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

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

Opuscule mathématique sur les trajectoires des géodésiques et des horocycles

Ce texte est une synthèse sur la dynamique topologique des flots géodésique et horocyclique sur les fibrés unitaires tangents des surfaces hyperboliques. La démarche choisie pour aborder ce thème est de relier la dynamique de ces flots à celle de l'action du groupe fondamental de la surface sur le demi-plan de Poincaré. Des exemples sont traités explicitement : la surface modulaire a été privilégiée pour son lien avec l'arithmétique et les quotients par les groupes de Schottky pour l'approche symbolique qu'ils permettent. Des applications à l'étude des actions linéaires et à la théorie des approximations diophantiennes sont développées

On the growth of nonuniform lattices in pinched negatively curved manifolds

We study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices. These objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold

Points de vue sur les valeurs aux entiers des formes quadratiques binaires

Papers from the Mathematical Days X-UPS held at the École Polytechnique, Palaiseau, May 9--10, 2007The primary focus of this largely expository paper is to present different points of view on the values assumed by binary quadratic forms, sometimes with irrational coefficients, at integer values. Three different points of view are taken. One is the traditional approximation of real numbers by continued fractions, a second brings in the actions of the group ${\rm SL}_2(\Bbb{Z})$, and the third involves topological considerations of quotient spaces