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10-21 juin 2024 PARIS (France)
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RésumésMINI-COURS
Consider an analytic symplectic map of the 2-dimensional annulus that has an invariant curve on which the dynamic is quasi-periodic (conjugated to an irrational rotation). Then, generically, the invariant curve is accumulated by other invariant analytic quasi-periodic curves ("KAM") but also by transverse homoclinic orbits associated to hyperbolic periodic orbits ("instability").
Plan of the lectures:
1- Introduction
2- Birkhoff normal form
3- Existence of invariant tori
4- Existence of hyperbolic periodic orbits
5- Existence of (transverse) homoclinic orbits
Luigi Chierchia Plan du cours: Kolmogorov’s Theorems on Hamiltonian dynamics [1954];
The non-torus set; optimal estimates for primary tori [1981].
Singular KAM theory: (1) the non- perturbative set and secondary integrable structures; (2) uniformization lemma; (3) the complex singularities of action-angle variables; (4) the singular twist theorem [2024].
References for the first part (Kolmogorov's 1954 Theorems):
(Kolmogorov's Theorem 1, 1954)
(Kolmogorov's Theorem 2, 1954; see, in particular, Sect 2)
References for Singular KAM Theory:
(KAM constants for primary tori)
(Averaging theory with minimal analyticity loss)
(Complex analytic properties of action-angle variables)
(Uniformization lemma for natural Hamiltonian systems at simple resonances)
(Singular KAM Theory: main results)
Danijela Damjanovic and Bassam Fayad
KAM techniques can be used to study the local rigidity of higher rank actions. We will present the case of affine $\Z^2$-actions on the torus.
We will discuss three manifestations of local rigidity in this context : 1) KAM-rigidity of simultaneously Diophantine torus translations 2) Smooth rigidity of hyperbolic or partially hyperbolic higher rank actions. 3) KAM rigidity of some parabolic actions that have all their rank one factors being Diophantine translations. Raphaël Krikorian
Michela Procesi I shall discuss the persistence of invariant tori for close-to-integrable Hamiltonian systems with a focus on Nonlinear Hamiltonian PDEs on the circle and particularly on the NonLinear Schrodinger equation with a convolution potential.
My main purpose is to show that for ``many'' potentials, ``many'' choices of Gevrey initial data give rise to almost-periodic solutions.
I shall first motivate the problem, describe relevant literature and give an idea of the main difficulties. In particular I shall give an operative definition of almost-periodic solution via the construction of infinite dimensional invariant tori.
Then I shall explain the scheme of the proof in simplified cases, explaining in broad terms the main tools such as the Moser counterterm theorem, the role of diophantine conditions and resummation techniques.
Finally I will give an overview of the proof focusing on the general strategy and highlighting open problems and further perspectives.
Masha Saprykina
Approximation-by-conjugation method (AbC) was introduced by D. V. Anosov and A. B. Katok in the 70-th.
By now it has become one of the main tools in constructing examples of smooth dynamics with prescribed chaotic properties. In this course we will present the general method and discuss several useful ideas for its application.
In the .pdf below you will find a more detailed plan and a list of papers that will be referred to during the lectures.
You can also find my notes for the first lecture. I plan to upload the notes for each lecture. A short plan of the course to be found here; Hand-written notes for Lecture 1 are here. Here are the notes for lecture 4.
************************************************ EXPOSÉS
Dario Bambusi A Nekhoroshev theorem for some (smoothing) perturbations of the Benjamin-Ono equation with initial data close to finite gap tori We consider the Benjamin Ono equation with periodic boundary conditions on a segment. We add a small Hamiltonian perturbation and consider the case where the corresponding Hamiltonian vector field is analytic as a map form energy space to itself. Let $\epsilon$ be the size of the perturbation. We prove that for initial data close in energy norm to an $N$-gap state of the unperturbed equation all the actions of the Benjamin Ono equation remain $\cO(\epsilon^{\frac{1}{2(N+1)}})$ close to their initial value for times exponentially long with $\epsilon^{-\frac{1}{2(N+1)}}$. The result is made possible by the use of Gerard-Kapeller's formulae for the Hamiltonian of the BO equation in Birkhoff variables. Joint work with Patrick Gerard
Pierre Berger Analytic pseudo-rotations on spheres, disks and annuli We introduce a way to perform the appproximation by conjugacy method of Anosov-Katok among surface analytic symplectomorphisms. This produces transitive analytic symplectomorphisms of the sphere, the disk and the cylinder with finite number of periodic points. This disproves a conjecture of Birkhoff (1941), and solves problems of Birkhoff (1927), Herman (1998), Fayad-Katok (2004) and Fayad-Krikorian (2018).
