RÉSUMÉS
des Prépublications
n°399
n°400 In this paper, we consider tilings of the hyperbolic 2-space, built with a finite number of polygonal tiles, up to affine transformation. To such a tiling
T, we associate a space of tilings: the continuous hull Omega(T) on which the
affine group acts. This space Omega(T) inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First we prove the finite harmonic measures of this laminated space correspond to finite invariant
measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally we give examples with an arbitrary number of ergodic invariant probability measures.
n°401
n°402 A lot of partial results are known about the Li\'enard differential equations~: \dot x= y -F_a^n(x), \dot y =-x. Here F_a^n is a polynomial of degree 2n+1, F_a^n(x)= \sum_{i=1}^{2n}a_ix^i+x^{2n+1}, where a = (a_1,\cdots,a_{2n}) \in \R^{2n}.
For instance, it is easy to see that for any a the related vector field X_a has just a finite number of limit cycles. This comes from the fact that X_a has a global return map on half-axis Ox=\{x \geq 0\}, and that this map is analytic and repulsive at infinity. It is also easy to verify that at most n limit cycles can bifurcate from the origin. For these reasons, Lins Neto, de Melo and Pugh have conjectured that the total number of limit cycles is also bounded by n, in the whole plane and for any value a.
In fact it is not even known if there exists a finite bound L(n) independent of a, for the number of limit cycles.
In this paper, I want to investigate this question of finiteness. I show that there exists a finite bound L(K,n) if one restricts the parameter in a compact K and that there is a natural way to put a boundary to the space of Liénard equations. This boundary is made of slow-fast equations of Liénard type, obtained as singular limits of the Liénard equations for large values of the parameter. Then the existence of a global bound L(n) can related to the finiteness of the number of limit cycles which bifurcate from slow-fast cycles of these singular equations.
n°403
n°404 In this paper, we discuss the remaining obstacles to prove Smale's conjecture about the C^1- density of hyperbolicity among surface
diffeomorphisms. Using a C1- generic approach, we
classify the possible pathologies that may obstruct
the C1- density of hyperbolicity. We show that there are
essentially two types of obstruction : (i) persistence of infinitely many
hyperbolic homoclinic classes and (ii) existence of a single homoclinic
class which robustly presents homoclinic tangencies.
In the course of our discussion, we obtain some related results about
C1- generic properties of surface diffeomorphisms involving homoclinic
classes, chain-recurrence classes, and hyperbolicity.
In particular, it is shown that on a connected surface
the C1-generic diffeomorphisms whose non-wandering sets have non-empty interior are the Anosov diffeomorphisms.
n°405
n°406 In this article, we begin a systematic study of conformal of codimension 1 foliations. We first define and study local conformal invariant. A case of particular interest is harmonic foliations of the plane. Then we study existence of totally umbilical and "Dupin" foliations on compact 3-manifolds of constant curvature.
n°407
n°408 This paper mainly discusses how to define Hamilton-Jacobi semigroup and prove it to be a solution of Hamilton-Jacobi inequality on path space PG over Lie group and on loop group LG. Then transportation cost inequlities are shown through the hypercontractivity of Hamilton-Jacobi semigroup on PG and LG. On loop group, we use the metric defined by M. Hino and J.A. Ramirez [12] as the cost function. We need firstly to prove some properties of this metric. These properties are not difficult to be obtained on Riemannian manifolds and on path space PG, but here they are more delicate. We also consider the Monge-Kantorovich problem on path space PG and give a solution.
n°409
n°410
n°411 Dans cette note, nous allons considérer la distance riemannienne sur le goupe des lacets. Un théorème de Rademacher est établi, ainsi qu'une inégalité de transport relative à une mesure de la chaleur pour cette distance riemannienne.
n°412 Generalizing Doering's 3-dimensional result, we proved in a joint work with Bonatti and Gourmelon that a C1-robustly transitive vector field, defined on any compact manifold of dimension d>2, admits a dominated splitting for the linear Poincaré flow. We prove here that the result still holds in the conservative 4-dimensional case : any Hamiltonian vector field, defined on a 4-dimensional compact symplectic manifold, which admits a robustly transitive regular enregy surface is Anosov on this regular energy surface. The proof of this result requires the use of a Hamiltonian version of Franks'Lemma, which we prove in this paper.
