11 research outputs found
On almost-sure versions of classical limit theorems for dynamical systems
The purpose of this article is to construct a toolbox, in Dynamical Systems,
to support the idea that ``whenever we can prove a limit theorem in the
classical sense for a dynamical system, we can prove a suitable almost-sure
version based on an empirical measure with log-average''. We follow three
different approaches: martingale methods, spectral methods and induction
arguments. Our results apply among others to Axiom A maps or flows, to systems
inducing a Gibbs-Markov map and to the stadium billiard.Comment: 41 pages; submitted v2: replaced the argument for Gibbs-Markov maps
with a general spectral argumen
Rare events, escape rates and quasistationarity: some exact formulae
We present a common framework to study decay and exchanges rates in a wide
class of dynamical systems. Several applications, ranging form the metric
theory of continuons fractions and the Shannon capacity of contrained systems
to the decay rate of metastable states, are given
Entropic Fluctuations in Statistical Mechanics I. Classical Dynamical Systems
Within the abstract framework of dynamical system theory we describe a
general approach to the Transient (or Evans-Searles) and Steady State (or
Gallavotti-Cohen) Fluctuation Theorems of non-equilibrium statistical
mechanics. Our main objective is to display the minimal, model independent
mathematical structure at work behind fluctuation theorems. Besides its
conceptual simplicity, another advantage of our approach is its natural
extension to quantum statistical mechanics which will be presented in a
companion paper. We shall discuss several examples including thermostated
systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric
spaces and Anosov diffeomorphisms.Comment: 72 pages, revised version 12/10/2010, to be published in Nonlinearit
Amenability of groups and -sets
This text surveys classical and recent results in the field of amenability of
groups, from a combinatorial standpoint. It has served as the support of
courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure.
The goals of the text are (1) to be as self-contained as possible, so as to
serve as a good introduction for newcomers to the field; (2) to stress the use
of combinatorial tools, in collaboration with functional analysis, probability
etc., with discrete groups in focus; (3) to consider from the beginning the
more general notion of amenable actions; (4) to describe recent classes of
examples, and in particular groups acting on Cantor sets and topological full
groups
Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties
37 pagesInternational audienceCompact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set. This provides a self-contained theory that not only reproduces all the known classical results but gives also new insights on the statistical properties of these systems