Marc Peigné

Principe variationnel et groupes Kleiniens

Let Γ be a non-elementary Kleinian group acting on a Cartan-Hadamard manifold X̃ ; denote by Λ(Γ) the non-wandering set of the geodesic flow (φt) acting on the unit tangent bundle T (X̃/Γ). When Γ is convex cocompact (i.e. Λ(Γ) is compact), the restriction of (φt) to Λ(Γ) is an Axiom A flow : therefore, by a theorem of Bowen-Ruelle, there exists a unique invariant measure on Λ(Γ) which has maximal entropy. In this paper, we study the case of an arbitrary Kleinian group Γ. We show that there exists a measure of maximal entropy for the restriction of (φt) to Λ(Γ) if and only if the Patterson-Sullivan measure is finite ; furthermore when this measure is finite, it is the unique measure of maximal entropy. By a theorem of Handel-Kitchens, the supremum of the measure-theoretic entropies equals the infimum of the entropies of the distances d on Λ(X) ; when Γ is geometrically finite, we show that this infimum is achieved by the Riemannian distance d on Λ(X).

On the Patterson-Sullivan measure of some discrete group of isometries

In this paper, we describe a large class of groups of isometries of thed-dimensional hyperbolic space. These groups may be non-geometrically finite but their Patterson-Sullivan measure is always finite.

Homologie des géodésiques fermées sur des variétés hyperboliques avec bouts cuspidaux

Abstract The presence of cusps in hyperbolic manifolds has an influence on the homological behavior of closed geodesics. This phenomenon is studied here for a class of geometrically finite manifolds obtained as quotients of the hyperbolic space by free products of abelian groups acting in a Schottky way. We get in particular an exact estimate for the number of closed geodesics in a fixed homology class which depends in a peculiar way of the Hausdorff dimension of the limit set with a transition at certain half-integer values.

Local limit theorems on some non unimodular groups

Let Gd be the semi-direct product of R*+ and Rd, d = 1 and let us consider the product group Gd,N = Gd x RN, N = 1. For a large class of probability measures µ on Gd,N, one prove that there exists ?(µ) I ]0,1] such that the sequence of finite measures {(n(N+3)/2 / ?(µ)n) µ*n}n = 1 converges weakly to a non-degenerate measure.

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 .

The survival probability of a critical multi-type branching process in i.i.d. random environment

Conditioned on the generating functions of offspring distribution, we study the asymp-totic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments by using limits theorems for products of positive random matrices. Under some certain assumptions, the survival probability is proportional to 1/ √ n.

Convergence and Counting in Infinite Measure@@@Convergence et comptage en mesure infinie

We construct convergent and divergent lattices in negative curvature and give a precise asymptotic description of the behavior of their counting function.

Analytic and probabilistic approaches to dynamics in negative curvature

1 S. Le Borgne: Martingales in Hyperbolic Geometry.- 2 F. Faure, M. Tsujii: Semi classical Approach for the Ruelle-Pollicott Spectrum of Hyperbolic Dynamics.

Conditioned Random Walk in Weyl Chambers and Renewal Theory

We present here the main result from [8] and explain how to use Kashiwara crystal basis theory to associate a random walk to each minuscule irreducible representation V of a simple Lie algebra; the generalized Pitman transform defined in [10] for similar random walks with uniform distributions yields yet a Markov chain when the crystal attached to V is endowed with a probability distribution compatible with its weight graduation. The main probabilistic argument in our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice [8]. We present some explicit examples, which can be computed using insertion schemes on tableaux described in [9].