Généralisant un résultat tridimensionnel de Doering, nous avons prouvé dans un article commun à Bonatti et Gourmelon que tout champ de vecteurs robustement transitif en topologie C1 défini sur une variété compacte de dimension d>2 admet une décomposition dominée pour le flot linéaire de Poincaré. Nous prouvons dans cet article que ce résultat est encore valable dans le cas conservatif en dimension 4 : si une fonction hamiltonienne définie sur une variété symplectique compacte de dimension 4 admet des surfaces d'énergies régulières robustement transitives, alors ces dernières sont hyperboliques. La démonstration de ce résultat requiert l'utilisation d'une version hamiltonienne du lemme de Franks pour les champs de vecteurs, dont nous proposons une preuve dans cet article.
n°413
n°414 We completely classify the topological and geometric structures of some series of closed connected orientable 3-manifolds introduced by Kim and Kostrikin in [20] and [21] as quotient spaces of certain polyhedral 3-cells by pairwise identifications of their boundary faces. Then we study further classes of closed orientable 3-manifolds arising from similar polyhedral schemata, and describe their topological properties.
n°415 In this work, we shall deal with the critical Sobolev isotropic Brownian flows on the sphere Sd. Based on previous works by O. Raimond and LeJan-Raimond (see Ann. Inst. H. Poincaré, 35 (1999), p. 313-354 and Ann. of Prob., 30 (2002), p. 826-873), we prove that the associated flow is a flow of homeomorphisms.
n°416 We define Fermionic Brownian motion of A. Rogers as a distribution in the manner of non-commutative differential geometry. We give a Ferminionic Feynman-Kac formula.
n°417
n°418 Based on Kostant's cohomological interpretation of the Amitsur-Levitzki theorem, we prove a super version of this theorem for the Lie superalgebras osp(1,2n). We conjecture that no other classical Lie superalgebra can satisfy an Amitsur-Levitzki type identity. We show several (super) identities for the standard super polynomials. Finally, a combinatorial conjecture on the standard skew super supersymmetric polynomials is stated.
n°419 The configuration space of the planar mechanism of a robot with n arms each of which has a rotational joint and a fixed end points is studied. Its topological type is given by a Morse theoretical way and a topological way.
n°420 Let (Y,X) denote a 3-dimensional Poincar\'e pair (PD3-pair). By the work of Eckmann, Mueller and Linnel we may suppose, up to a homotopy equivalence,
that the boundary X is a closed 2-manifold. We show that if a component of X fails to be \pi1-injective in Y then there is an essential simple loop in X which is nullhomotopic in Y. It follows that there is a finite
process of attaching 2-disks along essential simple loops on X, and filling spherical components of X, which transforms (Y,X) in to a PD3-pair (Y',X') with aspherical incompressible boundary X' and such that \pi 1(Y)=\pi 1(Y').
The PD3-pair (X',Y') then admits a canonical decomposition as a connected sum of a PD3-pair with virtually free (possibly finite or trivial) fundamental group and boundary a (possibly empty) disjoint union of projective planes, together with a finite number of aspherical PD3-pairs with incompressible boundary.
n°421 In this paper, we study the concept of weak prox-regularity introduced in Jourani [15] in Asplund spaces. We show that this notion includes prox-regular functions, functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension. We establish also that weak prox-regularity is equivalent to Fréchet regularity.
n°422
n°423 It is well known that the non-spiraling leaves of real analytic foliations of codimension 1 all belong to the same o-minimal structure. Naturally, the question arises if the same statement is true for non-oscillating trajectories of real analytic vector fields. We show, under certain assumptions, that such a trajectory generates an o-minimal and model complete structure together with the analytic functions. The proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi-analyticity result from which the main theorem follows. As applications, we present an infinite family of o-minimal structures such that any two of them do not admit a common extension, and we construct a non-oscillating trajectory of a real analytic vector field in dimension 5 that is not definable in any o-minimal extension of the reals.
n°424 This article deals with the transfer of a satellite between Keplerian orbits. We study the controllability properties of the system and make a preliminary analysis of the time optimal control using the maximum principle. Second order sufficient conditions are also given. Finally, the time optimal trajectory to transfer the system from an initial low orbit with large eccentricity to a terminal geostationary orbit is obtained numerically.
n°425 We study a gcd algorithm directed by Least Significant Bits, the
so--called LSB algorithm, and provide a precise average--case analysis of its main parameters [number of iterations, number of shifts,
etc\ldots]. This analysis is based on a precise study of the dynamical systems which provide a continuous extension of the algorithm, and, here,
it is proved convenient to use both a 2--adic extension and a real one. This leads to the framework of products of random matrices, and our
results thus involve a constant $\gamma$ which is the Lyapunov exponent of the set of matrices relative to the algorithm. The algorithm can be
viewed as a race between a dyadic hare with a speed of 2 bits by step and a ``real'' tortoise with a speed equal to $ \gamma/\log 2 \sim 0.05$
bits by step. Even if the tortoise starts before the hare, the hare easily catches up with the tortoise [unlike in Aesop's fable ... ], and
the algorithm terminates.
