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Let $f$ be a smooth symplectic diffeomorphism of $\mathbb{R}^2$ admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if $f$ is a perturbation of the time-1 map of a symplectic autonomous vector field, this separatrix is accumulated by a positive measure set of invariant circles. On the other hand, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles., Comment: 44 pages, 7 figures. To be appear in Ergodic Theory and Dynamical Systems (Katok Memorial Special Issue)

International audience; We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori.We show that a fixed point with Diophantine frequency vector omega (0) is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Russmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that omega (0) has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency omega (0).This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.

We develop a new KAM scheme that applies to SL(2, \({{\mathbb R}}\)) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg–Sinai’s theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

Nous presentons une preuve du dernier theoreme geometrique d'Herman qui affirme que, si un diffeomorphisme F de l'anneau possede la propriete d'intersection, alors toute courbe C°° F-invariante, sur laquelle le nombre de rotation de F est diophantien, est accumulee par un ensemble de mesure positive de courbes invariantes C°° sur lesquelles F est C°° -conjuguee a une rotation. Ceci implique en particulier la stabilite des points fixes elliptiques diophantiens des diffeomorphismes du plan qui preservent l'aire. Le caractere remarquable de ce theoreme est qu'il ne requiert aucune condition de torsion.

In this paper we introduce a new technique that allows us to investigate reducibility properties of smooth SL(2, R)-cocycles over irrational rotations of the circle beyond the usual Diophantine conditions on these rotations. For any given irrational angle on the base, we show that if the cocycle has bounded fibered products and if its fibered rotation number belongs to a set of full measure $\Sigma(\a)$, then the matrix map can be perturbed in the $C^\infty$ topology to yield a $C^\infty$-reducible cocycle. Moreover, the cocycle itself is almost rotations-reducible in the sense that it can be conjugated arbitrarily close to a cocycle of rotations. If the rotation on the circle is of super-Liouville type, the same results hold if instead of having bounded products we only assume that the cocycle is $L^2$-conjugate to a cocycle of rotations. When the base rotation is Diophantine, we show that if the cocycle is $L^2$-conjugate to a cocycle of rotations and if its fibered rotation number belongs to a set of full measure, then it is $C^\infty$-reducible. This extends a result proven in [5]. As an application, given any smooth SL(2, R)-cocycle over a irrational rotation of the circle, we show that it is possible to perturb the matrix map in the $C^\infty$ topology in such a way that the upper Lyapunov exponent becomes strictly positive. The latter result is generalized, based on different techniques, by Avila in [1] to quasiperiodic SL(2, R)-cocycles over higher-dimensional tori. Also, in the course of the paper we give a quantitative version of a theorem by L. H. Eliasson, a proof of which is given in the Appendix. This motivates the introduction of a quite general KAM scheme allowing to treat bigger losses of derivatives for which we prove convergence.

In this paper we investigate the exponential growth of products of two matrices . We prove, assuming A is a fixed hyperbolic matrix, that for Lebesgue almost every B, products of length n involving less than nα, 0 ≤ α 1.

We study the accumulation of an invariant quasi-periodic torus of a Hamiltonian flow by other quasi-periodic invariant tori. ¶ We show that an analytic invariant torus ${\mathcal{T}}_{0}$ with Diophantine frequency $\omega _{0}$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at ${\mathcal{T}}_{0}$ satisfies a Rüssmann transversality condition, the torus ${\mathcal{T}}_{0}$ is accumulated by Kolmogorov–Arnold–Moser (KAM) tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least $d+1$ that is foliated by analytic invariant tori with frequency $\omega _{0}$ . ¶ For frequency vectors $\omega _{0}$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian $H$ satisfies a Kolmogorov nondegeneracy condition at ${\mathcal{T}}_{0}$ , then ${\mathcal{T}}_{0}$ is accumulated by KAM tori of positive total measure. ¶ In four degrees of freedom or more, we construct for any $\omega _{0}\in{\mathbb{R}}^{d}$ , $C^{\infty}$ (Gevrey) Hamiltonians $H$ with a smooth invariant torus ${\mathcal{T}}_{0}$ with frequency $\omega _{0}$ that is not accumulated by a positive measure of invariant tori.

