Sharp polynomial bounds on the number of Pollicott-Ruelle resonances

Abstract

We give a sharp polynomial bound on the number of Pollicott-Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure-Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer-Sjöstrand in scattering theory and by Faure-Sjöstrand in the theory of Anosov flows.

18 pages; minor revision based on referee's comments. To appear in Erg. Theory Dyn. Syst

References in corpus (3)

• Anisotropic Sobolev spaces and dynamical transfer operators: C^infty foliations
• Contact Anosov flows and the FBI transform
• Fractal Weyl laws for asymptotically hyperbolic manifolds

Cited by in corpus (9)

• Pollicott-Ruelle resonances for open systems
• Resonance projectors and asymptotics for r-normally hyperbolic trapped sets
• Power spectrum of the geodesic flow on hyperbolic manifolds
• Fractal Weyl law for the Ruelle spectrum of Anosov flows
• Higher rank quantum-classical correspondence
• FBI Transform in Gevrey Classes and Anosov Flows
• Spectral analysis of morse-smale flows ii: resonances and resonant states
• A local trace formula for Anosov flows (with an appendix by Frédéric Naud)
• Fractal Weyl laws and wave decay for general trapping