RÉSUMÉS

des Prépublications

n°143 Let H be a germ of holomorphic diffeomorphism at 0 \in C . Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germ S at 0 , such that S(ze^{2 \pi i})=H\circ S(z) (1). If H_\lambda is an unfolding of diffeomorphisms depending on \lambda \in (C,0), with H₀ =Id, one introduces its ideal {\cal I}_H. It is the ideal generated by the germs of coefficients (a_i (\lambda ),0) at 0\in C^k, where H_\lambda (z)-z=\sum a_i (\lambda )zⁱ. Then one can find a parameter solution S_\lambda(z) of (1) which has at each point z₀ belonging to the domain of defintion of S₀, an expansion in series S_\lambda (z)=z+\sum b_i (\lambda )(z-z₀)ⁱ with (b_i,0)\in {\cal I}_H, for all i. This result may be applied to the bifurcation theory of vector fields of the plane. Let X_\lambda be an unfolding of analytic vector fields at 0\in R² such that this point is a hyperbolic saddle point for each \lambda . Let H_\lambda (z) be the holonomy map of X_\lambda at the saddle point and {\cal I}_H its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals \sigma , \tau , transversal to the separatrices of the saddle point, such that the difference between the transition map D_\lambda(z) and the identity is divisible in the ideal {\cal I}_H. Finally, suppose that X_\lambda is an unfolding of a saddle connection for a vector field X₀, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular point z\in \sigma , can also be computed at the singular point z=0. From this last observation it follows easily that the cylicity of the unfolding X_\lambda is finite and can be computed explicitly in terms of the Bautin ideal.

n°144

n°145 Conjugate unstable manifolds of saturated hyperbolic sets of Smale diffeomorphisms are characterised in terms of the combinatorics of their geometrised Markov partitions.

n°146

n°147 Nous disons que deux points périodiques hyperboliques d'un difféomorphisme f présentent une connexion persistante si, pour un ensemble de difféomorphismes dense dans un voisinage de f , ils appartiennent à un même ensemble transitif. Nous montrons que génériquement, deux points sont dans un même ensemble transitif si et seulement s'ils présentent une connextion persistante et que leurs classes homoclines sont égales. En conséquence, nous prouvons la généricité locale dans l'espace des C¹-difféomorphismes des variétés de dimension 3, du phénomène de Newhouse (co-existence d'une infinité d'attracteurs ou de répulseurs périodiques).

We say that two hyperbolic periodic points of a diffeomorphism f are persistently connected if there is a neighbourhood of f and a dense subset of it of diffeomorphism for which there is a transitive set containing such points. We prove that generically two points are in the same transitive set if and only if they are persistently connected and their homoclinic class are equal. As a consequence from our results we get the local genericity of the Newhouse's phenomenon (coexistence of infinitely many sinks or sources) for C¹-diffeomorphisms of three manifolds.

n°148 Dans [1], nous montrons que le germe d'un champ de vecteurs X , en dimension 3, le long d'un ensemble hyperbolique saturé K détermine de façon unique une variété à bord munie d'un flot (en quelque sorte la variété et le flot les plus simples réalisant le germe considéré) appelés modèle de (X,K). Ce modèle est alors caractérisé à équivalence topologique près par une information combinatoire finie : le type géométrique d'une partition de Markov de K. Dans cet article, nous donnons une construction explicite du modèle d'un type géométrique quelconque. Cette construction nous permet de dégager des propriétés des modèles et nous fournit des exemples montrant la différence entre les ensembles hyperboliques des flots en dimension 3 et ceux des suspensions de difféomorphismes de surfaces.