We show that for almost every frequency ?? ?? R\Q, for every C?O potential v : R/Z ?? R, and for almost every energy E the corresponding quasiperiodic Schr?Nodinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schr?Nodinger operator, and allows us to complete the proof of the Aubry-Andr?Le conjecture on the measure of the spectrum of the Almost Mathieu Operator.

Biographie; National audience; De nationalité franco-brésilienne, Artur Avila est né en 1979 à Rio de Janeiro. Sous la direction de Welington de Melo, il y a préparé sa thèse, à l’IMPA, et l’a soutenue en 2001. Il a ensuite fait un post-doctorat (2001-03) au Collège de France avec Jean-Christophe Yoccoz. Il a été recruté en 2003 comme Chargé de recherche au CNRS (LPMA, université Pierre et Marie Curie). Il est depuis 2009 Directeur de recherche au CNRS à l’IMJ, université Denis Diderot et détenteur de la chaire Arminio Fraga à l’IMPA.Artur Avila a reçu la médaille Fields à l’ICM de Séoul 2014.

Let us be given a compact “semi-simple” Lie group G with Lie algebra g, a regular element A ϵ g, a bounded interval Λ ⊂ R and a diophantine vector ω ϵ Rd; then if F ϵ Cω(Rd/Zd,g) is small enough, ω meaning here “real analytic”, for Lebesgue-a.e. λ ϵ Λ, the quasi-periodic system λA + F( ω1 2π t,…, ωd 2π t) , with frequency vector ω, is Floquet-reducible modulo some finite covering depending only on the group G. This theorem is a generalization of the one proved in [5].

Nous prouvons que dans l'ensemble $SW^0({\bf T}^1,SU(2))$ des difféomorphismes de ${\bf T}^1\times SU(2)$ de la forme $(\theta,y)\to (\theta+\alpha,A(\theta).y)$, $A\in C^{\infty}({\bf T}^1,SU(2))$, $\alpha\in{\bf T}^1$, les systèmes $(\alpha,A)$ réductibles, c'est-à-dire ceux qui admettent une écriture de la forme $A(\theta)= B(\theta+\alpha)A_0B(\theta)^{-1}$, $B\in C^{\infty}({\bf T}^1,SU(2))$, $A_0\in SU(2)$, forment une partie dense pour la $C^0$-topologie. Ce théorème est à comparer à un résultat de Herman [7], qui dit qu'il en est de même des systèmes non réductibles. Ceci démontre (une forme d')une conjecture de Rychlik (cf. [14]).

In this note we study the linear part of nonlinear holomorphic systems admitting the origin as a local attractor, and we show that this linear part is asymptotically stable. This is a positive answer to a question raised by Dayawansa and Martin in 1989.

Let $R_1, R_2,\ldots,R_m$ be rotations generating ${\mathbb{SO}}_{d+1}$ , $d\geq 2$ , and let $f_1, f_2,\ldots,f_m$ be small smooth perturbations of them. We show that $\{f_\alpha\}$ can be linearized simultaneously if and only if the associated random walk has zero Lyapunov exponents. As a consequence, we obtain stable ergodicity of actions of random rotations in even dimensions

We prove that given a in a set of total (Haar) measure in T 1 = R/Z, the set of A ∈ C ∞ (T 1 , SU(2)) for which the skew-product system (α, A): T 1 × SU(2) → T 1 × SU(2), (α, A)(θ,y) = (θ + α, A(θ)y) is reducible - that is, A(.) = B(.+ α)A 0 B(.) - 1 , for some A 0 ∈ SU(2), B ∈ C ∞ (T 1 , SU(2)),- is dense for the C°°-topology.

