Cohomology classes of dynamically non-negative Ck functions

Let T:x↦2x (mod 1) be the doubling map of the circle ?=ℝ/ℤ. We construct a trigonometric polynomial f:?→ℝ with the following property: ∫f  dμ≥0 for every T-invariant probability measure μ, so that f is cohomologous to a non-negative Lipschitz function, yet f is not cohomologous to any non-negative C ¹ function. Oblatum 28-VI-2001 & 4-X-2001?Published online: 18 January 2002     

Sur les retardateurs

A discrete-time retarder is a stochastic increasing function , whose distribution is invariant by conjugation by the translations of Z. Continuous-time retarders can be defined similarly.About these objects, which can be composed, we prove theorems of two kinds. First, we establish various inequalities about the composition and iteration of retarders whose delay is (p−1)-integrable. Then we prove a super-multiplicative ergodic theorem, which can be used to define the “cycle time” of a retarder.     

Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture

Given an Iterated Function System (IFS) of topical maps verifying some conditions, we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional, the average height, on some compact space of measures. We give general results to determine these extrema, and then apply them to two concrete problems. First, we give a new proof of the theorem that the densest heaps of two Tetris pieces are sturmian. Second, we construct an explicit counterexample to the Lagarias-Wang finiteness conjecture for the joint spectral radius of a set of matrices.

RÉSUMÉ. Etant donné un système itéré de fonctions (IFS) topicales, vérifiant certaines conditions, nous montrons que les questions d'optimisation asymptotique de la hauteur sont équivalentes à la recherche des extrema d'une fonctionnelle continue, la hauteur moyenne, sur un certain espace compact de mesures. Nous présentons des résultats généraux permettant de déterminer ces extrema, puis appliquons ces méthodes à deux problèmes concrets. Premièrement, nous redémontrons que les empilements les plus denses de deux pièces de Tetris sont sturmiens. Deuxièmement, nous construisons un contre-exemple effectif à la conjecture de finitude de Lagarias et Wang sur le rayon spectral joint d'un ensemble de matrices.

     

Un lemme de Mañé bilatéralA bilateral version of Mañé's lemma

We prove that, assuming some hyperbolicity on the dynamical system T:X→X and some regularity on $\text{f:X→}\text{R}$, there exists $\text{θ:X→}\text{R}$ in the same regularity class and such that α(f)?f?θ+θ°T?β(f), where α(f), β(f) are the infimum and the supremum of the averages of f along periodic orbits. To cite this article: T. Bousch, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 533–536.     

Fluorogenic Photo-Crosslinking of Glycan-Binding Protein Recognition Using a Fluorinated Azido-Coumarin Fucoside

Glycan recognition by glycan-binding proteins is central to the biology of all living organisms. The efficient capture and characterization of relatively weak non-covalent interactions remains an important challenge in various fields of research. Photoaffinity labeling strategies can create covalent bonds between interacting partners, and photoactive scaffolds such as benzophenone, diazirines and aryl azides have proved widely useful. Since their first introduction, relatively few improvements have been advanced and products of photoaffinity labeling remain difficult to detect. We report a fluorinated azido-coumarin scaffold which enables photolabeling under fast and mild activation, and which can leave a fluorescent tag on crosslinked species. Coupling this scaffold to an α-fucoside, we demonstrate fluorogenic photolabeling of glycan-protein interactions over a wide range of affinities. We expect this strategy to be broadly applicable to other chromophores and we envision that such “fluoro-crosslinkers” could become important tools for the traceable capture of non-covalent binding events.