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A microlocal toolbox for hyperbolic dynamics
Post-edited
Authors : Dyatlov, Semyon (Author of the conference)
CIRM (Publisher )
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
CIRM (Publisher )
Abstract :
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.
MSC Codes : 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
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 30/03/15 Conference Date : 11/03/15 Subseries : Research talks arXiv category : Dynamical Systems ; Analysis of PDEs Mathematical Area(s) : Dynamical Systems & ODE ; Geometry ; PDE Format : QuickTime (.mov) Video Time : 00:56:38 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2015-03-11_Dyatlov.mp4 
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Information on the Event
Event Title : Analysis and geometry of resonances / Analyse et géométrie des résonancesEvent Organizers : Guillarmou, Colin ; Hilgert, Joachim ; Pasquale, Angela ; Przebinda, Tomasz Dates : 09/03/15 - 13/03/15 Event Year : 2015 Event URL : http://www.math.univ-metz.fr/~pasquale/C...
Citation Data
DOI : 10.24350/CIRM.V.18727003Cite 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 |
Bibliography
- Dyatlov, S., & Guillarmou, C. (2014). Pollicott-Ruelle resonances for open systems.
- Dyatlov, S., & Zworski, M. (2015). Dynamical zeta functions for Anosov flows via microlocal analysis.