We propose in these notes a list of some old and new questions related to quasi-periodic dynamics. A main aspect of quasi-periodic dynamics is the crucial influence of arithmetics on the dynamical features, with a strong duality in general between Diophantine and Liouville behavior. We will discuss rigidity and stability in Diophantine dynamics as well as their absence in Liouville ones. Beyond this classical dichotomy between the Diophantine and the Liouville worlds, we discuss some unified approaches and some phenomena that are valid in both worlds. Our focus is mainly on low dimensional dynamics such as circle diffeomorphisms, disc dynamics, quasi-periodic cocycles, or surface flows, as well as finite dimensional Hamiltonian systems. In an opposite direction, the study of the dynamical properties of some diagonal and unipotent actions on the space of lattices can be applied to arithmetics, namely to the theory of Diophantine approximations. We will mention in the last section some problems related to that topic. The field of quasi-periodic dynamics is very extensive and has a wide range of interactions with other mathematical domains. The list of questions we propose is naturally far from exhaustive and our choice was often motivated by our research involvements.

In this thesis we are interested in the Lyapunov exponent of ergodic Schrödinger cocycles. These cocycles occur in the analysis of solutions to the Schrödinger equation where the potential is defined with ergodic dynamics. We study two distinct aspects related to the the Lyapunov exponent for different kinds of dynamics. First we focus on a large deviation theorem for quasi-periodic cocycles and then for potentials defined by the doubling map. We prove that estimates of Bourgain & Goldstein type are granted if an upper estimate involved in the theorem is true on a strip of the complex plane. Then we establish a new technique to prove this upper bound in the quasi-periodic setting, based on subharmonic arguments suggested by Avila, Jitomirskaya & Sadel. We adapt afterwards the method to the doubling map and prove a large deviation theorem for the inverse branches of this dynamics. In the second part, we establish an asymptotic development similar to the results of Figotin & Pastur and Sadel & Schulz-Baldes for the Lyapunov exponent of Schrödinger cocycles at small coupling when the potential is a mixture of quasi-periodic and random. The analysis distinguishes the cases when the energy is either diophantine or resonant with respect to the frequency of the quasi-periodic part of the potential., L'objectif de cette thèse est de traiter de deux aspects différents de la théorie de l'exposant de Lyapunov de cocycles de Schrödinger définis par une dynamique ergodique. Dans la première partie, on s'intéresse aux estimées de grandes déviations de type Bourgain & Goldstein pour des cocycles quasi-périodiques, puis pour ceux définis par le doublement de l'angle. Après avoir montré que seule une estimée par dessus sur une bande complexe est nécessaire pour avoir la minoration, on redémontre cette inégalité pour une dynamique quasi-périodique en utilisant des techniques de mouvement brownien en lien avec des fonctions sous-harmoniques. Ensuite on adapte la méthode au cas du doublement de l'angle pour lequel on prouve des estimées de grandes déviations sur les branches inverses de cette dynamique. Dans la deuxième partie sont étudiés des cocycles de Schrödinger dont la dynamique est une somme de dynamiques quasi-périodique et aléatoire. On démontre que, dans le régime perturbatif, les développements asymptotiques de l'exposant de Lyapunov attaché à ces cocycles sont similaires à ceux déjà démontrés dans le cas aléatoire par Figotin & Pastur ou Sadel & Schulz-Baldes. L'analyse se fait en fonction du caractère diophantien ou résonant de l'énergie par rapport à la fréquence diophantienne de la partie quasi-périodique du potentiel.

