International conference in Hamiltonian Dynamical Systems
in honor of Jean-Pierre Marco
Virtually at Observatoire de Paris, June 7–10 2021
Summaries of Lectures

For invariant submanifolds of dynamics that decrease a symplectic form, I will discuss these two notions and their relations. This is a joint work with Jacques Fejoz.

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The focus will be on two examples related to Arnold’s family of circle diffeomorphisms.

Introduction, on J.-P. Marco

The dynamics of the Rydberg electron in a circularly polarized microwave field is described by a Hamiltonian with two degrees of freedom depending on a parameter , which for is just the classical Kepler problem. For , the Hamiltonian system has two equilibria: (center-saddle for all ) and (center-center for small and complex-saddle otherwise). Therefore, there is a family of Lyapunov periodic orbits emanating from that form a normally hyperbolic invariant manifold (NHIM). In this talk, we compute the primary transverse homoclinic orbits of this NHIM (and therefore the associated scattering maps) by combining Poincaré-Melnikov methods with numerical methods. It should be noted that the transversality of these homoclinic orbits is exponentially small in (as in the case of the libration point of the R3BP).

This is a joint work with Mercè Ollé and Juan Ramon Pacha, from U. Politècnica de Catalunya.

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We will give unified proofs for results of the people mentioned in the title.

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In a joint work with Umberto Hryniewicz, we give a numerical criterion for right-handedness of dynamically convex Reeb flows on . As an application, we find an explicit constant such that if a Riemannian metric on the 2-sphere is -pinched for some , then its geodesic flow lifts to a right-handed flow on . The notion of right-handedness, introduced by E. Ghys, has strong dynamical implications: in particular, any finite collection of periodic orbits of such a flow binds an open book whose pages are global surfaces of section.

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We give a characterization of homeomorphisms of a closed surface of genus with no wandering point that have finitely many periodic points. The main result is the fact that there exists an integer such that the periodic points of are fixed and is isotopic to the identity relative to its fixed point set. The emblematic way to construct such an example is to start with the time one map of a flow of minimal direction for a translation surface, to add finitely many stopping points and to lift this map to a finite covering. The main result in the proof is that every homeomorphism with no wandering point, isotopic to a Dehn twist map, has infinitely many periodic points. Such a result was known for a generic area preserving diffeomorphism in the isotopy class. To extend this result, obtained with Martin Sambarino, to the general case, one needs to introduce a “forcing lemma” , very similar to a forcing result obtained with Fabio Tal in the case of maps isotopic to the identity.

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Lorsque l’évolution de l’état d’un système physique est mathématiquement décrite par le flot d’un champ de vecteurs hamiltonien sur une variété symplectique, un état de Gibbs au sens usuel est un état statistique stationnaire, souvent considéré par les physiciens comme un état d’équilibre thermodynamique, construit au moyen du hamiltonien du système considéré. Ce hamiltonien est un moment de l’action hamiltonienne locale du groupe à un paramètre des translations du temps. Dans son livre publié en 1902 Elementary principles in statistical mechanics, developed with especial reference to the rational foundation of thermodynamics, le célèbre savant américain Josiah Willard Gibbs (1839–1903), fondateur de la Mécanique statistique, a considéré aussi des états de Gibbs plus généraux utilisant, outre le hamiltonien, des intégrales premières telles que le moment cinétique total du système (qui n’est autre que le moment de l’action hamiltonienne du groupe des rotations). Bien plus récemment, le scientifique français Jean-Marie Souriau (1922–2012) a systématiquement étudié, dans son livre Structure des systèmes dynamiques publié en 1969 et dans plusieurs articles, les états de Gibbs construits au moyen d’un moment de l’action hamiltonienne d’un groupe de Lie général sur une variété symplectique. Il a d’autre part, aussi souvent que possible, utilisé la variété symplectique des mouvements d’un système hamiltonien plutôt que son espace des phases, et a envisagé les applications de ces notions en Physique et en Cosmologie. En utilisant les documents [1, 2, 3], je présenterai dans mon exposé les propriétés de ces états de Gibbs généraux, j’en donnerai des exemples et j’envisagerai leurs applications posssibles.

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Références

[1] Marle, C.-M., On Gibbs states of mechanical systems with symmetries. Preprint, 59 pages, arXiv:2012.00582v2, https://arxiv.org/pdf/2012.00582

[2] Marle, C.-M., On Gibbs states of mechanical sysyems with symmetries. JGSP 57 (2020) 45–85.

