Weak KAM theory - XXV+1 years later
Avignon Université, France
May 30 - June 3, 2022    (new dates)

Schedule

Monday, May 30th

Tuesday, May 31st

Wednesday, June 1st

Thursday, June 2nd

Friday, June 3rd

Abstracts

• Gabriele Benedetti - The dynamics of strong magnetic fields on surfaces: Periodic orbits, trapping regions, and a magnetic Bertrand theorem
In this talk, we study the motion of a charged particle on a closed surface under the effect of a strong magnetic field. By means of a Hamiltonian normal form, we will construct periodic orbits and trapping regions for the particle, and prove a Bertrand-type theorem on the rigidity of magnetic fields all of whose orbits are periodic. This is joint work with Luca Asselle.

• Patrick Bernard - Title
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• Piermarco Cannarsa - Dynamics and topology of singularities of solutions to Hamilton-Jacobi equations
The study of the structural properties of the set of points at which a solution u of a first order Hamilton-Jacobi equation fails to be differentiable has been investigated for a long time, starting fifty years ago with a seminal paper by W. Fleming. Later on, such a topic was put into the right perspective by the introduction of viscosity solutions by M.Crandall and P.-L. Lions. All these years have registered enormous progress in the comprehension of the way how singularities propagate, which is described, from the dynamical point of view, by generalised characteristics and, from the topological point of view, by intrinsic characteristics. In this talk, I will revisit some of the milestones of this theory and discuss recent results obtained in a joint work with Wei Cheng and Albert Fathi.

• Chong-Qing Cheng - Different ways to cross multiple resonance
The problem of Arnold diffusion is raised for nearly integrable Hamiltonian system with more than two degrees of freedom. As one of the key steps to construct diffusion, one has to deal with the problem of multiple resonance. In this talk, I shall show different ways to cross the multiple resonance.

• Wei Cheng - Commutators of Lax-Oleinik operators and applications
We analyse the commutators of Lax-Oleinik operators $$T^-_t\circ T^+_t$$ and $$T^+_t\circ T^-_t$$ which give rise to some new observation on both the weak KAM solutions and the structure of cut locus of weak KAM solutions. As an application we also give a simple and geometric proof of the propagations of singularities along generalized characterisitcs. We also remind some connections of such commutators to the Kantorovich dual in the theory of optimal transport in the context of weak KAM theory and Aubry-Mather theory initiated by Bernard and Buffoni. This is mainly based on our recent work with Jiahui Hong and Piermarco Cannarsa.

• Gonzalo Contreras - Homoclinics for geodesic flows of surfaces
We prove that the geodesic flow of a Kupka-Smale Riemannian metric on a closed surface has homoclinic orbits for all of its hyperbolic closed geodesics.

• Albert Fathi - Title
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• Jacques Fejoz - Title
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• Anna Florio - Measures with vanishing asymptotic Maslov index on cotangent bundles
In the Tonelli Hamiltonian flow setting, complete integrability (i.e., the existence of a foliation by invariant Lagrangian graphs) can be characterised by vanishing asymptotic Maslov index everywhere. Aubry-Mather theory assures the existence of minimizing measures, supported on "ghosts" of invariant submanifolds, which also have vanishing index. In a joint work with Marie-Claude Arnaud and Valentine Roos, we show the existence of invariant measures with vanishing asymptotic Maslov index in the more general setting of conformally symplectic flows on cotangent bundles, without any convexity hypothesis.

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• Vincent Humilière - Groups of area preserving homeomorphisms and their non-simplicity, XXV+XVII years later
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• Elena Kosygina - Title
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• Jessica E. Massetti - On the persistence of periodic tori for symplectic twist maps
Invariant tori that are foliated by periodic points are at the core of the fragility of integrable systems since they are somehow extremely easy to break, in counterposition to the generic robustness of the quasi-periodic ones considered by KAM theory. On the other hand, the investigation of rigidity of integrable twist maps, i.e. to understand to which extent it is possible to deform a map in a non-trivial way preserving some (or all) of its features, is related to important questions and conjectures in dynamics. In this talk I shall discuss the persistence of Lagrangian periodic tori for symplectic twist maps of the 2d-dimensional annulus and a rigidity property of completely integrable ones. This is based on a joint work with Marie-Claude Arnaud and Alfonso Sorrentino.

• Ana Rechtman - Broken books and Reeb dynamics in dimension 3
Giroux’s correspondence gives, in particular, for every contact structure on a closed 3-manifold an adapted open book decomposition. Hence, it exists a Reeb vector field that is tangent to the binding and transverse to the interior of the pages. For this vector field, each page is a Birkhoff section and the dynamics of its flow can be studied from the first return map to the Birkhoff section. An open question, is whether every Reeb vector field admits a Birkhoff section. In collaboration with V. Colin and P. Dehornoy, we proved that every non-degenerate Reeb vector field on a closed 3-manifold is adapted to a broken book decomposition (a generalisation of open book). This construction gives a system of transverse surfaces with boundary and allows to establish results on the dynamics of the vector field, for example, we proved that any non-degenerate Reeb vector field has either two or infinitely many periodic orbits.

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• Xifeng Su - Title
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• Philippe Thieullen - Zero temperature chaotic convergence of Gibbs measures in a two-dimensional lattice for a finite range potential
A finite range potential is an energy function at the zero site that depends on a finite number of nearby sites. A Gibbs measure is a probability measures that is associated to every finite range potential at a given temperature. The zero chaotic convergence is the phenomenon of non convergence of the Gibbs measures by letting the temperature goes to zero. The accumulation points may be interpreted as minimizing measures as in Aubry-Mather theory. It is known from Brémont that the phenomenon of chaotic convergence does not exist in 1d lattices. Following a previous result of Chazottes-Hochman, we prove that it is not any more true in dimension 2. This work has been done in collaboration with S. Barbieri, R. Bissacot, G. Dalle Vedove Nosaki.

• Maxime Zavidovique - Weak KAM solutions of conservative Twist maps of the annulus
On the occasion of this birthday, we will come back to the roots of Aubry-Mather and Weak KAM theory: exact conservative twist maps of the annulus. We will explain that there exists a continuous choice of weak KAM solutions with respect to the cohomology class. Moreover we will provide a rather precise description of how those weak KAM solutions can be constructed. More precisely, we will show that all twist maps resemble the classical example of the pendulum: at rational rotation numbers, all corresponding pseudo-graphs are included in 2 extremal pseudo-graphs (the separatrices in the case of the pendulum).

• Ke Zhang - Title
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