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Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Geometry
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
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- xxi; 185 p.
ISBN 978-2-85629-303-4
Séminaires et congrès , 0019
Localisation : Collection 1er étage
opérateur de transfert # produit de matrices aléatoires stationnaires # spectre de Lyapunov # algorithme de Jacobi-Perron # algorithme des fractions continues multidimensionnelles # approximation diophantienne simultanée # dimension de Hausdorff # espace localement symétrique # flot unipotent # théorème de Ratner # espace homogène p-adique # points de Heegner # billards polygonaux
11Gxx ; 11H16 ; 11H46 ; 11J06 ; 11J13 ; 11J17 ; 11J25 ; 11J70 ; 11J83 ; 11K50 ; 11K55 ; 11K60 ; 22E40 ; 22E46 ; 28A80 ; 30F40 ; 37A15 ; 37A17 ; 37A30 ; 37D50 ; 37H15 ; 53C35
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Research talks;Analysis and its Applications;Dynamical Systems and Ordinary Differential Equations;Mathematical Physics
In this talk we explain how the Fibonacci trace map arises from the Fibonacci substitution and leads to a unified framework in which a variety of models can be studied. We discuss the associated foliations, hyperbolic sets, stable and unstable manifolds, and how the intersections of the stable manifolds with the model-dependent curve of initial conditions allow one to translate dynamical into spectral results.
81Q10 ; 81Q35 ; 37D20 ; 37D50
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Special events;Lagrange Days;Dynamical Systems and Ordinary Differential Equations;Mathematical Physics
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
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Research schools;Combinatorics;Dynamical Systems and Ordinary Differential Equations
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
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Research talks;Dynamical Systems and Ordinary Differential Equations
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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Research talks;Dynamical Systems and Ordinary Differential Equations;Probability and Statistics
We investigate the diffusion and statistical properties of Lorentz gas with cusps at flat points. This is a modification of dispersing billiards with cusps. The decay rates are proven to depend on the degree of the flat points, which varies from $n^{-a}$, for $ a\in (0,\infty)$. The stochastic processes driven by these systems enjoy stable law and have super-diffusion driven by Lévy process. This is a joint work with Paul Jung and Françoise Pène.
We investigate the diffusion and statistical properties of Lorentz gas with cusps at flat points. This is a modification of dispersing billiards with cusps. The decay rates are proven to depend on the degree of the flat points, which varies from $n^{-a}$, for $ a\in (0,\infty)$. The stochastic processes driven by these systems enjoy stable law and have super-diffusion driven by Lévy process. This is a joint work with Paul Jung and Françoise ...
37D50 ; 37A25 ; 60F05 ; 37D25
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Research talks;Dynamical Systems and Ordinary Differential Equations;Probability and Statistics
We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Then we discuss applications such as proving various mixing properties of infinite measure preserving systems. Based on joint work with Dmitry Dolgopyat.
We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Then we discuss applications such as proving various mixing properties of infinite measure preserving systems. Based on joint work with Dmitry ...
37A50 ; 37D50 ; 60F05 ; 37D20
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Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry
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
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Research schools;Analysis and its Applications;Dynamical Systems and Ordinary Differential Equations
We will discuss an approach to the statistical properties of two-dimensional dispersive billiards (mostly discrete-time) using transfer operators acting on anisotropic Banach spaces of distributions. The focus of this part will be our recent work with Mark Demers on the measure of maximal entropy but we will also survey previous results by Demers, Zhang, Liverani, etc on the SRB measure.
37D50 ; 37C30 ; 37B40
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Research schools;Analysis and its Applications;Dynamical Systems and Ordinary Differential Equations
We will discuss an approach to the statistical properties of two-dimensional dispersive billiards (mostly discrete-time) using transfer operators acting on anisotropic Banach spaces of distributions. The focus of this part will be our recent work with Mark Demers on the measure of maximal entropy but we will also survey previous results by Demers, Zhang, Liverani, etc on the SRB measure.
37D50 ; 37C30 ; 37B40
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Research schools;Analysis and its Applications;Dynamical Systems and Ordinary Differential Equations
We will discuss an approach to the statistical properties of two-dimensional dispersive billiards (mostly discrete-time) using transfer operators acting on anisotropic Banach spaces of distributions. The focus of this part will be our recent work with Mark Demers on the measure of maximal entropy but we will also survey previous results by Demers, Zhang, Liverani, etc on the SRB measure.
37D50 ; 37C30 ; 37B40
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- 61 p.
ISBN
Bonner mathematische schriften , 0328
Localisation : Publication 1er étage
système dynamique # entropie topologique positive # billard # billard convexe # plan hyperbolique # anneau non concentrique # analyse locale
37B40 ; 37D50 ; 37C35 ; 37C27 ; 37B10
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- vii; 193 p.
ISBN 978-0-8218-4282-9
Memoirs of the american mathematical society , 0927
Localisation : Collection 1er étage
pocessus de diffusion # mouvement brownien # théorème limite
37D50 ; 34C29 ; 60F17
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- 316 p.
ISBN 978-0-8218-4096-2
Mathematical surveys and monographs , 0127
Localisation : Collection 1er étage
dynamique # système dynamique # comportement chaotique des systèmes # théorie ergodique # théorie de la mesure # probabilités billiards # système hyperbolique non-uniforme # équilibre # exposant de Lyapunov # billiard de Burnimovich
37D50 ; 37D25 ; 37A25 ; 37N05 ; 82B99
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- 176 p.
ISBN 978-0-8218-3919-5
Student mathematical library , 0030
Localisation : Collection 1er étage
géométrei # système dynamique # billard # calacul de variations # système hyperbolique avec des singlantes # équation d'hamilton # mécanique statistique # chaos # théorie de Morse # théorème de récurrence de Poincaré # 4 problème de Hilbert # courbe plane
37-01 ; 37D50 ; 37J10 ; 51-02 ; 70H05 ; 82C05
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