Groupoïde d'homotopie d'un feuilletage riemannien et réalisation symplectique de certaines variétés de poisson

The purpose of this paper is to prove the existence of a symplectic realization for a large classe of regular Poisson manifolds with riemannian two dimensional characteristic foliation. To do so, we will show that the homotopy groupoid of a riemannian foliation is locally trivial

Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces

This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this flow assuming that the natural affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G. A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of PSL(2,R) and a Lie group G, which places us within the homogeneous framework investigated by M. Ratner. Since our purpose is to deal with non-homogeneous situations, we do not use Ratner's famous Orbit-Closure Theorem, but we give an elementary proof. We show that this special situation arises for homogeneous Riemannian and Lie foliations, reintroducing the foliation point of view. Examples and counter-examples take an important place in our work, in particular, the very instructive case of the hyperbolic torus bundles over the circle. Our aim in writing this text is to offer to the reader an accessible introduction to a subject that was intensively studied in the algebraic setting, although there still are unsolved geometric problems.Comment: Final version to appear in Expositiones Mathematica

Horocycle flow on flat projective bundles: from topology to measures

In this paper we study some topological and measurable aspects of the dynamics of the foliated horocycle flow on flat projective bundles over hyperbolic surfaces. If $\rho : \Gamma \to {\rm PSL}(n+1,\mathbb{R})$ is a representation of a non-elementary Fuchsian group $\Gamma$, the unit tangent bundle $Y$ associated to the flat projective bundle defined by $\rho$ admits a natural action of the affine group $B$ obtained by combining the foliated geodesic and horocycle flows. If the image $\rho(\Gamma)$ satisfies Conze-Guivarc'h condition, namely strong irreducibility and proximality, the dynamics of the $B$-action is captured by the proximal dynamics of $\rho(\Gamma)$ on $\mathbb{R}{\rm P}^n$ (Theorem A). In fact, the dynamics of the foliated horocycle flow on the unique $B$-minimal subset of $Y$ can be described in terms of dynamics of the horocycle flow on its non-wandering set in the unit tangent bundle $X$ of the base surface $S= \Gamma \backslash \mathbb{H}$ (Theorem B). Assuming the existence of a continuous limit map induced by $\rho$, an one-to-one correspondence between conservative ergodic invariant measures can be stated for the foliated horocycle flow on the proximal part of the non-wandering set in $Y$ and the horocycle flow on the non-wandering set in $X$ (Theorem C).Comment: 11 page

Suppressors of selection

Inspired by recent works on evolutionary graph theory, an area of growing interest in mathematical and computational biology, we present the first known examples of undirected structures acting as suppressors of selection for any fitness value $r > 1$. This means that the average fixation probability of an advantageous mutant or invader individual placed at some node is strictly less than that of this individual placed in a well-mixed population. This leads the way to study more robust structures less prone to invasion, contrary to what happens with the amplifiers of selection where the fixation probability is increased on average for advantageous invader individuals. A few families of amplifiers are known, although some effort was required to prove it. Here, we use computer aided techniques to find an exact analytical expression of the fixation probability for some graphs of small order (equal to $6$, $8$ and $10$) proving that selection is effectively reduced for $r > 1$. Some numerical experiments using Monte Carlo methods are also performed for larger graphs.Comment: New title, improved presentation, and further examples. Supporting Information is also include

Sur l'intégration symplectique de la structure de poisson singuliére Λ = (x2 + y2) ∂/∂x Λ ∂/∂y de R2

The purpose of this note is to give an example of a singular Poisson structure on R2 which admits a symplectic realization by a Lie groupoid

A method for validating Rent's rule for technological and biological networks

Rent’s rule is empirical power law introduced in an effort to describe and optimize the wiring complexity of computer logic graphs. It is known that brain and neuronal networks also obey Rent’s rule, which is consistent with the idea that wiring costs play a fundamental role in brain evolution and development. Here we propose a method to validate this power law for a certain range of network partitions. This method is based on the bifurcation phenomenon that appears when the network is subjected to random alterations preserving its degree distribution. It has been tested on a set of VLSI circuits and real networks, including biological and technological ones. We also analyzed the effect of different types of random alterations on the Rentian scaling in order to test the influence of the degree distribution. There are network architectures quite sensitive to these randomization procedures with significant increases in the values of the Rent exponents