n°426 The paper deals with the bifurcation of relaxation oscillations in two dimensional slow-fast systems. The most generic case in studied by means of geometric singular perturbation theory, using blow up at contact points. It reveals that the bifurcation goes through a continuum of transient canard oscillations, controlled by the slow divergence integral along the critical curve. The theory is applied to polynomial Liénard equations, showing that the cyclicity near a generic coallescence of two relaxation oscillations does not need to be limited to two, but can by arbitrarily high.
n°427 The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions of Order Two and COnjugate times), available on the web, and apply it to the minimal time problem of orbit transfer, and to the attitude control problem of a rigid spacecraft. This algorithm involves both normal and abnormal cases.
n°428 We consider the family of polynomials in $\C[x,y,z]$ of the form
$x^2y-z^2-xq(x,z)$. Two such polynomials $P_1$ and $P_2$ are equivalent if there is an automorphism $\varphi^*$ of $\C[x,y,z]$ such
that $\varphi^*(P_1)=P_2$. We give a complete classification of the equivalence classes of these polynomials in the algebraic and
analytic category. As a consequence, we find the following results. There are explicit examples of inequivalent
polynomials $P_1$ and $P_2$ such that the zero set of $P_1+c$ is isomorphic to the zero set of $P_2+c$ for all $c\in\C$. There
exist polynomials which are algebraically inequivalent but analytically equivalent. There exist polynomials which are algebraically inequivalent but when considered as polynomials in $\C[x,y,z,w]$ become equivalent. This last result answers a problem posed in \cite{ml-shp}. Finally, we get a complete
classification of $\C^+$-actions on $\C^3$ which are defined by a triangular locally nilpotent derivation of the form $x^2\d/\d z+(2z+xq(x,z))\d/\d y$.
n°429 The aim of this work is to prove that one can easily reconstruct
surfaces and approximate medial axis of smooth submanifolds in $\R^k$ with topological guaranties. Our results are based upon elementary
results in critical point theory for distance functions. Given a smooth compact submanifold $S$ of $\R^k$ and a compact approximation
$\KK$ of it, we first study the location of the critical points of the distance function to $\KK$. Second, we prove
that under quite general hypothesis the connected components of the boundary of union of balls centered on $\KK$ is isotopic to $S$.
Third, given a ball $\BB$ containing the convex hull of $S \cup \KK$, we prove, under the same kind of hypothesis, that a subset (known
as the $\lambda$-medial axis) of the medial axis of $\BB \setminus \KK$ has the homotopy type of $\R^k \setminus S$. It turns out that if
$\KK$ is a finite sample of points, this subset is an easy to compute subcomplex of the Vorono\" \i \ diagram of $\KK$.
n°430 We show that a particular free-by-cyclic group $G$ has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. Starting from a fixed presentation 2-complex we define a family of non-positively curved piecewise Euclidean ``model'' spaces for $G$, and show that whenever the group acts properly by isometries on any proper 2-dimensional CAT(0) space $X$ there exists a $G$-equivariant map from the universal cover of one of the model spaces to $X$ which is locally isometric off the 0-skeleton and injective on vertex links.
From this we deduce bounds on the relative translation lengths of various elements of $G$ acting on any such space $X$ by first studying the geometry of the model spaces. By taking HNN-extensions of $G$ we then produce an infinite family of 2-dimensional hyperbolic groups which do not act properly by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6.
n°431 We show that a large class of right-angled Artin groups
(in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group
$\DiffD$ of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends results of
Benaim and Gambaudo who gave quasi-isometric embeddings of $F_n$ and $\Z^n$ for all $n>0$. As a consequence we
are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the group $\DiffD$. Examples include hyperbolic surface groups, some HNN-extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3-manifold.
n°432 This paper is essentially an introduction to, and a survey of, multisequences, maps from the sets of maximal elements of some special trees. However it also contains some new concepts and results. Multisequences find application in numerous problems related to sequentiality, like characterizations of sequential and subsequential spaces, preservation of sequentiality and of Fréchetness by various operations, estimates of sequential order of products.
n°433 Regular and irregular pretopologies are studied. In particular, for every ordinal there exists a topology such that the series of its partial (pretopological) regularizations has length of that ordinal. Regularity and topologicity of standard pretopologies on cascades can be characterized in terms of their states, so that their study for such spaces reduces to that of a combinatorics of states. For example, if an iterated partial regularization rk \pi is topological for k>0 then r\pi is a regular topology. Irregularity of pretopologies of countable character can be characterized in terms of sequential cascades with standard irregular pretopologies.
n°434
n°435 Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural non trivial supertrace and an associated non degenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples.
n°436 We revisit graded Lie algebras related to Deformation Theory, examining among others, Gerstenhaber, Gerstenhaber-Nijenhuis, Schouten and super Poisson brackets. We give new applications to identities of standard polynomials, deformation theory of quadratic Lie algebras, cyclic cohomology of quadratic Lie algebras, generalized Lie algebras, generalized Poisson brackets and so on.