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Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.[-]

Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties ...[+]

81Q50 ; 37N20 ; 35P20 ; 58J51 ; 58J50 ; 37D40

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We identify natural conditions on group actions on trees which imply that the induced action on the boundary is (Borel/measure) hyperfinite. We will consider the differences between the Borel and measurable versions, and discuss different notions of amenability which arise in the proofs.

03E15 ; 54H05 ; 37D40

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The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study, this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and define the measure of maximal entropy via Patterson-Sullivan construction. In the fourth lecture, we will show that the measure of maximal entropy is unique and we will use the cross ratio function to prove the geodesic flow is mixing with respect to the constructed measure. In the last lecture, we will prove the prime geodesic theorem of the surface. This is based on two joint work with Gerhard Knieper and Vaughn Climenhaga.[-]

The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study, this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and ...[+]

37C35 ; 37D40 ; 53C22

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The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study,this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and define the measure of maximal entropy via Patterson-Sullivan construction. In the fourth lecture, we will show that the measure of maximal entropy is unique and we will use the cross ratio function to prove the geodesic flow is mixing with respect to the constructed measure. In the last lecture, we will prove the prime geodesic theorem of the surface. This is based on two joint work with Gerhard Knieper and Vaughn Climenhaga.[-]

The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study,this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and ...[+]

37C35 ; 37D40 ; 53C22

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The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study, this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and define the measure of maximal entropy via Patterson-Sullivan construction. In the fourth lecture, we will show that the measure of maximal entropy is unique and we will use the cross ratio function to prove the geodesic flow is mixing with respect to the constructed measure. In the last lecture, we will prove the prime geodesic theorem of the surface. This is based on two joint work with Gerhard Knieper and Vaughn Climenhaga.[-]

The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study, this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and ...[+]

37C35 ; 37D40 ; 53C22

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This is a work with Yves Colin de Verdière, Charlotte Dietze and Maarten De Hoop, motivated by recent works by M. De Hoop on inverse problems for sound wave propagation in gas giant planets. On such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with a boundary whose metric blows up near the boundary. With appropriate variable changes, we can reduce the study of the Laplacian?Beltrami to that of a kind of sub-Riemannian Laplacian. In this talk, I will explain how to approach the spectral analysis of such operators, and in particular how to calculate WeylÕs law.[-]

This is a work with Yves Colin de Verdière, Charlotte Dietze and Maarten De Hoop, motivated by recent works by M. De Hoop on inverse problems for sound wave propagation in gas giant planets. On such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with a boundary whose metric blows up near the boundary. With appropriate variable changes, we can reduce the study of ...[+]

11F72 ; 58C40 ; 53C22 ; 37D40 ; 53C65 ; 35R30

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I will present results on the dynamics of horocyclic flows on the unit tangent bundle of hyperbolic surfaces, density and equidistribution properties in particular. I will focus on infinite volume hyperbolic surfaces. My aim is to show how these properties are related to dynamical properties of geodesic flows, as product structure, ergodicity, mixing, ...

37D40

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37D40 ; 37A50 ; 37D10

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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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37D40 ; 58J50

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Documents 37D40 32 résultats

Semiclassical behaviour of quantum eigenstates - lecture 2 - Rivière, Gabriel (Auteur de la Conférence) | CIRM H Nouveau

Amenability and hyperfiniteness for group actions on trees - Spaas, Pieter (Auteur de la Conférence) | CIRM H Nouveau

Geodesic flow on surfaces without conjugate points - part 1 - War, Khadim Mbacke (Auteur de la Conférence) | CIRM H Nouveau

Geodesic flow on surfaces without conjugate points - part 2 - War, Khadim Mbacke (Auteur de la Conférence) | CIRM H Nouveau

Geodesic flow on surfaces without conjugate points - part 3 - War, Khadim Mbacke (Auteur de la Conférence) | CIRM H Nouveau

From gas giant planets to the spectral theory of subelliptic Laplacians - Trélat, Emmanuel (Auteur de la Conférence) | CIRM H Nouveau

Horocyclic flows on hyperbolic surfaces - Part I - Schapira, Barbara (Auteur de la Conférence) | CIRM H Nouveau

Precise statistical properties of the geodesic flow on periodic hyperbolic manifolds - Thomine, Damien (Auteur de la Conférence) | CIRM H Nouveau

On the ergodicity of billiards in non-rational polygons - Forni, Giovanni (Auteur de la Conférence) | CIRM H Nouveau

Correspondence between Ruelle resonances and quantum resonances for non-compact Riemann surfaces - Guillarmou, Colin (Auteur de la Conférence) | CIRM H Nouveau