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.

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.

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.

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.

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.

We will be interested in viscosity solutions of the evolution Hamilton-Jacobi equation \[\partial_t U+H(x,\partial_xU)=0.\] Here we think of the case where \(U:[0,+\infty[\times M\to {\bf R}\), where \(M\) is a manifold. If \(M\) is compact, as has been known for a long time, the maximum principle yields uniqueness for a given initial condition \(U|_{\{0\}\times M}\). This in turn implies the representation by a Lax-Oleinik type formula. When \(M\) is not compact, the global maximum principle does not immediately hold. Hitoshi Ishii and his coworkers obtained results about several years ago under some restrictions when \(M={\bf R}^n\). Basically the restrictions are about controlled growth at infinity. We will explain that under the hypothesis that \(H\) is Tonelli, all continuous solutions of the evolution Hamilton-Jacobi equation above satisfy the Lax-Oleinik representation even for non-compact \(M\). This of course will imply uniqueness for a given initial condition. Moreover, we will also show that if any pointwise finite \(U\) is given by the Lax-Oleinik representation, it is automatically continuous and therefore a viscosity solution.

Conformal symplectic dynamical systems on a symplectic manifold form a one-dimensional extension of symplectic systems, for which the symplectic form is transformed colinearly with itself. In this context, we examine how the isotropy of an invariant submanifold relates to the entropy of the dynamics it carries. Central to our study is an inequality which may be seen as a refinement of Yomdin's inequality. This is a joint work with Marie-Claude Arnaud.

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.

A global surface of section (GSS) reduces the study of a non-singular flow in 3D to that of a surface diffeomorphism. In this talk I will present some existence statements within the class of Reeb flows. Firstly, I will focus on the question of finding a GSS spanned by a given collection of periodic orbits, with no genericity assumptions, and state results obtained in collaboration with Salomão and Wysocki. These statements have applications in Celestial Mechanics. Secondly, I will present recent existence statements for rational GSS’s (Birkhoff sections) under genericity assumptions, and explain how to use them to prove that \(C^\infty\) generically a Reeb flow on a closed 3-manifold has positive topological entropy. This is fruit of joint work with Colin, Dehornoy and Rechtman, and is based on broken book decompositions.

Recently constructed counterexamples show that homogenization of viscous and non-viscous Hamilton-Jacobi equations in stationary ergodic random media can fail in dimensions 2 or higher if the momentum part of the Hamiltonian has a strict saddle point (while the Hamiltonian is super-linear as \(|p|\to\infty\)). It is expected that in one space dimension the non-convexity of the Hamiltonian should not be an obstacle to homogenization. We shall discuss a homogenization result for a class of one-dimensional viscous Hamilton-Jacobi equations. The Hamiltonian is assumed to be of the form \(G(p)+V(x,\omega)\), where \(G\) has the shape of a double well and \(V\) is bounded and Lipschitz. The talk is based on the joint work with Andrea Davini (Sapienza Università di Roma) and Atilla Yilmaz (Temple University).

This talk will focus on the dynamics of the positive energy levels of the Newtonian \(N\)-body problem. We will show that hyperbolic motions (i.e. motions of the form \(x(t)=ta+o(t)\), being \(a=(a_1,\dots,a_N)\) a configuration without collisions) define geodesic rays for the Jacobi-Maupertuis metric. We will show that the associated Busemann function of such a motion only depends, up to a constant, on the limit shape \(a\). These Busemann functions are global viscosity solutions of the Hamilton-Jacobi equation \(H(x,d_xu)=h\) where \(h>0\) is the energy level of the motion. We will use them to show the existence of hyperbolic motions \(x(t)\) for any choice of the constant energy \(h>0\), the limit shape \(a\), and the initial configuration \(x_0=x(0)\). Joint work with Andrea Venturelli.

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.

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.

Hamilton-Jacobi PDE associated with space-time stationary Hamiltonian functions are used to study stochastic growth phenomena in statistical mechanics, and turbulence in hydrodynamics. Lagrangian techniques in Aubry-Mather theory for action-minimizing trajectories, PDE techniques of weak KAM theory, and probabilistic methods related to first/last passage percolation problems have been employed to study long-time behavior of solutions. Most notably, a unique invariant measure has been constructed for any prescribed average velocity for some important examples of Hamiltonian functions. In this talk I will discuss a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such Gibbsian solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating particles.

We relate a time-dependent Hamilton-Jacobi equation posed on the n-dimensional torus to an optimal transport problem. The Hamiltonian satisfies rather general assumptions, and it is not Tonelli. We provide a simple proof of Kantorovich duality theorem and of a Brenier-Benamou type formula adapted to the framework. We further investigate the role of Mather measures in this context.

We will aim to mention some "weak KAM" ideas on non-compact problems and discuss them in two parts. In the first part, we intend to give a geometric and dynamical exposition of a remarkable example attributed to F. Almgren and explain its relevance in the global theory of geodesic flows and some global problems such as homogenization in quasi-periodic media. This is joint work with Rafael de la Llave. In the second part, we will introduce some special distance-like functions on a complete, connected, locally compact, non-compact geodesic space. We will show the existence of such functions and explore several properties of them. This is a joint work with Xiaojun Cui and Liang Jin.

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.

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).

It is well known that the viscous Hamilton-Jacobi equation on a compact domain converges exponentially fast to a stationary solution. (For example, Sinai proved this for a random potential on the torus in the late 80s). However, the a priori exponent decreases to 0 as the viscosity decreases to 0. On the other hand, in the zero viscosity case, the solution converges exponentially fast if the associated Lagrangian system admits a unique, hyperbolic minimizing orbit. We will show that for a generic "kicked" potential, the uniform convergence rate carry over to systems with small viscosity. This is a joint work with Konstantin Khanin and Lei Zhang.