On the geometry of bifurcation currents for quadratic rational maps

to appear in Ergodic Th. and Dyn. Syst.International audienceWe describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1, 1)-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current

Ramification current, post-critical normality and stability of holomorphic endomorphisms of Pk\mathbb P^k

In the context of holomorphic families of endomorphisms of $\mathbb P^k$, we prove that stability in the sense of [BBD18] is equivalent to a summability condition for the post-critical mass and to the convergence of a suitably defined ramification current. This allows us to both simplify the approach of [BBD18] and better relate stability to post-critical normality.Comment: 15 page

Dip coating with colloids and evaporation

International audienceThere is a growing interest in coating hard and soft substrates with colloids, with numerous applications to optics and microelectronics [1]. A possibility to realize these substrates is to use dip coating with evaporation [2], i.e. to remove at constant speed a plate from a bath of colloids while drying occurs. This leads to several undesired effects: defects, heterogeneous deposition, fracture and de-lamination [1,3]. The problem is also difficult to model as three divergences may coexist at the contact line (CL) receding on the substrate [4-5] (and even in a advancing case [6]) : (1) divergence of viscous stresses, (2) divergence of evaporation as in the well known "coffee stain" effect [7-9], (3) and divergence of colloid concentration. In a recent paper we modeled the hydrodynamics in the vicinity of a moving, evaporating, contact line [4], and we found that in the dip coating case there should exist two different regimes at respectively low and high plate velocity, in which the deposed mean thickness should respectively decrease and increase with the plate velocity. This should lead to a minimum of the deposed thickness for a critical intermediate velocity. Up to a recent thesis in our group [5], this effect has never been evidenced in a dip coating experiment, though similar behaviors were found for deposition of phospholipids [10], and for colloids in a rather specific two-plate geometry (meniscus receding in a Hele-Shaw cell) [11-12]. We present here evidences in favour of this effect, revealed by this work, and we correct the model of ref.[4] which contained a mistake. A sketch of the experimental set up is suggested on Fig.1. A clean glass plate is plunged inside a colloidal suspension and removed from this bath at constant speed (V ranging between 50 µm/s and 5 cm/s), while deposition and evaporation takes place on the glass. We used silica suspensions (Klebosol silica sluries 50R50, 30R25 and 30R12) with three different particle diameters (12 nm, 25 nm and 50 nm), and two different volume concentrations (φ 0 =5% and 10%). The glass plate is cleaned and prepared before each experiment by the following protocole. First the glass surface is rub with a abrasive cerium oxid suspension (concentration 20%), cleaned with pure water, ethanol, and again pure water, and then let to dry. A plasma treatment is then imposed to the glass

Dip coating with colloids and evaporation

International audienceThere is a growing interest in coating hard and soft substrates with colloids, with numerous applications to optics and microelectronics [1]. A possibility to realize these substrates is to use dip coating with evaporation [2], i.e. to remove at constant speed a plate from a bath of colloids while drying occurs. This leads to several undesired effects: defects, heterogeneous deposition, fracture and de-lamination [1,3]. The problem is also difficult to model as three divergences may coexist at the contact line (CL) receding on the substrate [4-5] (and even in a advancing case [6]) : (1) divergence of viscous stresses, (2) divergence of evaporation as in the well known "coffee stain" effect [7-9], (3) and divergence of colloid concentration. In a recent paper we modeled the hydrodynamics in the vicinity of a moving, evaporating, contact line [4], and we found that in the dip coating case there should exist two different regimes at respectively low and high plate velocity, in which the deposed mean thickness should respectively decrease and increase with the plate velocity. This should lead to a minimum of the deposed thickness for a critical intermediate velocity. Up to a recent thesis in our group [5], this effect has never been evidenced in a dip coating experiment, though similar behaviors were found for deposition of phospholipids [10], and for colloids in a rather specific two-plate geometry (meniscus receding in a Hele-Shaw cell) [11-12]. We present here evidences in favour of this effect, revealed by this work, and we correct the model of ref.[4] which contained a mistake. A sketch of the experimental set up is suggested on Fig.1. A clean glass plate is plunged inside a colloidal suspension and removed from this bath at constant speed (V ranging between 50 µm/s and 5 cm/s), while deposition and evaporation takes place on the glass. We used silica suspensions (Klebosol silica sluries 50R50, 30R25 and 30R12) with three different particle diameters (12 nm, 25 nm and 50 nm), and two different volume concentrations (φ 0 =5% and 10%). The glass plate is cleaned and prepared before each experiment by the following protocole. First the glass surface is rub with a abrasive cerium oxid suspension (concentration 20%), cleaned with pure water, ethanol, and again pure water, and then let to dry. A plasma treatment is then imposed to the glass