The germ of a 3-vector field X along an hyperbolic saturated set K determines uniquely a manifold with boundary carrying a flow (roughly speaking the simpliest manifold and flow carrying the germ considered above) called the model of (X,K). We have seen in [1] that this model is caracterized up to topological equivalence by a finite combinatorial information : the geometrical type of a Markov partition of K. In this article, we give an explicit construction of the model of any geometrical type. From this construction follow properties of models and examples which show the difference existing between hyperbolic sets of 3-flows and hyperbolic sets of suspensions of diffeomorphisms of surfaces.

n°149

n°150 The paper treats about multiple limit cycle bifurcations in singular perturbation problems of planar vector fields. The results deal with any number of parameters. Proofs are based on the techniques introduced in the AMS-memoir : "Canard Cycles and Center Manifolds". Although the presentation is limited to generalized Liénard equations \varepsilon {\ddot x}+\alpha (x,c){\dot x}+\beta (x,c)=0 it is clear that the method can also be used in a more general context.

n°151 Flat sub-Riemannian structures are local approximations - nilpotentizations - of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3,4 and 5.

n°152

n°153 We give affine estimates for the number of zeros of complete Abelian integrals f_{\delta (h)}\omega on a continuous family of real cycles {\delta (h)} of a quartic elliptic Hamiltonian. These estimates only depend on the degree of the polynomial 1-form \omega .

Nous donnons des bornes affines pour le nombre de zéros d'intégrales Abéliennes \int _{\delta (h)} \omega sur une famille continue de cycles réels {\delta (h)} d'un Hamiltonien quartique elliptique. Ces estimations dépendent seulement du degré de la 1-forme polynomiale \omega .

n°154 Let {\cal A} be a central arrangement of hyperplanes in C ⁿ, let M({\cal A}) be the complement of {\cal A}, and let {\cal L}({\cal A}) be the intersection lattice of {\cal A}. For X in {\cal L}({\cal A}) we set {\cal A}_X ={H\in {\cal A} ; H\supseteq X}, and {\cal A}/X={H/X ; H\in {\cal A}_X }, and {\cal A}^X ={H\cap X ; H\in {\cal A}\backslash {\cal A}_X}. We exhibit natural embedings of M({\cal A}_X) in M({\cal A}) that give rise to monomorphismes from \pi ₁ (M({\cal A}_X)) to \pi ₁(M({\cal A})). We call the images of these monomorphismes intersection subgroups of type X and prove that they form a conjugacy class of subgroups of \pi ₁ (M({\cal A})).
Recall that X in {\cal L}({\cal A}) is modular if X+Y is an element of {\cal L} ({\cal A}) for all Y in {\cal L} ({\cal A}). We call X in {\cal L}({\cal A}) supersolvable if there exists a chain 0\subseteq X₁ \subseteq ... \subseteq X_d =X in {\cal L}({\cal A}) such that X_\mu is modular and dim X_\mu =\mu for all \mu =1, ... ,d. Assume that X is supersolvable and view \pi ₁ (M({\cal A}_X)) as an intersection subgroup of type X of \pi ₁ (M({\cal A})). Recall that the commensurator of a subgroup S in a group G is the set of a in G such that S \cap aSa^-1 has finite index in both S and aSa^-1. The main result of this paper is the characterization of the centralizer, the normalizer, and the commensurator of \pi ₁ (M({\cal A}_X)) in \pi ₁ (M({\cal A})). More precisely, we exhibit an embeding of \pi ₁(M({\cal A}^X)) in \pi ₁ (M({\cal A})) and prove :
1) \pi ₁ (M({\cal A}^X))\cap \pi ₁ (M({\cal A}_X))={1} and \pi ₁ (M({\cal A}^X)) is included in the centralizer of \pi ₁ (M({\cal A}_X)) in \pi ₁ (M({\cal A})) ;
2) the normalizer is equal to the commensurator and is equal to \pi ₁ (M({\cal A}^X)) times \pi ₁ (M({\cal A}_X)) ;
3) the centralizer is equal to \pi ₁ 1 (M({\cal A}^X )) times the center of \pi ₁(M({\cal A}_X)).
Our study starts with an investigation of the projection p:M({\cal A}\rightarrow M({\cal A}/X) induced by the projection Cⁿ\rightarrow Cⁿ /X. We prove in particular that this projection is a locally trivial C^\infty fibration if X is modular, and deduce some exact sequences involving the fundamental groups of the complements of {\cal A}, of {\cal A}/X, and some (affine) arrangement {\cal A}^X_z₀.

