François Béguin

Aperiodic oscillatory asymptotic behavior for some Bianchi spacetimes

We study the asymptotic behavior of type A vacuum Bianchi spacetimes. It has been conjectured that this behavior is driven by a certain circle map, called the Kasner map. As a step towards this conjecture, we prove that some orbits of the Kasner map do indeed attract some solutions of the system of ODEs which describes the behavior of type A vacuum Bianchi spacetimes. The orbits of the Kasner map for which we can prove such a result are those which are not periodic and which do not accumulate on any periodic orbit. This shows the existence of Bianchi spacetimes with a aperiodic oscillatory asymptotic behavior.

Denjoy constructions for fibered homeomorphisms of the torus

We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincare-like classification by Jager and Stark for this class of maps, we demonstrate that transitive but non-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification. Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that in some of the examples we construct, the unique minimal set is a Cantor set and its intersection with each vertical fibre is uncountable and nowhere dense (but may contain isolated points). We also prove that minimal sets of the latter kind cannot occur when the dynamics are given by the projective action of a quasiperiodic SL(2, R)-cocycle. More precisely, we show that for a quasiperiodic SL(2, ℝ)-cocycle, any minimal proper subset of the torus either is a union of finitely many continuous curves or contains at most two points on generic fibres.

Pseudo-rotations of the closed annulus: variation on a theorem of J Kwapisz

Consider a homeomorphism h of the closed annulus , isotopic to the identity, such that the rotation set of h is reduced to a single irrational number ? (we say that h is an irrational pseudo-rotation). For every positive integer n, we prove that there exists a simple arc ? joining one of the boundary components of the annulus to the other, such that ? is disjoint from its n first iterates under h. As a corollary, we obtain that the rigid rotation of angle ? can be approximated by homeomorphisms that are conjugate to h. The first result stated above is an analogue of a theorem of Kwapisz dealing with diffeomorphisms of the two-torus; we give some new, purely two-dimensional, proofs of this theorem, that work both for the annulus and for the torus case.

Flots de Smale en dimension 3: presentations finies de voisinages invariants d'ensembles selles

Abstract Given a vector field X on a compact 3-manifold, and a hyperbolic saddle-like set K of that vector field, we consider all the filtering neighbourhood of K: by such, we mean any submanifold which boundary is tranverse to X, the maximal invariant of which is equal to K and which intersection with every orbit of X is connected. Up to topological equivalence, there is only a finite number of such neighbourhoods. We give a finite combinatorial presentation of the global dynamics on any such neighbourhood. A key step is the construction of a unique model of the germ of X along K; this model is, roughly speaking, the simplest three-dimensional manifold and the simplest Smale flow exhibiting the germ considered above. Then, we give a combinatorial description of the surgeries leading back to the initial filtering neighbourhoods.