This thesis addresses the rigorous derivation of mean-field results for the continuous time dynamics of heterogeneous large neural networks. In our models, we consider firing-rate neurons subject to additive noise. The network is fully connected, with highly random connectivity weights. Their variance scales as the inverse of the network size, and thus conserves a non-trivial role in the thermodynamic limit. Moreover, another heterogeneity is considered at the level of each neuron. It is interpreted as a spatial location. For biological relevance, a model considered includes delays, mean and variance of connections depending on the distance between cells. A second model considers interactions depending on the states of both neurons at play. This last case notably applies to Kuramoto's model of coupled oscillators. When the weights are independent Gaussian random variables, we show that the empirical measure of the neurons' states satisfies a large deviations principle, with a good rate function achieving its minimum at a unique probability measure, implying averaged convergence of the empirical measure and propagation of chaos. In certain cases, we also obtained quenched results. The limit is characterized through a complex non Markovian implicit equation in which the network interaction term is replaced by a non-local Gaussian process whose statistics depend on the solution over the whole neural field. We further demonstrate the universality of this limit, in the sense that neuronal networks with non-Gaussian interconnections but sub-Gaussian tails converge towards it. Moreover, we present a few numerical applications, and discuss possible perspectives., Cette thèse porte sur l'obtention rigoureuse de limites de champ moyen pour la dynamique continue de grands réseaux de neurones hétérogènes. Nous considérons des neurones à taux de décharge, et sujets à un bruit Brownien additif. Le réseau est entièrement connecté, avec des poids de connections dont la variance décroît comme l'inverse du nombre de neurones conservant un effet non trivial dans la limite thermodynamique. Un second type d'hétérogénéité, interprété comme une position spatiale, est considéré au niveau de chaque cellule. Pour la pertinence biologique, nos modèles incluent ou bien des délais, ainsi que des moyennes et variances de connections, dépendants de la distance entre les cellules, ou bien des synapses dépendantes de l'état des deux neurones post- et présynaptique. Ce dernier cas s'applique au modèle de Kuramoto pour les oscillateurs couplés. Quand les poids synaptiques sont Gaussiens et indépendants, nous prouvons un principe de grandes déviations pour la mesure empirique de l'état des neurones. La bonne fonction de taux associée atteint son minimum en une unique mesure de probabilité, impliquant convergence et propagation du chaos sous la loi "averaged". Dans certains cas, des résultats "quenched" sont obtenus. La limite est solution d'une équation implicite, non Markovienne, dans laquelle le terme d'interactions est remplacé par un processus Gaussien qui dépend de la loi de la solution du réseau entier. Une universalité de cette limite est prouvée, dans le cas de poids synaptiques non-Gaussiens avec queues sous-Gaussiennes. Enfin, quelques résultats numérique sur les réseau aléatoires sont présentés, et des perspectives discutées.

In this PhD thesis we study quasiperiodic cocycles in semi-simple compact Lie groups. For the greatest part of our study, we will focus ourselves to one-frequency cocyles. We will prove that $C^{\infty }$-reducible cocycles are dense in the $C^{\infty }$ topology, for a full measure set of frequencies. We will firstly define two invariants of the dynamics, which we will call energy and degree and which give a preliminary distinction between reducible and non-reducible cocycles. We will then take up the proof of the density theorem. We will show that an algorithm of renormalization converges to perturbations of simple models, indexed by the degree. Finally, we will analyse these perturbations using methods inspired by K.A.M. theory. In this context we will prove that if a $C^{\infty }$ cocycle is measurably reducible to a diophantine constant, it is actually $C^{\infty }$-reducible., Cette thèse porte sur l'étude des cocycles quasi-périodiques à valeurs dans des groupes de Lie compacts semi-simples. Nous nous restreindrons au cas des cocycles à une fréquence. Nous démontrons que, pour un ensemble de mesure de Lebesgue pleine de fréquences, l'ensemble des cocycles $C^{\infty }$ qui sont $C^{\infty }$-réductibles sont $C^{\infty }$-denses. Le premier pas sera l'obtention de deux invariants de la dynamique, qu'on appellera énergie et degré, qui distinguent en particulier les cocycles réductibles des cocycles non-réductibles. On entamera ensuite la démonstration du théorème principal. Nous démontrons dans un second temps qu'un algorithme dit de renormalisation permet de ramener l'étude de tout cocycle à celle des perturbations de modèles simples indexés par le degré. Nous analysons ensuite ces perturbations par des méthodes inspirés de la théorie K.A.M.. En particulier, nous démontrons qu'un cocycle $C^{\infty }$ mesurablement réductible à une constante diophantienne est alors $C^{\infty }$-réductible.