We compute the divisor of Selberg's zeta function for convex co-compact, torsion-free discrete groups acting on a real hyperbolic space of dimension n + 1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = nH together with the Euler characteristic of X compacti ed to a manifold with boundary. If n is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identi ed using work of Bunke and Olbrich.

The divisor of Selberg's zeta function for Kleinian groups

To measure the dependence between a real-valued random variable X and a σ-algebra , we consider four distances between the conditional distribution function of X given and the distribution function of X. The coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions and dynamical systems. Starting from a new covariance inequality, we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence. We obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis, under a minimal condition on the coefficients. Using recent results, we show that our coefficients may be also used to obtain various exponential inequalities, a concentration inequality for Lipschitz functions, and a Berry-Esseen type inequality.

New dependence coefficients. Examples and applications to statistics

We prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hattractors.

/pdf/large-deviations-in-non-uniformly-hyperbolic-dynamical-2vtyr021zv.pdf

Large deviations in non-uniformly hyperbolic dynamical systems

With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many existence, uniqueness and finiteness results of Gibbs measures. We give many applications, to the Variational Principle, the counting and equidistribution of orbit points and periods, the unique ergodicity of the strong unstable foliation and the classification of Gibbs densities on some Riemannian covers.

/pdf/equilibrium-states-in-negative-curvature-3fkdc2klow.pdf

Equilibrium States in Negative Curvature

Let X be a real-valued random variable and 
$$M$$

 a σ-algebra. We show that the minimum 
$${\mathbb{L}}^1$$

 -distance between X and a random variable distributed as X and independant of 
$$M$$

 can be viewed as a dependence coefficient τ(
 
$$M$$

 ,X) whose definition is comparable (but different) to that of the usual β-mixing coefficient between 
$$M$$

 and σ(X). We compare this new coefficient to other well known measures of dependence, and we show that it can be easily computed in various situations, such as causal Bernoulli shifts or stable Markov chains defined via iterative random maps. Next, we use coupling techniques to obtain Bennett and Rosenthal-type inequalities for partial sums of τ-dependent sequences. The former is used to prove a strong invariance principle for partial sums.

Coupling for τ-Dependent Sequences and Applications

In this paper, we study the behaviour of the Poincare series of a geometrically finite group Γ of isometries of a riemannian manifoldX with pinched curvature, in the case when Γ contains parabolic elements. We give a sufficient condition on the parabolic subgroups of Γ in order that Γ be of divergent type. When Γ is of divergent type, we show that the Sullivan measure on the unit tangent bundle ofX/Γ is finite if and only if certain series which involve only parabolic elements of Γ are convergent. We build also examples of manifoldsX on which geometrically finite groups of convergent type act.

Séries de poincaré des groupes géométriquement finis

Études spectrales d'opérateurs de transfert et applications

On a class of hyperbolic manifolds with infinite volume, we give an asymptotic estimate for the number of closed geodesics in a given homology class. We show that, in certain cases, the existence of parabolic transformations in the fundamental group Γ of these manifolds has an effect on this estimate. This happens when the Hausdorff dimension of the limit set of Γ is less than 3/2. The geometrical meaning of this critical value remains to be understood.

Marc Peigné

Papers

Séries de poincaré des groupes géométriquement finis

Études spectrales d'opérateurs de transfert et applications

Closed geodesics in homology classes on hyperbolic manifolds with cusps