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## Program

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9

Marco

Bessi

10

Iturriaga

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11-11:30

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11:30

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12:30-14

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14

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Villani

15

Bolotin

Yoccoz

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16-16:30

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16:30

de la Llave

### Marie-Claude Arnaud  (Université d'Avignon)

Slides of the Talk [PDF - 1 Mo]
Boundaries of instability zones for symplectic twist maps
We will give an example of a C^2 symplectic twist map f of the annulus that has an essential invariant curve C such that:
- C is not differentiable;
- the dynamics of f restricted to C is Denjoy;
- C is at the boundary of an instability zone for f.
This is the first known example of an explicit boundary of an instability zone with an irrational rotation number.

### Victor Bangert (Universität Freiburg)

Area Growth and Rigidity of Surfaces without Conjugate Points
For complete Riemannian manifolds M the condition to have no conjugate points is equivalent to the following strong minimality property of all of its geodesics: If c : R → M is a geodesic and if s < t ∈ R, then c|[s, t] is the (up to parametrization) unique shortest curve connecting c(s) and c(t) in the homotopy class of c|[s,t]. In particular, if the fundamental group of M is free abelian, this is the minimality property of orbits studied by J. Mather for much more general variational principles. Starting with E. Hopf’s result (1948) that every Riemannian 2-torus without conjugate points is flat, there is long list of rigidity results for Riemannian manifolds without conjugate points. We treat complete, 2-dimensional, Riemannian planes and cylinders without conjugate points. These were first studied by K. Burns and G. Knieper (1991). Since there are complete Riemannian metrics of negative Gaussian curvature on these manifolds one has to impose additional conditions in order to have a chance to prove flatness of metrics without conjugate points. A natural and weak invariant that distinguishes between flat and negatively curved metrics is the (asymptotic) area growth.
We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.
This is a joint work with P. Emmerich.

### Patrick Bernard (Université Paris-Dauphine)

Abstract diffusion mechanisms
I will discuss abstract mechanisms that allow to build wandering orbits in convex Hamiltonian systems. The first mechanism of that kind was introduced by John Mather in 93. In conjuction with averaging methods, these mechanisms have be used to prove the existence of Arnold diffusion in several classes of systems.

### Ugo Bessi  (Università degli Studi Roma Tre)

A version of the Vlasov equation with viscosity
The Vlasov equation studies the motion of a group of particles which are governed by an external potential and a mutual interaction. Recently, Gangbo, Tudorascu and coworkers have developed a version of Aubry-Mather theory which works for the Vlasov equation; on the other side, a few years ago D. Gomes added a viscosity term to Aubry-Mather theory. We shall try to see if the ideas of Gomes can be combined with those of Gangbo and Tudorascu to give a viscous version of Vlasov.

### Sergey Bolotin (University of Wisconsin at Madison)

Shadowing heteroclinic chains of a critical manifold.
We consider a  Hamiltonian system with a nondegenerate normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$.
We show that under certain conditions a chain of heteroclinic solutions corresponding to a hyperbolic orbit of the scattering map of $M$ can be shadowed by a solution on $H^{-1}(\eps)$ which passes $O(\sqrt\eps)$ close to $M$.  An applications to the Poincaré second species solutions of the 3 body problem will be given.

### Chong-Qing Cheng (Nanjing University)

Arnold diffusion in nearly integrable Hamiltonian systems

### Richard Moeckel (University of Minnesota)

Geometry of the reduced and regularized three-body problem
For the planar three-body problem, it is possible to combine a global symplectic reduction with Levi-Civita type regularization of all three binary collisions and blow-up of the triple collision singularity to obtain a complete flow.    Most of the ingredients in this procedure have been known for a long time, but I will describe a nice geometrical approach to the complete result, emphasizing the size-shape splitting of the configuration space.   The regularized and unregularized shape spaces are both two-spheres and are related by a beautiful four-to-one branched covering map first discovered by Lemaitre.

### Leonid Polterovich (University of Chicago)

Symplectic topology of the Poisson bracket
I shall discuss "hard" symplectic invariants coming from the Poisson bracket. Applications include function theory, Hamiltonian dynamics and quantum mechanics. In parts the talk is based on a joint work with Michael Entov and Lev Buhovsky.