Massimiliano Berti Zoll Magnetic Flows on the Two-Torus: A Nash--Moser Construction
I will present a novel construction of infinite-dimensional family of smooth magnetic systems on the two-torus which are Zoll, meaning that all the unit-speed magnetic geodesics are periodic. The metric and the magnetic field of such systems are arbitrarily close to the flat metric and to a given constant magnetic field. This is the first existence result of this kind, which extends to the magnetic setting a famous result by Guillemin which holds for the geodesics flow on the two-sphere. The proof is based on a Nash-Moser implicit function theorem. This is a joint work with L. Asselle and G. Benedetti.
Spencer Durham Cohomological equations over parabolic base
A crucial step in a KAM scheme is solving a cohomological equation coming from the dynamics. We will review the cohomological equation and it's connection to the conjugacy problem. We will show how to solve the cohomological equation over a linear parabolic action and the limitations inherent to this process. Finally, we will see the construction of an example of precipitous drop of regularity in the conjugacy problem. This is joint work with Bassam Fayad.
David Fisher Finiteness of totally geodesic hypersurfaces in variable negative curvature When can a negatively curved manifold admit infinitely many totally geodesic submanifolds of dimension at least two?
I will explain some motivations for this question coming from different parts of mathematics. I will also explain a proof of the fact that a compact manifold with a real-analytic negatively curved metric admits only finitely many totally geodesic hypersurfaces, unless it is a hyperbolic manifold. And also state a more general conjecture. This is joint work with Simion Filip and Ben Lowe.
Lingrui Ge Sharp arithmetic spectral results for the type I operator
We will talk about sharp phase transition in frequency for the type I operators, which implies the robustness of the Aubry-Andre-Jitomirskaya phase transition conjecture. This is a joint work with Svetlana Jitomirskaya.
Benoît Grébert Existence of infinite dimensional invariant tori for analytic nonlinear Schrödinger equations without external parameters
We consider the nonlinear Schrödinger equation on the one dimensional torus without external parameters
$$i\partial_t u = \partial_x^2 u + f(|u|^2)u, \quad x\in \mathbb{T}$$
where $f$ an analytic function satisfying $f'(0) \neq 0$. We prove the existence of a large family of infinite dimensional KAM invariant tori.
In particular we prove that most of the Kuksin-Pöschel’s finite dimensional invariant tori are accumulated by infinite dimensional invariant tori. This result is obtained by exploiting the regularisation effects generated by the dispersion, which then allows us to implement an iterative scheme à la Pöschel.
In collaboration with Joackim Bernier and Tristan Robert
Marcel Guardia Transfer of energy in Hamiltonian systems on infinite lattices
In this talk I will explain a recent result about the existence of transfer of energy orbits in a chain of infinitely many weakly coupled pendulums. This is a Hamiltonian system posed on an infinite lattice with formal or convergent Hamiltonian. We develop geometric and functional tools to perform an Arnold diffusion mechanism in infinite dimensional phase spaces. In this way we construct solutions that move the energy of the pendulums along any prescribed path in the lattice. This is a joint work with Filippo Giuliani. Svetlana Jitomirskaya Dual Lyapunov exponents and the robust ten martini problem
The Hofstadter butterfly, a plot of the band spectra of almost Mathieu operators at rational frequencies, has become a pictorial symbol of the field of quasiperiodic operators and has gained renewed prominence through experimental study of moire materials. It is visually clear from this plot that for all irrational frequencies the spectrum must be a Cantor set, a statement that has been dubbed the ten martini problem. It has been established for the almost Mathieu operators, exploiiting various special features of this family. We will discuss a recently developed robust method allowing to establish it for a large class of one-frequency quasiperiodic operators, including nonperturbative analytic neighborhoods of several popular explicit families. The proof builds on the recently developed concept of dual Lyapunov exponents and partial hyperbolicity of the dual cocycles . Based on joint papers with L. Ge, J. You, and Q. Zhou.
