Abstract. Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we establish a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.

In this paper we prove that any C 1 vector field defined on a three-dimensional manifold can be approximated by one that is uniformly hyperbolic, or that exhibits either a homoclinic tangency or a singular cycle. This proves an analogous statement of a conjecture of Palis for diffeomorphisms in the context of C 1-flows on three manifolds. For that, we rely on the notion of dominated splitting for the associated linear Poincaré flow.  2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-On prouve que tout champ de vecteurs C 1 défini sur une variété de dimension trois peut être approché par un qui est uniformément hyperbolique ou bien par un qui présente soit une tangence homocline soit un cycle singulier. Ceci prouve, dans le contexte des flots C 1 sur les variétés de dimension trois, l'analogue d'une conjecture de Palis concernant les difféomorphismes. On s'appuie sur la notion de décomposition dominée pour le flot linéaire de Poincaré associé.  2003 Éditions scientifiques et médicales Elsevier SAS

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We study the dynamics of billiard models with a modified collision rule: the outgoing angle from a collision is a uniform contraction, by a factor λ , of the incident angle. These pinball billiards interpolate between a one-dimensional map when λ = 0 and the classical Hamiltonian case of elastic collisions when λ = 1. For all λ < 1, the dynamics is dissipative, and thus gives rise to attractors, which may be periodic or chaotic. Motivated by recent rigorous results of Markarian, Pujals and Sambarino [MPS08], we numerically investigate and characterise the bifurcations of the resulting attractors as the contraction parameter is varied. Some billiards exhibit only periodic attractors, some only chaotic attractors, and others have coexistence of the two types.

Abstract. Singular hyperbolicity is a weaker form of hyperbolicity that is found on any C 1 -robust transitive set with singularities of a flow on a three-manifold, like the Lorenz Attractor, [MPP]. In this work we are concerned in the dynamical properties of such invariant sets. For instance, we obtain that if the attractor is singular hyperbolic and transitive, the set of periodic orbits is dense. Also we prove that it is the closure of a unique homoclinic class of some periodic orbit. A corollary of the first property is the existence of an SRB measure supported on the attractor. These properties are consequences of a theorem of existence of unstable manifolds for transitive singular hyperbolic attractors, not for the whole set but for a subset which is visited infinitely many times by a resiudal subset of the attractor. Here we give a complete proof of this theorem, in a slightly more general context. A consequence of these techniques is that they provide a sufficient condition for the C 1 -robust transitivity.