Semyon Dyatlov: A microlocal toolbox for hyperbolic dynamics

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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+λ)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.

Recording during the meeting: "Analysis and geometry of resonances" the March 11, 2015 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

Видео Semyon Dyatlov: A microlocal toolbox for hyperbolic dynamics канала Centre International de Rencontres Mathématiques