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A microlocal toolbox for hyperbolic dynamics

De Semyon Dyatlov

Apparaît également dans la collection : Analysis and geometry of resonances / Analyse et géométrie des résonances

I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between shift to higher frequencies and escaping in the physical space. I will show meromorphic continuation of the resolvent of $X$; the poles, known as Pollicott-Ruelle resonances, describe exponential decay of correlations. As an application, I will prove that the Ruelle zeta function continues meromorphically for flows on non-compact manifolds (the compact case, known as Smale's conjecture, was recently settled by Giulietti-Liverani- Pollicott and a simple microlocal proof was given by Zworski and the speaker). Joint work with Colin Guillarmou.

Informations sur la vidéo

Données de citation

  • DOI 10.24350/CIRM.V.18727003
  • Citer cette vidéo Dyatlov, Semyon (11/03/2015). A microlocal toolbox for hyperbolic dynamics. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18727003
  • URL https://dx.doi.org/10.24350/CIRM.V.18727003

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