n°155 We consider partially hyperbolic diffeomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundle E^uu (uniformly expanding) and a subbundle E^c, dominated by E^uu. We prove that if the central direction E^c is mostly contracting for the diffeomorphism (negative Lyapunov exponents), then the ergodic Gibbs u-states are the Sinai-Ruelle-Bowen measures, there are finitely many of them, and their basins cover a full measure subset. If the strong-unstable leaves are dense, there is a unique Sinai-Ruelle-Bowen measure. We describe some applications of these results, and we also introduce a construction of robustly transitive diffeomorphisms in dimension larger than three, having no uniformly hyperbolic (neither contracting nor expanding) invariant subbundles.

n°156 On étudie le problème des modèles minimaux (au sens de Zariski) pour les feuilletages sur les surfaces algébriques complexes, et on caractérise les feuilletages qui n'ont pas de modèle minimal. On applique cela à l'étude des difféomorphismes polynomiaux du plan.

n°157 Il s'agit principalement d'une exposition du travail de M. Mc Quillan autour de la conjecture de Green-Griffiths sur les courbes entières dans les surfaces de type général.

n°158 In [AC₂], A'Campo associates a link in S³ to any proper generic immersion of a disjoint union of arcs into a 2-disc. We give a simple algorithmic way to produce, from the immersion, a representative braid for such links. As a by-product we get a minimal representative braid for any algebraic link, from a divide associated to a real deformation of the polynomial defining the link.

n°159 This paper is a continuation of a series of papers of the authors, dealing with contact sub-Riemannian metrics on R³. We study the special case of contact metrics that correspond to isoperimetric problems on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at least locally. It is equivalent to study the associated sub-Riemannian spheres of small radius. It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics, that we studied in our previous papers (although, there is a certain symmetry). Thanks to the classification of spheres, conjugate-loci and cut-loci, done in these papers, we conclude immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem of minimizing length, for prescribed area), these results do not apply. Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities). In an addendum, we say a few words about : 1) the singularities that can appear in general for the Dido problem, and 2) the motion of particles in a nonvanishing constant magnetic field.

n°160 Consider a sub-riemannian geometry (U,D,g) where U is a neighborhood of 0 in R³, D is a Martinet type distribution identified to ker \omega, \omega being the 1-forme : g=dz-{y²\over 2}dx, q=(x,y,z) and g is a metric on D which can be taken in the normal form : g=a(q)dx ² + c(q)dy ², a=1+yF(q), c=1+G(q), G _x=y=0 =0. In a previous article we analyze the flat case : a=c=1 ; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three one parameter deformations of the flat case which clarify the role of the three parameters \alpha , \beta , \gamma in the gradated normal form of order 0 where : a=(1+\alpha y)², c=(1+\beta x+\gamma y)². More generally this analysis provides an explanation of the role of abnormal minimizers in SR-geometry.

n°161 The Ricci tensor has been computed in severals infinite dimensional situations. In this work, we shall be interested in the case of the central extension of loop groups and in the asymptotic behaviour of the Ricci tensor on free loops when the Riemannian metrics varies.

n°162 We study sub-Riemannian (Carnot-Caratheodory) metrics defined by noninvolutive distributions on real-analytic Riemannian manifolds. We establish a connection between regularity properties of these metrics and nonoccurence of length minimizing abnormal geodesics. Utilizing the results of the previous study of abnormal length minimizers accomplished by the authors in [Annales IHP, Analyse nonlinéaire, vol. 13, n°6, pp. 635-690] we describe in this paper two classes of the germs of distributions (called 2-generating and medium fat) such that the corresponding sub-Riemannian metrics are subanalytic. To characterize the vastness of these classes of distributions we deterine the dimensions of the manifolds on which generic.

n°163 Geometric Singular Perturbation theory, up to now, seems to deal only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.

n°164 Let {\widetilde R} be an o-minimal expansion of the field of real numbers. We show that if {\widetilde R} has analytic cell decomposition, then its Pfaffian closure {\cal P} ({\widetilde R}) also has analytic cell decomposition. In particular, if {\widetilde R} has analytic Whitney stratification, then so does {\cal P} ({\widetilde R}).

