A spectral-like decomposition for transitive Anosov flows in dimension three

Given a (transitive or non-transitive) Anosov vector field $X$ on a closed three-dimensional manifold $M$, one may try to decompose $(M,X)$ by cutting $M$ along two-tori transverse to $X$. We prove that one can find a finite collection $\{T_1,\dots,T_n\}$ of pairwise disjoint, pairwise non-parallel incompressible tori transverse to $X$, such that the maximal invariant sets $\Lambda_1,\dots,\Lambda_m$ of the connected components $V_1,\dots,V_m$ of $M-(T_1\cup\dots\cup T_n)$ satisfy the following properties: 1, each $\Lambda_i$ is a compact invariant locally maximal transitive set for $X$, 2, the collection $\{\Lambda_1,\dots,\Lambda_m\}$ is canonically attached to the pair $(M,X)$ (i.e., it can be defined independently of the collection of tori $\{T_1,\dots,T_n\}$), 3, the $\Lambda_i$'s are the smallest possible: for every (possibly infinite) collection $\{S_i\}_{i\in I}$ of tori transverse to $X$, the $\Lambda_i$'s are contained in the maximal invariant set of $M-\cup_i S_i$. To a certain extent, the sets $\Lambda_1,\dots,\Lambda_m$ are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition $V_1,\dots,V_m$, equipped with the restriction of the Anosov vector field $X$, are "almost unique up to topological equivalence".Comment: 22 pages, 4 figure

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, Application to the Minkowski problem in the Minkowski space

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in the 3-dimensional Minkowski space for datas that are invariant under the action of a co-compact Fuchsian group

Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique

Mary Rees has constructed a minimal homeomorphism of the 2-torus with positive topological entropy. This homeomorphism f is obtained by enriching the dynamics of an irrational rotation R. We improve Rees construction, allowing to start with any homeomorphism R instead of an irrational rotation and to control precisely the measurable dynamics of f. This yields in particular the following result: Any compact manifold of dimension d>1 which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy. More generally, given some homeomorphism R of a (compact) manifold and some homeomorphism h of a Cantor set, we construct a homeomorphism f which "looks like" R from the topological viewpoint and "looks like" R*h from the measurable viewpoint. This construction can be seen as a partial answer to the following realisability question: which measurable dynamical systems are represented by homeomorphisms on manifolds

Denjoy constructions for fibred homeomorphisms of the torus

We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincar\'e-like classification for this class of maps of Jaeger-Stark, 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 later 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 strict subset of the torus either is a union of finitely many continuous curves, or contains at most two points on generic fibres

Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on AdS3AdS_3

International audienceWe prove that any maximal globally hyperbolic spacetime locally modelled on the anti-de Sitter space of dimension $3$, and admitting a closed Cauchy surface, admits a time function $\tau$, such that every fiber $\tau^{-1}(t)$ is a spacelike surface with constant mean curvature $t$

The Nymph Architect of the Cicada <em>Guyalna chlorogena</em>: Behaviours and Ecosystem

At the beginning of the last year of its larval life, the nymph of Guyalna chlorogena builds, from a vertical well, which is the result of a verticalization process from a deep horizontal gallery, a clay turret 20 to 40 cm high which appears as a regulating device of the physico-chemical conditions inside the burrow. The construction of the turret is remarkable for its finish. The nymph maintains, repairs and rebuilds it if necessary. It opens and closes it under certain circumstances. Before moulting, the nymph comes out at the top, opening it according to a set protocol and time schedule, using its chitins’ forelegs. The burrow is associated in a commensal relationship with arborescent Fabaceae species (of the Tachigali genus) through its nutrition mode, the suction of the elaborated sap in fine roots, close to the meristems