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H 2 A microlocal toolbox for hyperbolic dynamics

Auteurs : Dyatlov, Semyon (Auteur de la Conférence)
CIRM (Editeur )

 Loading the player... geodesic flow correlation meromorphic continuation applications to asymptotics open hyperbolic system Anosov and Axiom A flows Ruelle zeta function anisotropic Sobolev space microlocal approach to hyperbolic flows microlocal propagation estimates open billiards Questions

Résumé : 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.

Codes MSC :
35P25 - Scattering theory
37D20 - Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D50 - Hyperbolic systems with singularities (billiards, etc.)
53D25 - Geodesic flows
35B34 - Resonances in solutions of PDE

 Informations sur la Vidéo Langue : Anglais Date de publication : 30/03/15 Date de captation : 11/03/15 Collection : Research talks ; Partial Differential Equations ; Dynamical Systems and Ordinary Differential Equations ; Geometry Format : QuickTime (.mov) Durée : 00:56:38 Domaine : Dynamical Systems & ODE ; Geometry ; PDE Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-03-11_Dyatlov.mp4 Informations sur la rencontre Nom de la rencontre : Analysis and geometry of resonances / Analyse et géométrie des résonancesOrganisateurs de la rencontre : Guillarmou, Colin ; Hilgert, Joachim ; Pasquale, Angela ; Przebinda, TomaszDates : 09/03/15 - 13/03/15 Année de la rencontre : 2015 URL Congrès : http://www.math.univ-metz.fr/~pasquale/C... Citation Data DOI : 10.24350/CIRM.V.18727003 Cite this video as: Dyatlov, Semyon (2015). A microlocal toolbox for hyperbolic dynamics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18727003 URI : http://dx.doi.org/10.24350/CIRM.V.18727003

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