Raphaël KRIKORIAN - Academia.edu (original) (raw)

Papers by Raphaël KRIKORIAN

arXiv (Cornell University), 2013

We develop a "local theory" of multidimensional quasiperiodic SL(2,R)\SL(2,\R)SL(2,R) cocycles which are not ... more We develop a "local theory" of multidimensional quasiperiodic SL(2,R)\SL(2,\R)SL(2,R) cocycles which are not homotopic to a constant. It describes a C1C^1C1-open neighborhood of cocycles of rotations and applies irrespective of arithmetic conditions on the frequency, being much more robust than the local theory of SL(2,R)\SL(2,\R)SL(2,R) cocycles homotopic to a constant. Our analysis is centered around the notion of monotonicity with respect to some dynamical variable. For such {\it monotonic cocycles}, we obtain a sharp rigidity result, minimality of the projective action, typical nonuniform hyperbolicity, and a surprising result of smoothness of the Lyapunov exponent (while no better than Hölder can be obtained in the case of cocycles homotopic to a constant, and only under arithmetic restrictions). Our work is based on complexification ideas, extended "à la Lyubich" to the smooth setting (through the use of asymptotically holomorphic extensions). We also develop a counterpart of this theory centered around the notion of monotonicity with respect to a parameter variable, which applies to the analysis of SL(2,R)\SL(2,\R)SL(2,R) cocycles over more general dynamical systems and generalizes key aspects of Kotani Theory. We conclude with a more detailed discussion of one-dimensional monotonic cocycles, for which results about rigidity and typical nonuniform hyperbolicity can be globalized using a new result about convergence of renormalization.

HAL (Le Centre pour la Communication Scientifique Directe), 2008

In this note we investigate the exponential growth of products of two matrices A, B ∈ SL(2, R). W... more In this note we investigate the exponential growth of products of two matrices A, B ∈ SL(2, R). We prove, assuming A is a fixed hyperbolic matrix, that for Lebesgue almost every B, products of length n involving less than n α , 0 ≤ α < 1/2 matrices B are uniformly bounded from below by γ n for some γ > 1.

De nationalite franco-bresilienne, Artur Avila est ne en 1979 a Rio de Janeiro. Sous la direction... more De nationalite franco-bresilienne, Artur Avila est ne en 1979 a Rio de Janeiro. Sous la direction de Welington de Melo, il y a prepare sa these, a l’IMPA, et l’a soutenue en 2001. Il a ensuite fait un post-doctorat (2001-03) au College de France avec Jean-Christophe Yoccoz. Il a ete recrute en 2003 comme Charge de recherche au CNRS (LPMA, universite Pierre et Marie Curie). Il est depuis 2009 Directeur de recherche au CNRS a l’IMJ, universite Denis Diderot et detenteur de la chaire Arminio Fraga a l’IMPA. Artur Avila a recu la medaille Fields a l’ICM de Seoul 2014.

HAL (Le Centre pour la Communication Scientifique Directe), Oct 1, 2014

BiographieNational audienceDe nationalité franco-brésilienne, Artur Avila est né en 1979 à Rio de... more BiographieNational audienceDe 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

arXiv (Cornell University), Sep 21, 2021

Let f be a smooth symplectic diffeomorphism of R 2 admitting a (non-split) separatrix associated ... more Let f be a smooth symplectic diffeomorphism of 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.

HAL (Le Centre pour la Communication Scientifique Directe), 2003

arXiv (Cornell University), Jun 26, 2003

We show that for almost every frequency α ∈ R\Q, for every C ω potential v : R/Z → R, and for alm... more We show that for almost every frequency α ∈ R\Q, for every C ω potential v : R/Z → R, and for almost every energy E the corresponding quasiperiodic Schrödinger 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 Schrödinger operator, and allows us to complete the proof of the Aubry-André conjecture on the measure of the spectrum of the Almost Mathieu Operator.

arXiv (Cornell University), Jun 3, 2019

It is well known that a real analytic symplectic diffeomorphism of the 2d-dimensional disk (d ě 1... more It is well known that a real analytic symplectic diffeomorphism of the 2d-dimensional disk (d ě 1) admitting the origin as a non-resonant elliptic fixed can be formally conjugated to its Birkhoff Normal Form, a formal power series defining a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Perez-Marco's theorem [34] is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when d " 1 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real-analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus.

The Annals of Mathematics, 2001

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... more 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.

Ergod Theor Dyn Syst, 1999

We develop a new KAM scheme that applies to SL(2, R) cocycles with one frequency, irrespective of... more We develop a new KAM scheme that applies to SL(2, 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.

Proceedings of the International Congress of Mathematicians (ICM 2018), 2019

We propose in these notes a list of some old and new questions related to quasi-periodic dynamics... more 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 som...

Non UBCUnreviewedAuthor affiliation: Université Pierre et Marie CurieFacult

Astérisque, 2004

Etant donnee une fonction reguliere de moyenne nulle sur le tore de dimension 2, il est facile de... more Etant donnee une fonction reguliere de moyenne nulle sur le tore de dimension 2, il est facile de voir que ses integrales ergodiques au-dessus d'un flot de translation « generique » sont bornees. Il y a une dizaine d'annees, A. Zorich a observe numeriquement une croissance en puissance du temps de ces integrales ergodiques au-dessus de flots d'Hamiltoniens (non-exacts) « generiques » sur des surfaces de genre superieur ou egal a 2, et Kontsevich et Zorich ont propose une explication (conjecturelle) de ce phenomene par l'analyse du flot de Teichimiller sur l'espace des modules des differentielles abeliennes. Le but de l'expose est de presenter quelques idees de la preuve que G. Forni a donnee de la conjecture de Kontsevich et Zorich.

Ergodic Theory and Dynamical Systems, 2021

Let f be a smooth symplectic diffeomorphism of mathbbR2{\mathbb R}^2mathbbR2 admitting a (non-split) separatrix... more Let f be a smooth symplectic diffeomorphism of mathbbR2{\mathbb R}^2mathbbR2 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. However, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.