Konstantin Khanin Obstructions to higher smoothness rigidity for circle maps with singularities
I will discuss the problem of smooth rigidity for circle maps with singularities. While in the KAM setting C^1-rigidity implies higher smoothness, it is generally not true for the maps with critical behavior. I will explain why C^2 rigidity should not hold generically in the case of critical circle maps and present rigorous result in this direction for circle maps with breaks.
The talk is based on a joint work with N. Goncharuk and Yu. Kudryashov.
Philipp Kunde (anti-)classification results in Dynamical Systems and Ergodic Theory Jessica Massetti On linearization of infinite dimensional vector fields
Given an infinite dimensional vector field $X = \lambda_j x_j (\partial / \partial x^j)$ with $j\in {\bf Z}$, $\lambda_j \in {\bf C}$, where the frequencies $\lambda_j$ may satisfy infinitely many resonant relations, we discuss the linearization of holomorphic perturbations $Y = X + P$ and show that $Y$ can be put in some appropriate normal form such that, when restricted to the resonant manifold, the flow is linear with the same characteristic exponents $\lambda_j$. This is a joint work with M. Procesi and L. Stolovitch.
Carlos Matheus Elliptic dynamics on certain relative character varieties
In this talk, we shall discuss the dynamics of the action of a hyperbolic element of SL(2,Z) on certain levels of the SU(2) and SU(3) character varieties of once-punctured torii. This is a joint work with G. Forni, W. Goldman and S. Lawton.
Jaime Paradela Diaz Global instability in strongly degenerate real-analytic hamiltonians
Laurent Stolovitch Local rigidity of actions of isometries on compact Riemannian manifolds
In this joint work with Zhiyan Zhao (Nice), we consider perturbations of isometries of a compact Riemannian manifold $M$. We prove that, under some conditions, a finitely presented group of such small enough perturbations is analytically or smoothly conjugate on (analytic or smooth) $M$ to the same group of isometry it is a perturbation of.
The result generalizes the rigidity theorems of Arnold, Herman, Yoccoz, Moser, etc. about circle diffeomorphisms which are small perturbations of rotations. The proof relies on a ``Diophantine-like" condition, relating the actions of the isometry group and the eigenvalues of the Laplace-Beltrami operator. Frank Trujillo Inverse Problems in Analytic KAM Theory
According to classical KAM theory, a sufficiently small perturbation of a non-degenerate integrable Hamiltonian system admits a collection of invariant tori, whose restricted dynamics are conjugate to those of a rotation by a Diophantine vector. In this talk, we will discuss the following inverse problem: To what extent are the perturbed systems determined by their associated collections of invariant tori? We shall see that this collection completely characterizes the perturbed Hamiltonian, and show some of the dynamical implications on systems sharing large collections of invariant tori.
Dmitrii Turaev Averaging in slow-fast Hamiltonian systems with mixed fast dynamics We suggest that the non-ergodicity of the fast subsystem should enhance the equilibration in slow-fast Hamiltonian systems.
Zhenqi Wang Local rigidity of higher rank partially hyperbolic algebraic actions
We give a complete solution to the local classification program of higher rank partially hyperbolic algebraic actions. We show $C^\infty$ local rigidity of abelian ergodic algebraic actions for symmetric space examples, twisted symmetric space examples and automorphisms on nilmanifolds. The method is a combination of representation theory, harmonic analysis and a KAM iteration. The method does not require any specific information from representation theory. Jiangong You Discontinuity of Lyapunov exponent of quasiperiodic cocycles: smoothness vs arithmetic
Continuity of Lyapunov exponents of quasi-periodic cocycles depends sensitively on the smoothness of the cocycles and the arithmetic of the frequency. In general, smoothness supports continuity, while some kind of arithmetic property supports discontinuity. In this talk, we will discuss how smoothness and arithmetic influence the continuity of Lyapunov exponents, in particular, we will give examples of discontinuity of Lyapunov exponents in finite smooth, infinite smooth as well as the Gevrey spaces, which implies that the Gevrey space $G^2$ is the transition space for the continuity of Lyapunov exponents for a full measure set of frequency. The talk is based on recent joint work with Jinhao Liang and Kai Tao.
Qi Zhou Quantitative structured almost reducibility and its applications
We propose a method called quantitative structured almost reducibility for analytic quasiperiodic $SL(2,\R)$-cocycles, which allows us to deal with infinitely many normal frequency resonances. Also we give will its dynamical and spectral applications.
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