For a large class of tilings of R d , including the Penrose tiling in dimension 2 as well as the icosahedral ones in dimension 3, the continuous hull Ω T of such a tiling T inherits a minimal R d -lamination structure with flat leaves and a transversal Γ T which is a Cantor set. In this case, we show that the continuous hull can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact d-manifolds. Truncated sequences furnish better and better finite approximations of the asymptotic dynamical system and the algebraic topological features related to this sequence reflect the dynamical properties of the R d -action on the continuous hull. In particular the set of positive invariant measures of this action turns to be a convex cone, canonically associated with the orientation, in the projective limit of the d th -homology groups of the branched manifolds. As an application of this construction we prove a gap-labelling theorem:Consider the C * -algebra A T of Ω T , and the group K 0 (A T ), then for every finite R d -invariant measure µ on Ω T , the image of the group K 0 (A T ) by the µ-trace satisfies:where µ t is the transverse invariant measure on Γ T induced by µ and C(Γ T , Z) is the set of continuous functions on Γ T with integer values.

For any compact oriented surface we consider the group of diffeomorphisms of which preserve a given area form. In this paper we show that the vector space of homogeneous quasi-morphisms on this group has infinite dimension. This result is proved by constructing explicitly and for each surface an infinite family of independent homogeneous quasi-morphisms. These constructions use simple arguments related to linking properties of the orbits of the diffeomorphisms.

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Abstract. -A braid defines a link which has a signature. This defines a map from the braid group to the integers which is not a homomorphism. We relate the homomorphism defect of this map to Meyer cocycle and Maslov class. We give some information about the global geometry of the gordian metric space.Résumé (Tresses et signatures). -Une tresse définit un entrelacs qui possède une signature. Ceci définit une application du groupe des tresses vers les entiers qui n'est pas un homomorphisme. Nous relions le défaut d'homomorphisme de cette application au cocycle de Meyer età la classe de Maslov. Nous donnons quelques informations sur la géométrie globale de l'espace métrique gordien.

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