[3] Marle, C.-M., Examples of Gibbs states of mechanical sysyems with symmetries. JGSP 58 (2020) 55–79.

The evolution with time of a physical system is often mathematically described by the flow of a Hamiltonian vector field on a symplectic manifold. In this framework, a Gibbs state in the usual sense is a statistical state built with the Hamiltonian of the system (which is the moment map of the local Hamiltonian action of the one-parameter group of translations in time). For physicists, Gibbs states are states of thermodynamic equilibrium. The famous American scientist Josiah Willard Gibbs (1839–1903), founder of Statistical Mechanics, considered in his book Elementary principles in statistical mechanics, developed with especial reference to the rational foundation of thermodynamics, published in 1902, more general Gibbs states. These Gibbs states involve, together with the Hamiltonian, other first integrals such as the total angular momentum of the system (which is a moment map of the Hamiltonian action of the group of rotations). Much more recently, the French scientist Jean-Marie Souriau (1922–2012) thoroughly discussed Gibbs states built with a moment map of the Hamiltonian action of a general Lie group on a symplectic manifold. He presented his results in his book Structure des systèmes dynamiques, published in 1969, and in several articles, and proposed several applications in Physics and in Cosmology. Moreover he often used the symplectic manifold of motions of a mechanical system rather than its phase space. Using results presented in my papers [1, 2, 3], I will present in my talk the properties of these general Gibbs states, give several examples and discuss their possible applications in Physics.

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References

[1] Marle, C.-M., On Gibbs states of mechanical systems with symmetries. Preprint, 59 pages, arXiv:2012.00582v2, https://arxiv.org/pdf/2012.00582

[2] Marle, C.-M., On Gibbs states of mechanical sysyems with symmetries. JGSP 57 (2020) 45–85.

[3] Marle, C.-M., Examples of Gibbs states of mechanical sysyems with symmetries. JGSP 58 (2020) 55–79.

Existence of almost periodic solutions (i.e. solutions which are limit, in the uniform topology in time, of quasi-periodic functions) for evolution PDEs is a tough problem, with a lot of open questions. Very few results are known on this topic and most of them deal with the construction of very regular solutions for semilinear parameter dependent PDEs (mainly the Nonlinear Schrödinger (NLS) equation). In this talk I shall discuss some recent results on the existence of finite regularity solutions of this kind for the translation invariant NLS. Actually our solutions tipically solve the equation only in a weak sense, which constitutes, as far as we know, the first result on weak almost-periodic or quasi-periodic solutions for a PDE.

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The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier-Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, in the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Etnyre and Ghrist more than two decades ago.

Time permitting, we end up this talk using the contact mirror to address an apparently different question: What kind of physics might be non-computational? The universality result above yields the Turing completeness of the steady Euler flows on a 17-dimensional sphere. But, can this result be improved? We sketch lower dimensional constructions and relate them to polygonal billiards. This talk is based on several works with R. Cardona and D. Peralta-Salas (some of them with F. Presas).

The construction of Gevrey examples of unstable Hamiltonian or exact-symplectic systems is a line of research that had started in collaboration with Michel Herman and Jean-Pierre Marco in 1999. Recently, with Bassam Fayad and Jean-Pierre Marco [Asterique 416 (2020), 321-340], we have constructed the first example of a symplectic diffeomorphism of that has a non-resonant elliptic fixed point and a non-trivial orbit converging to it in forward time. By reversing time, we get a Lyapunov unstable fixed point. This is another example of a Gevrey non-analytic unstable system.

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Over the last years there has been an increasing interest in the study of the Hamilton–Jacobi Equation on networks and related questions. These problems, in fact, involve a number of subtle theoretical issues and have a great impact in the applications in various fields, for example to data transmission, traffic management problems, etc… While locally — i.e., on each branch of the network (arcs) —, the study reduces to the analysis of 1-dimensional problems, the main difficulties arise in matching together the information converging at the juncture of two or more arcs, and relating the local analysis at a juncture with the global structure/topology of the network.

In this talk I shall discuss several results related to the global analysis of this problem, obtained in collaboration with Antonio Siconolfi (Univ. of Rome La Sapienza); more specifically, we developed analogues of the so-called Weak KAM theory and Aubry–Mather theory in this setting. The salient point of our approach is to associate to the network an abstract graph, encoding all of the information on the complexity of the network, and to relate the differential equation to a discrete functional equation on the graph.

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