Cet article comprend deux parties indépendantes. On démontre d’abord que pour tout feuilletage dont les singularités sont des selles ayant au moins 3 séparatrices, sur une surface fermée orientable de genre $g\ge 2$, le cône des mesures transverses invariantes se plonge comme un sous-espace isotrope dans l’espace $ℳℱ$ des feuilletages mesures muni de sa structure symplectique linéaire par morceaux, définie par Thurston. On en déduit une nouvelle démonstration d’un résultat essentiellement du à Katok, qui...

Étant donnée une fonction régulière de moyenne nulle sur le tore de dimension $2$, il est facile de voir que ses intégrales ergodiques au-dessus d’un flot de translation “générique”sont bornées. Il y a une dizaine d’années, A. Zorich a observé numériquement une croissance en puissance du temps de ces intégrales ergodiques au-dessus de flots d’hamiltoniens (non-exacts) “génériques”sur des surfaces de genre supérieur ou égal à $2$, et Kontsevich et Zorich ont proposé une explication (conjecturelle) de...

Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f\times f²\times \cdots \times f^m:X^m\to X^m$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^m$ under the action of $f\times f²\times \cdots \times f^m$. We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii)....

The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C¹ interior of the set of all diffeomorphisms on a $C^{\infty }$ closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C¹ interior is finite if and only if f is Morse-Smale.

Si deux systèmes dynamiques de dimension 1 et de classe $C^r$ sont $C^1$-conjugués, dans quelles conditions sont-ils $C^r$-conjugués ? Par “système dynamique de dimension 1”, nous entendons ici un feuilletage de codimension 1 ou une application du cercle dans lui-même. Nous donnons des conditions très faibles pour que la réponse à la question précédente soit positive.

We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup $P_t\phi$ does exist for any bounded $\phi$; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.

In this paper we introduce a concept of Schauder basis on a self-similar fractal set and develop differential and integral calculus for them. We give the following results: (1) We introduce a Schauder/Haar basis on a self-similar fractal set (Theorems I and I'). (2) We obtain a wavelet expansion for the L²-space with respect to the Hausdorff measure on a self-similar fractal set (Theorems II and II'). (3) We introduce a product structure and derivation on a self-similar fractal set (Theorem III)....

We consider the problem of Arnold’s diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincaré-Melnikov function.

For a positive integer n and R>0, we set $$B_R^n=\left\{x\in {ℝ}^n|{∥x∥}_{\infty }<R\right\}$$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $$h\left(r\right)=\frac{1}{2}{r}_1^2+...\frac{1}{2}{r}_{n-1}^2+r_n$$ on $${𝕋}^n\times B_R^n$$ , 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 $${𝕋}^n\times B_R^n$$ , and setting $${\epsilon }_j:={∥h-H_j∥}_{C^0\left(V\right)}$$ the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface,...

A small perturbation of a rational function causes only a small perturbation of its periodic orbits. We show that the situation is different for transcendental maps. Namely, orbits may escape to infinity under small perturbations of parameters. We show examples where this "diffusion to infinity" occurs and prove certain conditions under which it does not.