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Documents  37D50 | enregistrements trouvés : 5

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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 $(X + \lambda)u = f$ , 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. 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 ...

37D50 ; 53D25 ; 37D20 ; 35B34 ; 35P25

Multi angle  Billiard and rigid rotation
Treschev, Dmitry (Auteur de la Conférence) | CIRM (Editeur )

Can a billiard map be locally conjugated to a rigid rotation? We prove that the answer to this question is positive in the category of formal series. We also present numerical evidence that for "good" rotation angles the answer is also positive in an analytic category.
billiard systems # integrable Hamiltonian systems # normal form convergence # small divisors # elliptic fixed point # analytic conjugacy

37D50 ; 70H06

Multi angle  Flat surfaces and combinatorics
Goujard, Élise (Auteur de la Conférence) | CIRM (Editeur )

Billiards in polygons are related to dynamics of the linear flow on flat surfaces. Through some examples of counting problems on flat surfaces and on moduli spaces of flat surfaces, we will see how combinatorics can lead to interesting dynamical results in this setting.

30F30 ; 32G15 ; 37D50

We will present a geometric criterion for the ergodicity of the billiard flow in a polygon with non-rational angles and discuss its application to the Diophantine case.

37D40 ; 37D50 ; 30F10 ; 30F60 ; 32G15

Tiling billiards is a dynamical system where beams of light refract through planar tilings. It turns out that, for a regular tiling of the plane by congruent triangles, the light trajectories can be described by interval exchange transformations. I will explain this surprising correspondence, give related results, and show computer simulations of the system.

37D50 ; 37B50

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