23 citations · 116 across the 28 of their papers we have counts for
32 papers
Positive measure of effective quasi-periodic motion near a Diophantine torus
Abed Bounemoura, Gerard Farré
It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus , invariant by a real-analytic Hamiltonian system, is always accumulated by…
Optimal linearization of vector fields on the torus in non-analytic Gevrey classes
Abed Bounemoura
We study linear and non-linear small divisors problems in analytic and non-analytic regularity. We observe that the Bruno arithmetic condition, which is usually attached to non-lin…
Some remarks on the Classical KAM Theorem, following P{ö}schel
Abed Bounemoura
We propose a slight correction and a slight improvement on the main result contained in "A lecture on Classical KAM Theorem" by J. P{ö}schel.
Reducibility of ultra-differentiable quasi-periodic cocycles under an adapted arithmetic condition
Abed Bounemoura, Claire Chavaudret, Shuqing Liang
We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which…
Positive measure of KAM tori for finitely differentiable Hamiltonians
Abed Bounemoura
Consider an integer and real numbers and . Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which…
Some remarks on the optimality of the Bruno-R{ü}ssmann condition
Abed Bounemoura
We prove that the Bruno-R{ü}ssmann condition is optimal for the analytic preservation of a quasi-periodic invariant curve for an analytic twist map. The proof is based on Yoccoz's…
Hamiltonian perturbation theory for ultra-differentiable functions
Abed Bounemoura, Jacques Féjoz
Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well…
KAM, -Gevrey regularity and the -Bruno-Rüssmann condition
Abed Bounemoura, Jacques Féjoz
We prove a new invariant torus theorem, for -Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the -Bruno-Rüssmann condition, and which reduces…
Double exponential stability of quasi-periodic motion in Hamiltonian systems
Abed Bounemoura, Bassam Fayad, Laurent Niederman
We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is doubly exponentially stable in…
Generic perturbations of linear integrable Hamiltonian systems
Abed Bounemoura
In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invaria…
Double exponential stability for generic real-analytic elliptic equilibrium points
Abed Bounemoura, Bassam Fayad, Laurent Niederman
We consider the dynamics in a neighborhood of an elliptic equilibrium point with a Diophantine frequency of a symplectic real analytic vector field and we prove the following resul…
Nekhoroshev's estimates for quasi-periodic time-dependent perturbations
Abed Bounemoura
In this paper, we consider a Diophantine quasi-periodic time-dependent analytic perturbation of a convex integrable Hamiltonian system, and we prove a result of stability of the ac…
Ergodization time for linear flows on tori via geometry of numbers
Abed Bounemoura
In this paper, we give a new, short, simple and geometric proof of the optimal er-godization time for linear flows on tori. This result was first obtained by Bourgain, Golse and We…
A note on micro-instability for Hamiltonian systems close to integrable
Abed Bounemoura, Vadim Kaloshin
In this note, we consider the dynamics associated to an epsilon-perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and…
Instability of resonant invariant tori in Hamiltonian systems
Abed Bounemoura
In this article, we consider the dynamics in a neighborhood of a quasi-periodic torus which is invariant by a Hamiltonian flow, we discuss several notions of stability and we prove…
Non-degenerate Liouville tori are KAM stable
Abed Bounemoura
In this short note, we prove that a quasi-periodic torus, with a non-resonant frequency (that can be Diophantine or Liouville) and which is invariant by a sufficiently regular Hami…
The classical KAM theorem for Hamiltonian systems via rational approximations
Abed Bounemoura, Stephane Fischler
In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-Rüssmann conditi…
Generic diffusion for a class of non-convex Hamiltonians with two degrees of freedom
Abed Bounemoura, Vadim Kaloshin
In this paper, we study small perturbations of a class of non-convex integrable Hamiltonians with two degrees of freedom, and we prove a result of diffusion for an open and dense s…
Effective stability for slow time-dependent near-integrable Hamiltonians and application
Abed Bounemoura
The aim of this note is to prove a result of effective stability for a non-autonomous perturbation of an integrable Hamiltonian system, provided that the perturbation depends slowl…
The KAM theorem through Dirichlet's box and Khintchine's transference principles
Abed Bounemoura
In this paper, we give a new proof of the classical KAM theorem which avoids small divisors and relies on two basic principles of Diophantine approximation: Dirichlet's box and Khi…
Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians
Abed Bounemoura
This paper is a sequel to "Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians", in which we gave a new construction of resonant normal forms with a…
Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians
Abed Bounemoura
In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based…
A Diophantine duality applied to the KAM and Nekhoroshev theorems
Abed Bounemoura, Stephane Fischler
In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a conse…
A geometrical proof of the persistence of normally hyperbolic submanifolds
Pierre Berger, Abed Bounemoura
We present a simple, computation free and geometrical proof of the following classical result: for a diffeomorphism of a manifold, any compact submanifold which is invariant and no…