Temps d'instabilité pour les perturbations de systèmes intégrables analytiques

For a positive integer n and $R>0$, we set $B_R^n=\{x\in {\mathbb{R}}^n|6x{6}_{\infty }<R\}$. Given $n?4$ and $R>1$ we construct a sequence of analytic perturbations $(H_j)$ of the completely integrable Hamiltonian $?(r)=\frac{1}{2}(r_1^2+?+r_{n?1}^2)+r_n$ on ${\mathbb{T}}^n\times B_R^n$, with unstable orbits for which we can estimate the time of drift in the action space. These functions $H_j$ are analytic on a fixed complex neighborhood V of ${\mathbb{T}}^n\times B_R^n$, and if ${?}_j\text{:}=6H_j??6_{C^0(V)}$ the time of drift of these orbits is smaller than $\exp (c{(1/{?}_j)}^{1/2(n?3)})$ for a fixed constant $c>0$. Our unstable orbits pass close to a doubly resonant surface, so the result is almost optimal since the stability exponent for such orbits is $1/2(n?2)$. To cite this article: J.-P. Marco, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Un théorème sur les anneaux primaires et complètement intégralement clos

Sans résumé     

Temps d'instabilité des systèmes hamiltoniens initialement hyperboliques

We compute the time of drift along a transition chain for an initially hyperbolic Hamiltonian system. We prove that this time is polynomial with respect to the opposite of the perturbing parameter.     

Systèmes fortement hyperboliques et systèmes symétriques

a is a matrix valued 4 × 4 strongly hyperbolic operator; we state the following result: if the reduced dimension of a is superior or equal to 8, a is symmetric in a convenient basis.     

Systèmes de sortie ℱ Dt prévisibles

Summary Given a random closed setM, adapted to a filtration ( _t ), we construct a local time ofM which is both ( _t ) adapted and ( _Dt ) predictable, whereD _t =inf{s>t: sM}. Similarly an exit system, both ( _t ) optional and ( _Dt ) predictable, is associated withM and with the process representing the future at each timet. The paper is motivated by the markovian case, where the general results are applied to Ray processes.     

Persistance d’orbites périodiques relatives dans les systèmes hamiltoniens symétriques

It was known to Poincaré that a non-degenerate periodic orbit in a Hamiltonian system persists to nearby energy-levels. In this Note, we consider the analogous problem for relative periodic orbits in symmetric Hamiltonian systems. We show that non-degenerate relative periodic orbits also persist when shifting to nearby values of the energy-momentum map, under the hypothesis that the group of symmetries acts freely.     

Systèmes dynamiques topologiques associés à des automates récurrents

Summary To any subshift of A (with A a finite set) corresponds canonically a language L on A, together with the class of finite or infinite automata recognizing L.Two classes of invariant measures on X are distinguished: recurrent and transient measures (with respect to the automaton). In most instances, the maximum entropy of recurrent measures may be algebraically computed, and one can tell whether there exists a unique recurrent measure with maximum entropy, or none. No such properties may be stated about transient measures; some examples are given showing that they are responsible for the existence of more than one invariant measure on X with topological entropy.

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Membre du Laboratoire Associé au CNRS n 224: «Processus stochastiques et Applications     

Systèmes quasi-orthogonaux et théorème central limite presque sur

We prove, in the case of irrational rotations, the existence of a generalized moment version of the Almost Sure Central Limit Theorem (ASCLT), with a speed of convergence. In our strategy, we use the notion of quasi-orthogonal system, and purpose a Gaussian randomization technic, new at least in this context, which should also permit to be able to treat problems of comparable nature.     

Existence d'orbites périodiques complètement elliptiques des Hamiltoniens convexes présentant certaines symétries

Using certain quadratic forms associated to symplectic endomorphisms which we compare with the Clarke-Ekeland dual action functional, we prove: THEOREM. — Let H be a C²-Hamiltonian defined on R²ⁿ, strictly convex, proper and invariant under a certain symplectic rational positive and non-degenerate rotation (this is defined in the introduction); then, every hypersurface of H contains a completely elliptic periodic orbit. This generalizes the result of G. Dell'Antonio, B. D'Onofrio and I. Ekeland contained in [1].     

Loi de Weyl presque s?re pour un Syst��me Diff��rentiel en Dimension 1

We consider quite general differential operators on the circle with a small random lower order perturbation. We embrace two points a view, the semiclassical and the high energy limits. We show (a) in the semiclassical limit, that the eigenvalues inside a subdomain of the pseudospectrum are distributed according to a Weyl law with a probability close to 1, (b) that the large eigenvalues obey a Weyl law almost surely.     

Les SurfacesW (Weingarten) à Lignes de Courbure planes dans les deux Systèmes

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