International conference in Dynamical Systems and Geometry
in honor of Alain Chenciner
June 2023
Summaries of Lectures

Albouy, Alain

Les régularisations de Levi-Civita

Au cours d’une collision binaire du problème des $n$ corps, l’accélération gravitationnelle et la vitesse tendent vers l’infini. La régularisation de Levi-Civita transforme ce mouvement singulier en un mouvement régulier où toutes les variables restent finies, et qui continue après la collision. Les auteurs modernes citent à ce sujet un article de Levi-Civita publié en 1920 dans Acta Mathematica. Ils ajoutent que la régularisation décrite est limitée au problème plan, et qu’il a fallu attendre 1965 et la publication de Kustaanheimo et Stiefel pour avoir une régularisation dans l’espace. Ce n’est pas vrai. Levi-Civita régularise le problème dans l’espace. Sa régularisation plane date de 1904. Il explique en 1916, puis en 1920, qu’il a enfin trouvé une régularisation tridimensionnelle. Ses formules sont plus simples que celles de Kustaanheimo et Stiefel. Elles ont été ignorées par tous, à de notables exceptions près (à leur tour ignorées) : Wintner, dans son livre de 1941, §415, Siegel, dans son livre de 1956, §7, et Moser, dans son article de 1970, p. 615. La construction de Levi-Civita présente un défaut que nous allons expliquer et corriger. Pour motiver cette correction, nous posons la question : existe-t-il une régularisation qui rende le flot régulier au voisinage d’une collision binaire parabolique?

Je remercie Giovanni Gronchi de m’avoir informé du sujet de traité par Levi-Civita.

Bennequin, Daniel

Spaces of semantic informations. Examples and applications

In most of his works, Alain Chenciner, even using a lot of analysis and algebra, gave frequently the preference to geometrical arguments. He also manifested his particular interest for the Shannon’s theory of information. For instance he gave courses on Shannon’s theorems about entropy. Then we are happy to present to him, at the occasion of his (forthcoming) anniversary, the elements of a new theory of "semantic information", where the analog of entropy is given by spaces, that are obtained by purely geometrical constructions from the structures in the language. The starting point was the study of Carnap and Bar-Hillel in 1952, immediately after the discoveries of Shannon, looking for the forms of meaning in very simple artificial languages. However, these authors finally followed the statistical approach of Shannon. The new ingredient we use now, is the natural action of categories on these languages (equivalently, the fact that the languages in question belong to certain topos). This fact appeared recently in the study of the functioning of deep neural networks (Arxiv, 2022). For justifying our definitions and constructions, an application in psychology is presented, about the learning of classifications by concepts, where dynamical aspects appear.

(Joint work with J-C. Belfiore, Huawei, Paris)

Chaperon, Marc

Classification $C^r$ ($r\ge 0$) des germes de champs holomorphes : quelques remarques

Alain Chenciner m’ayant « commandé » un livre pour Springer, j’ai été amené à revenir sur un vieux travail, mon meilleur, effectué sous sa direction, et où l’on voit entre autres que le plus souvent les « petits dénominateurs » du théorème de linéarisation de Siegel n’ont guère de réalité géométrique. Le nouveau point de vue apporte un peu plus de symétrie, au prix de quelques problèmes supplémentaires.

Krikorian, Raphaël

Locally integrable perturbations of real analytic conservative diffeomorphisms

I shall consider the following problem. Let $f$ be a real analytic conservative (for us this will mean symplectic or having the intersection property) diffeomorphism of $(R^2,0)$ admitting the origin as an elliptic fixed point. Is it accumulated in the real analytic topology by real analytic conservative diffeomorphisms that are locally integrable at the origin i.e. integrable on some (small) neighborhoods of the origin (how small depending on the perturbation)? Our main result is that if $f$ satisfies a twist condition this is possible. The proofs (which are different in the symplectic and the intersection property cases) rely on a holomorphic KAM theorem, sheaf theoretic flavored Nash-Moser theory and arguments of deformations of integrable complex structures.

Laskar, Jacques

Chaotic diffusion in the solar system

Laudenbach, François

About ${\mathrm{\Gamma }}_4=0$

This cryptic title names a famous theorem proved by Jean Cerf in the late sixties. It states that every diffeomorphism of the 3-dimensional sphere extends to the 4-ball. And hence, there are no twisted 4-spheres.

The actual 1968 theorem of Cerf states that every orientation preserving diffeomorphism of the 3-sphere is isotopic to the identity, and hence ${\mathrm{\Gamma }}_4=0$ as an immediate corollary.

In his 1992 article to the memory of Claude Godbillon and Jean Martinet, Yakov Eliashberg gave a proof of ${\mathrm{\Gamma }}_4=0$ using the tools of the time in contact geometry and pseudo holomorphic curves, without passing by ${\pi }_0({\mathrm{Diff}}_{+}{S}^3)=0$.

