Recurrence and genericity

We prove a C¹-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C¹-generic diffeomorphisms. For instance, C¹-generic conservative diffeomorphisms are transitive. To cite this article: C. Bonatti, S. Crovisier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Une condition suffisante de minimalité pour les géodésiques de la métrique de HoferA sufficient minimality condition for geodesics of the Hofer's metric

We give a new sufficient condition for a Hamiltonian H to generate a length minimizing geodesic of the Hofer's metric on the group of Hamiltonian diffeomorphisms on $\text{R}{}^{\mathrm{2n}}$. This condition is related to the spectra of the linearized maps of the flow {φ^H_t} generated by H at the fixed points of the flow. In addition we show that if φ₀, φ₁ are two diffeomorphisms linked by such a geodesic, then the Hofer's distance between φ₀ and φ₁ is the same as Viterbo's one. To cite this article: J. Le Crapper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 359–364.     

Quasi-morphismes et invariant de Calabi

In this paper, we give two elementary constructions of homogeneous quasi-morphisms defined on the group of Hamiltonian diffeomorphisms of certain closed connected symplectic manifolds (or on its universal cover). The first quasi-morphism, denoted by CalS, is defined on the group of Hamiltonian diffeomorphisms of a closed oriented surface S of genus greater than 1. This construction is motivated by a question of M. Entov and L. Polterovich [M. Entov, L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30 (2003) 1635-1676]. If U⊂S is a disk or an annulus, the restriction of CalS to the subgroup of diffeomorphisms which are the time one map of a Hamiltonian isotopy in U equals Calabi's homomorphism. The second quasi-morphism is defined on the universal cover of the group of Hamiltonian diffeomorphisms of a symplectic manifold for which the cohomology class of the symplectic form is a multiple of the first Chern class.     

Version kählérienne d'une conjecture de Robert J. Zimmer

Let M be a compact Kähler manifold. Let G be a connected simple real Lie group. Let Γ be a lattice in G. We prove the following: if the R-rank of G is strictly larger than the complex dimension of M any morphism from Γ to the group of holomorphic diffeomorphisms of M has finite image. This is a particular case in a conjecture of Robert J. Zimmer     

L'invariant de Calabi pour les homéomorphismes quasiconformes du disqueThe Calabi invariant for quasiconformal mappings of the unit disk

We prove that the Calabi invariant for the symplectic diffeomorphisms of the unit disk with compact support is well defined for quasiconformal maps and depends continuously with respect to these homeomorphisms in the quasiconformal topology. To cite this article: P. Ha??ssinsky, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 635–638.     

Actions de groupes de Kazhdan sur le cercle

We prove that every homomorphism from a discrete Kazhdan group to the group of orientation-preserving diffeomorphisms of the circle of class C^1+α (α>1/2) has a finite image.     

Quasi-morphisme de Calabi sur les surfaces de genre supérieur

We construct a homogeneous quasi-morphism ${\mathfrak{Cal}}_S$ on the group of Hamiltonian diffeomorphisms of a (closed, connected, oriented) surface S of genus greater or equal to 2, with the following property. For each connected open set U in S diffeomorphic to a disk or to an annulus, the restriction of ${\mathfrak{Cal}}_S$ to the subgroup of diffeomorphisms which are the time 1 map of a Hamiltonian isotopy in U, equals Calabi's homomorphism. To cite this article: P. Py, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the flat remainder in normal forms of families of analytic planar saddles

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter. To cite this article: P. Bonckaert, F. Verstringe, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Extension to BV functions of the large deformation diffeomorphisms matching approach

The image matching within the framework of large deformations via diffeomorphisms is extended to the space of bounded variation functions. Thanks to a semi-differentiation lemma, which is the central new result of this article, we derive the geodesic equations for a general penalty term and we describe the associated momentum. To cite this article: F.-X. Vialard, F. Santambrogio, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Modèles pour les applications fibrées et les polysystèmes

We introduce a fibered version of the classical ‘horseshoe’ of Smale, that one can use to analyze the dynamical properties of diffeomorphisms with a normally hyperbolic invariant manifold and suitable homoclinic intersections. Our abstract model gives rise to fibered dynamics over the Bernoulli shift and to the more particular case of polysystems that we define here, which can be seen as the simultaneous iteration of several maps on the same space in any prescribed order. To cite this article: J.-P. Marco, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Symplectic flows on the open ball

We construct a complete, infinitesimally symplectic vectorfield on the open unit ball in R⁴ with no zeros. This shows that there are symplectic diffeomorphisms of the ball with no fixed points.     

Embedding smooth diffeomorphisms in flows

In this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces. First we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain a class of local Ck diffeomorphisms for k∈N∪{∞,ω} which admit embedding vector fields with some smoothness. Finally we prove that for any k∈N∪{∞} under the coefficient topology the subset of local Ck diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local Ck diffeomorphisms.     

The set of axiom A diffeomorphisms with no cycles

LetM be aC ^∞ closed manifold and Diff¹ (M) be the space of diffeomorphisms ofM endowed with theC ¹ topology. This paper contains an affirmative answer to the following conjecture raised by Mañé, which is an extension of the stability and Ω-stability conjectures of Palis and Smale, as follows: theC ¹ interior of the subset of diffeomorphism such that all the periodic points are hyperbolic is characterized as the set of diffeomorphisms satisfying Axiom A and the no-cycles condition. Moreover, it is showed that theC ¹ interior of the set of all Kupka-Smale diffeomorphisms coincides with the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition.     

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of \(d\) -dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.     

Compactification des champs de chtoucas de Drinfeld

Another way to construct the compactifications of the stacks of Drinfeld's shtukas introduced by Lafforgue is presented. The method is based on the variation of GIT quotients studied by Thaddeus and Dolgachev–Hu. To cite this article: Tuan Ngo Dac, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Effondrement de quotients aléatoires de groupes hyperboliques avec torsion

We show that random quotients of hyperbolic groups with ‘harmful’ torsion collapse at densities smaller than expected. To cite this article: Y. Ollivier, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Estimation des résidus de l'amplitude de diffusion pour des perturbations de longue portée

We study the semi-classical behavior when h→0, of the residues of the scattering amplitude associated to the Schrödinger operator P(h)=?h²Δ+V(x) for long-range perturbations V(x). For resonances close to the real axis and under a separation condition, we give an estimate of each residue in terms of the parameter h and the imaginary part of the resonance. To cite this article: L. Michel, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Classes de Hirzebruch et classes de Chern motiviques

Let X be a complex algebraic variety. We define and study new theories of characteristic classes, defined on the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. One of them, $T_y$ is a homology class version of the motivic measure and generalizes the corresponding Hirzebruch characteristic. It unifies the Chern class transformation of Schwartz and MacPherson, the Todd class transformation of Baum–Fulton–MacPherson and the L-class transformation of Cappell–Shaneson. To cite this article: J.-P. Brasselet et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Existence de solutions pour une classe de systèmes non linéaires de Boussinesq

We give a few existence results of solutions for a class of Boussinesq systems, with suitable conditions on the right-hand side of the momentum equation, the forcing term depending on temperature. To cite this article: A. Attaoui, C. R. Acad. Sci. Paris, Ser. I 346 (2008).