n°165 Consider a sub-Riemannian geometry (U,D,g) where U is a neighborhood of O in R ³, D is a Martinet type distribution identified to Ker \omega , \omega =dz-{y²\over 2}dx, q=(x,y,z) and g is a metric on D which can be taken in the normal form : a(q)dx²+c(q)dy², a=1+yF(q), c=1+G(q), G_{\mid x=y=0} =0. In a previous article we analyzed the flat case : a=c=1 ; we showed that the set of geodesics is integrable using elliptic integrals of the first and second kind ; moreover we described the sphere and the wave front near the abnormal direction using the exp-log category. The objective of this article is to analyze the transcendence we need to compute the sphere and the wave front of small radius in the abnormal direction and globally when we consider the gradated normal form of order 0 : a=(1+\alpha y)², c=(1+\beta x+\gamma y)², where \alpha , \beta , \gamma are real parameters.

n°166 We estimate the decay of correlations for some Markov maps on a countable states space. A necessary and sufficient condition is given for the transfer operator to be quasi-compact on the space of locally Lipschitz functions. In the non quasi-compact case, the decay of correlations depends on the contribution to the transfer operator of the complementary of finitely many cylinders. Estimates are given for some non uniformly expanding maps and for birth-and-death processes.

n°167 La dynamique oscillante d'un champ de vecteurs analytique en dimension trois s'organise autour d'un nombre fini d'axes de tourbillonnement lorsqu'elle ne se délocalise pas par des éclatements de point.

The oscillating dynamics of an analytic vector field in dimension three is organized around a finite number of twister axis when it is localisable by point blowing-ups.

n°168 In this paper, we summarize everything we know about subriemannian geometry in the contact case. Most of the work concerns the three dimensional case.
First, we study invariants and normal forms for contact subriemannian metrics, and we obtain complete results in any dimension.
Second, we give a 3-d full, local, generic classification of the associated conjugate loci, cut loci, spheres and wave fronts. We also care about the special subriemannian metrics that are related to isoperimetric problems on a 2-d riemannian manifold. We show that the case of a "generic isoperimetric problem", is, roughly speaking, the same as the case of a general (non isoperimetric) metric (in fact, there are differences, but they are very tight).
Third, we care about a very old particular isoperimetric problem : the Dido problem on a 2-d riemannian manifold (i.e. the problem of minimizing the perimeter for prescribed area). This classical problem is so degenerate that the previous results do not apply. We describe the solutions of this problem (in the case of a generic riemannian metric), at the local level (i.e. when the prescribed area is small).
The results concerning the generic case, and the generic isoperimetric case have already been published by the authors. What is completely new is the solution of the generic Dido problem. Here, we mainly state the results. Detailed computations and proofs (that are rather long and tedious) can be found in our previous papers [2], [10], [4], and in the preprint [6].

n°169 Pour des applications non markoviennes, monotones par morceaux de l'intervalle, associées à un potentiel, on prouve que la loi du temps d'entrée dans un cylindre, renormalisée par la mesure de ce cylindre, converge vers une loi exponentielle pour presque tous les cylindres. Ce résultat permet ensuite de montrer que les fluctuations de R_n, temps de premier retour dans un cylindre, suivent une loi lognormale.

n°170 The purpose of this paper is an approach of the problems of compositionnal inverse in the logarighmic-exponential field. A new derivation family is defined, and allows us to give an explicit necessary condition of inversibility. Their existence is based on the knowledge of the functions which are the exponential of another one in a finite transcendental extension of the fields of rationnal functions.

Cet article aborde les problèmes d'inversion de fonctions pour la composition dans le corps des fonctions exponentio-logarithmique. La construction d'une nouvelle famille de dérivations permet de donner une condition nécessaire explicite d'inversibilité. L'existence de celles-ci passe par la connaissance des fonctions qui sont l'exponentielle d'une autre dans un corps, extension de transcendance finie du corps des fractions rationnelles.

n°171 L'article présente l'histoire de la géométrie intégrale, en la divisant en trois périodes : les débuts et son interaction avec la théorie de la mesure, les relations entre informations partielles obtenues par projection et sections et la géométrie riemannienne, et enfin les inégalités utilisant la topologie. A la fin nous mentionnons des questions et quelques résultats de géométrie intégrale conforme.

The article surveys three periods of integral geometry : the birth of geometric probabilities, local riemannian invariants and their relations with projections and sections, integral geometry and topology. At the end we mention questions and first results in conformal integral geometry.