Semyon Dyatlov, Frédéric Faure, and Colin Guillarmou

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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.

Article information

Anal. PDE, Volume 8, Number 4 (2015), 923-1000.

Received: 28 October 2014
Revised: 30 January 2015
Accepted: 6 March 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Pollicott–Ruelle resonances hyperbolic manifolds


Dyatlov, Semyon; Faure, Frédéric; Guillarmou, Colin. Power spectrum of the geodesic flow on hyperbolic manifolds. Anal. PDE 8 (2015), no. 4, 923--1000. doi:10.2140/apde.2015.8.923.

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