Jean-Marie Lion

Analytic stratification in the Pfaffian closure of an o-minimal structure

Introduction. Let U ⊆ R be open andω = a1dx1+·· ·+andxn a nonsingular, integrable 1-form onU of classC1, and let be the foliation onU associated toω. A leafL ⊆ U of is aRolle leaf if anyC1 curveγ : [0,1] → U with γ (0),γ (1) ∈ L is tangent to at some point, that is, ω(γ (t))(γ ′(t)) = 0 for somet ∈ [0,1]. Note that while a leaf of is in general only an immersed manifold, any Rolle leaf of is an embedded and closed submanifold of U . Throughout this paper, we fix an arbitrary o-minimal expansion R̃ of the field of real numbers. Whenever U and a1, . . . ,an are definable iñR , then a leaf of is called a leafover R̃ . We useR̃1 to denote the expansion of̃ R by all Rolle leaves over R̃ . For example, the expansion Ran of the real field generated by all globally semianalytic sets is o-minimal; in fact the sets definable in Ran are exactly the globally subanalytic sets (see [7], [4]). Building on Khovanski ı̆’ theory of fewnomials [10] and subsequent work by Moussu and Roche [14], Lion and Rolin [12] showed that (Ran)1 is also o-minimal. Adapting the various ideas involved to the general o-minimal setting, Speissegger [15] proved the following statement.

Champs de vecteurs analytiques et champs de gradients

A theorem of Łojasiewicz asserts that any relatively compact solution of a real analytic gradient vector field has finite length. We show here a generalization of this result for relatively compact solutions of an analytic vector field X with a smooth invariant hypersurface, transversally hyperbolic for X, where the restriction of the field is a gradient. This solves some instances of R. Thoms Gradient Conjecture. Furthermore, if the dimension of the ambient space is three, these solutions do not oscillate (in the sense that they cut an analytic set only finitely many times); this can also be applied to some gradient vector fields.

Wandering domains for infinitely renormalizable diffeomorphisms of the disk

Denjoys theorem and counter-example for circle maps have counterparts for infinitely renormalizable diffeomorphisms of the 2-disk.

The theorem of the complement for nested sub-Pfaffian sets

Let R be an o-minimal expansion of the real field, and let L(R) be the language consisting of all nested Rolle leaves over R. We call a set nested subpfaffian over R if it is the projection of a boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we prove that the complement of a nested subpfaffian set over R is again a nested subpfaffian set over R. As a corollary, we obtain that if R admits analytic cell decomposition, then the pfaffian closure P(R) of R is obtained by adding to R all nested Rolle leaves over R, a one-stage process, and that P(R) is model complete in the language L(R).

Théorème de Gabrielov et fonctions log-exp-algébriques

Resume Nous obtenons le theoreme de Wilkie sur les fonctions log-exp-algebriques du theoreme du complementaire ≪ explicite ≫ de Gabrielov, et de notre presentation geometrique du theoreme de van den Dries, Macintyre et Marker sur les fonctions log-exp-analytiques.

Volumes transverses aux feuilletages définissables dans des structures o-minimales

Let Fλ be a family of codimension p foliations defined on a family Mλ of manifolds and let Xλ be a family of compact subsets of Mλ. Suppose that Fλ, Mλ and Xλ are definable in an o-minimal structure and that all leaves of Fλ are closed. Given a definable family Ωλ of differential p-forms satisfaying iZ Ωλ = 0 forany vector field Z tangent to Fλ, we prove that there exists a constant A > 0 such that the integral of on any transversal of Fλ intersecting each leaf in at most one point is bounded by A. We apply this result to prove that p-volumes of transverse sections of Fλ are uniformly bounded.

Une propriété des solutions non spiralantes d’équations différentielles analytiques du plan

Resume On montre que le contact entre deux courbes integrales singulieres et non spiralantes des deux equations de Pfaff analytiques du plan est au plus exponentiellement petit.