In this talk, I would like to explain the proof that I recently wrote of Cerf’s 1968 theorem. This immediately reduces to an isotopy theorem for foliations of $S^2\times [0,1]$ tangent to the boundary. Surprisingly enough, this geometric setup is easy to deal with. The key relies on a convenient series of Dehn’s surgeries which kill all obstructions without changing the initial isotopy problem.

Levi, Mark

Sharp Arnold tongues and fragile Frenkel-Kontorova equilibria

Back in 1983 Arnold pointed out an interesting effect: if the perturbation terms in a circle map is a trigonometric polynomial, then Arnold tongues are exponentially sharp (in terms of the period of the resonance). And the lower is the power of the polynomial, the sharper are the tongues. Arnold also rediscovered a similar effect for instability zones of Mathieu-type equations, an effect originally discovered by D. Levi and J. B. Keller. I will describe another result of similar flavor for area-preserving cylinder maps, also pointing out some perhaps unexpected implications for the Frenkel-Kontorova model. This talk is based on joint work with Jing Zhou.

Marco, Jean-Pierre

Recent results on the sources of the polynomial entropy

We will first review the notion of polynomial entropy with some applications to Hamiltonian dynamics and Riemannian geometry. We will then examine two "sources" for the polynomial entropy, namely the notion of one-way horseshoe introduced by S. Roth, Z. Roth and L. Snoha (which can be seen as a counterpart of the usual Smale horseshoes for the topological entropy), and the torsion function for fibered systems (introduced in a joint work with Flavien Grycan), which provides a new conjugacy invariant. Finally, time-permitting, we will discuss some new applications of the torsion to dynamical versions of the Birkhoff conjecture for billiards.

Massetti, Jessica

Long time behavior of Sobolev norms: normal form and energy methods

We discuss the problem of long time behavior of solutions of two given PDEs defined on a compact manifold. This talk will be twofold.

On the one hand, I shall discuss exponential type stability times in the degenerate context of the beam equation with mass in 1 space dimension. A key ingredient is a suitable Diophantine condition (weaker than the original one proposed by Bourgain) that enables one to perform a Birkhoff Normal Form procedure.

On the other hand, with the aim of relaxing the requirement on the size/regularity of initial data arising from the BNF, we discuss a different approach on the completely resonant NLS on tori. A key ingredient is some energy method based on paradifferential calculus and suitable tame estimates. The control over finite but long times on high Sobolev norms requires only conditions on the low ones. This discussion is based on recent results in collaboration with Roberto Feola.

Ortega, Rafael

Stable fixed points of area preserving maps

Some known results on the topological properties of stable fixed points in the plane will be reviewed. They are related to questions of the type: are stable fixed points always isolated? can we compute their fixed point index? Several consequences for Hamiltonian systems in two degrees of freedom can be deduced. For example, stable closed orbits always appear in families.

Sauzin, David

Divergence of geometric normalizations for an elliptic fixed point in the plane (joint work with Alain Chenciner and Qiaoling Wei)

Classically, for a local analytic diffeomorphism $F$ of $(R^2,0)$ with a non-resonant elliptic fixed point (eigenvalues $\exp (\pm 2\pi i\omega )$ with $\omega$ real irrational), one can find formal normalizations, i.e. formal conjugacies to a formal diffeomorphism invariant under the group of rotations. Less demanding is the notion of a "geometric normalization" that we introduce: this is a formal conjugacy to a formal diffeomorphism which maps any circle centered at $0$ to a circle centered at $0$. Geometric normalizations are not unique, but they correspond in a natural way to a unique formal invariant foliation (any leaf is mapped to a leaf by $F$). Suppose that $\omega$ is super-Liouville. We then show that, generically, all geometric normalizations are divergent, so there is no analytic invariant foliation. This is a sequel—or rather a prequel—to [A. Chenciner, D. Sauzin, S. Sun & Q. Wei: Elliptic fixed points with an invariant foliation: Some facts and more questions, RCD 2022, Vol. 27], where we had considered the exceptional situation where $F$ leaves invariant an analytic foliation and then shown the generic divergence of formal normalizations.

Tokieda, Tadashi

PDF/T2.pdf

Venturelli, Andrea

Buseman functions and hyperbolic motions in the Newtonian N-body problem

Given a non-collision configuration $a$ on the N-body problem, we construct a special viscosity solution $b$ of the Hamilton-Jacobi equation so that all calibrated curves of $b$ are hyperbolic motions (as defined by J. Chazy) with limit shape given by $a$. Moreover, we show that such a viscosity solution is unique up to an additive constant. This results implies that for any initial configuration $x_0$ there is a hyperbolic motion $x$ starting at $x_0$ at time $t=0$ and with limit shape given by $a$, and moreover, this hyperbolic motion is unique for almost every initial configuration $x_0$.

Joint work with Ezequiel Maderna.

Zhao, Lei

Mechanical transformations and integrable mechanical billiards

In this talk, I shall discuss several integrable billiards with Hooke, Kepler, two-center and Lagrange potentials on constant curvature spaces and their links via conformal and projective transformations. The talk is based on several joint works with A. Takeuchi (Augsburg).