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A Course In Functional Analysis

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

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/104348/1/0347-3.pdf

複素解析的特異葉層構造 (Topology of Foliations)

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

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

Nons nous interessons au spectre des longueurs associe a une variete de courbure negative. Nous demontrons que le spectre des longueurs d'une surface n'est pas inclus dans un sous-groupe discret de ℝ. Nous comparons egalement le spectre des longueurs de differentes structures Riemanniennes sur une meme variete.

Remarques sur le spectre des longueurs d'une surface et comptages

Soient X une variete de Hadamard de courbure ≤ -1 et Γ un groupe d'isometries non elementaire. Nous montrons qu'il y a equivalence entre la non-arithmeticite du spectre des longueurs de Γ\X le melange topologique du flot geodesique et l'existence d'une feuille dense pour le feuilletage fortement stable.

/pdf/topologie-du-feuilletage-fortement-stable-3sax19d9nd.pdf

Françoise Dal'Bo

Papers

Séries de poincaré des groupes géométriquement finis

Remarques sur le spectre des longueurs d'une surface et comptages

Topologie du feuilletage fortement stable

Études spectrales d'opérateurs de transfert et applications

Some negatively curved manifolds with cusps, mixing and counting