m

F Nous contacter


0

Documents  37D40 | enregistrements trouvés : 21

O

-A +A

P Q

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.
Déposez votre fichier ici pour le déplacer vers cet enregistrement.

We consider the one-parameter families of transfer operators for geodesic flows on negatively curved manifolds. We show that the spectra of the generators have some "band structure" parallel to the imaginary axis. As a special case of "semi-classical" transfer operator, we see that the eigenvalues concentrate around the imaginary axis with some gap on the both sides.

37C30 ; 37D40 ; 53D25 ; 81Q50

Déposez votre fichier ici pour le déplacer vers cet enregistrement.
Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

We will discuss old and recent results on topological and measurable dynamics of diagonal and unipotent flows on frame bundles and unit tangent bundles over hyperbolic manifolds. The first lectures will be a good introduction to the subject for young researchers.

37D40 ; 37A17 ; 37A25

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

In this talk, we will prove the projective equidistribution of integral representations by quadratic norm forms in positive characteristic, with error terms, and deduce asymptotic counting results of these representations. We use the ergodic theory of lattice actions on Bruhat-Tits trees, and in particular the exponential decay of correlation of the geodesic flow on trees for Hölder variables coming from symbolic dynamics techniques.

20E08 ; 11J61 ; 37A25 ; 20G25 ; 37D40

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is uniquely ergodic in almost every direction. We call this class of surface essentially finite.
(joint work with Anja Randecker).
The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is ...

37D40 ; 51A40 ; 37A25

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

We prove a couple of general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of push-forwards of horocycle measures under the geodesic flow and a new short proof of a theorem of Chaika and Eskin on Birkhoff genericity in almost all directions for the Teichmüller geodesic flow.
We prove a couple of general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of ...

37D40 ; 37C40 ; 37A17

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

I will survey recent results on the generic properties of probability measures invariant by the geodesic flow defined on a nonpositively curved manifold. Such a flow is one of the early example of a non-uniformly hyperbolic system. I will talk about ergodicity and mixing both in the compact and noncompact setting, and ask some questions about the associated frame flow, which is partially hyperbolic.

37B10 ; 37D40 ; 34C28 ; 37C20 ; 37C40 ; 37D35

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Over the last few years I developed (partly jointly with coauthors) dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for certain hyperbolic surfaces/orbifolds L \ H with cusps (both of finite and infinite area; arithmetic and non-arithmetic).
Both types of transfer operators arise from discretizations of the geodesic flow on L \ H. The eigenfunctions with eigenvalue 1 of slow transfer operators characterize Maass cusp forms. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. The Fredholm determinant of the fast transfer operators equals the Selberg zeta function, which yields that the zeros of the Selberg zeta function (among which are the resonances) are determined by the eigenfunctions with eigenvalue 1 of the fast transfer operators. It is a natural question how the eigenspaces of these two types of transfer operators are related to each other.
Over the last few years I developed (partly jointly with coauthors) dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for certain hyperbolic surfaces/orbifolds L \ H with cusps (both of finite and infinite area; arithmetic and non-arithmetic).
Both types of transfer operators arise from discretizations of the geodesic flow on L \ H. The eigenfunctions with eigenvalue 1 of slow transfer ...

37C30 ; 11F03 ; 37D40

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

I will explain how one can get a complete description of the correlation spectrum of a Morse-Smale flow in terms of the Lyapunov exponents and of the periods of the flow. I will also discuss the relation of these results with differential topology.
This a joint work with Nguyen Viet Dang (Université Lyon 1).

37D15 ; 58J51 ; 37D40

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.
These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...

37D35 ; 37D40 ; 37C40 ; 37D25

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.
These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...

37D35 ; 37D40 ; 37C40 ; 37D25

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.
These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...

37D35 ; 37D40 ; 37C40 ; 37D25

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

We study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted regular periodic orbits are equidistributed relative to these unique equilibrium states.
We study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted ...

37D35 ; 37D40 ; 37D25

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

In this talk, we will discuss various growth rates associated to Anosov flows and their covers. The topological entropy of an Anosov flow on a compact manifold is realised as the exponential growth rate of its periodic orbits. If we pass to a regular cover of the manifold then we can consider a corresponding growth rate for the lifted flow. This growth is bounded above by the topological entropy but if the cover is infinite then the growth rate may be strictly smaller. For abelian covers, this phenomenon admits a precise description in terms of a variational principle. More recent work, joint with Rhiannon Dougall, considers more general infinite covers.
In this talk, we will discuss various growth rates associated to Anosov flows and their covers. The topological entropy of an Anosov flow on a compact manifold is realised as the exponential growth rate of its periodic orbits. If we pass to a regular cover of the manifold then we can consider a corresponding growth rate for the lifted flow. This growth is bounded above by the topological entropy but if the cover is infinite then the growth rate ...

37D20 ; 37D35 ; 37D40 ; 37B40

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

For negatively curved Riemannian manifolds, Margulis gave an asymptotic formula for the number of closed geodesics with length below a given threshold. I will describe joint work with Gerhard Knieper and Khadim War in which we obtain the corresponding result for surfaces without conjugate points by first proving uniqueness of the measure of maximal entropy and then following the approach of recent work by Russell Ricks, who established the asymptotic estimates in the setting of CAT(0) geodesic flows.
For negatively curved Riemannian manifolds, Margulis gave an asymptotic formula for the number of closed geodesics with length below a given threshold. I will describe joint work with Gerhard Knieper and Khadim War in which we obtain the corresponding result for surfaces without conjugate points by first proving uniqueness of the measure of maximal entropy and then following the approach of recent work by Russell Ricks, who established the ...

53D25 ; 37D40

Z