### Tere Seara (Universitat Politècnica de Catalunya)

Oscilatory orbits in the circular restricted planar three body problem for big values of the Jacobi constant and any mass ratio.
In this work we consider the circular restricted three body problem. In this problem the Jacobi constant is a first integral. On the other hand, the restricted three body problem can be considered as a perturbation of the two body problem for small mass ratio, where the total energy is also a first integral.
In 1980 J. Llibre and C. Sim\'o \cite{LlibreS80} proved the existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits  which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they  had to assume that  the ratio  between the masses of the two primaries was exponentially small with respect to the angular momentum (or the Jacobi constant). In the present work, we generalize their work proving the existence of oscillatory motions for any value of the mass ratio.
In fact, we will show that, for big values of the Jacobi constant and for any mass ratio, there exist transversal intersections between the stable and unstable manifolds of infinity which guarantee the existence of a symbolic dynamics that  creates the so called oscillatory orbits. The main achievement is to rigorously prove the existence of these orbits without assuming the mass ratio small and therefore, this transversality  can not be checked by using classical perturbation theory respect to the mass ratio.The main point in our approach  is that, for big values of the Jacobi constant, the restricted three body problem is a fast and small perturbation of a two body problem. This makes the transversality of the invariant manifolds to be exponentially small, but non zero, for big Jacobi constant and any mass ratio.
This is a joint work with M. Guardia and P. Martin.

### Alfonso Sorrentino (University of Cambridge)

Rigidity of Birkhoff billiards
Despite being conceptually very simple, mathematical billiards presents a very rigid dynamics, which is completely determined by the geometry of the boundary and therein encoded. Trying to understand the extent of such rigidity and its implications to the dynamics is a formidable task, which lies behind many intriguing questions and conjectures.In this talk I shall discuss the following problem: given two strictly convex billiards whose maps are conjugate near the boundary (i.e. for small angles), how are their shapes related?  In a joint work with Vadim Kaloshin we prove that if the conjugacy is sufficiently smooth, then the two domains must be similar, i.e. they are the same up to a rescaling and an isometry.Our interest in this problem was arisen by a question posed by Guillemin and Melrose, which could be rephrased as follows: do the lengths of periodic orbits  characterize the shape of the billiard domain? Or in a more colourful way: can one "hear" the shape of a billiard? I shall describe how we plan to use our result to provide an affirmative answer to a version of this question (work in progress).

### Dmitry Treschev (Steklow Institute, Moscow)

Polymorphisms are multivalued (Lebesgue) measure preserving self-maps of the interval [0,1]. This class of dynamical systems was introduced by Vershik. Later polymorphisms appeared in the problem of destruction of an adiabatic invariant after crossing through a separatrix. We plan to discuss ergodic properties and typical singularities of "adiabatic" polymorphisms.

### Jean-Christophe Yoccoz (Collège de France)

Linearization of nonlinear perturbations of interval exchange maps
We present a smooth linearization result, joint work with S.Marmi and P.Moussa, for nonlinear perturbations of interval exchange maps or nonlinear perturbations of linear flows on translation surfaces. We discuss in particular the diophantine condition on the unperturbed map or flow, as well as some related open questions.

### Lai-Sang Young (Courant Institute, NYU)

Time-dependent billiards with slowly moving scatterers
We propose a model of Sinai billiards with moving scatterers, in which the locations of the scatterers may be shifted by small amounts between collisions. Our main result is the exponential loss of memory of initial data, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on convergence to quasi-stationary states for time-dependent dynamical systems with slowly varying coefficients.

### Ke Zhang (Institute for Advanced Studies, Princeton)

Arnold Diffusion via Invariant Cylinders and Mather Variational Method
The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian dynamics on a typical energy surface has a dense orbit. This question is wide open. In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian with degree of freedom n>2 and conjectured that this is a generic phenomenon, nowadays, called Arnold diffusion. A proof of Arnold's conjecture in two and half degrees of freedom has been announced by J. Mather in 2003. Based on two works: one joint with V. Kaloshin and P. Bernard, another with V. Kaloshin, we provide a proof for two and half degrees of freedom using an alternative approach. Our approach is based on constructing a net of normally hyperbolic invariant cylinders and a version of Mather variational method.
Honorary degree
On 2012,June 5th at 5:30 pm, John N. Mather was awarded an honorary degree from ENS de Lyon

Last update September 27, 2012 - Archived Oct 1, 2015
Ens de Lyon
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