Xavier Bressaud

Necessary and Sufficient Conditions to be an Eigenvalue for Linearly Recurrent Dynamical Cantor Systems

We give necessary and sufficient conditions to have measurable and continuous eigenfunctions for linearly recurrent Cantor dynamical systems. We also construct explicitly an example of linearly recurrent system with nontrivial Kronecker factor and a trivial maximal equicontinuous factor.

Rate of approximation of minimizing measures

For T a continuous map from a compact metric space to itself and f a continuous function, we study the minimum of the integral of f with respect to the members of the family of invariant measures for T and in particular the rate at which this minimum is approached when the minimum is restricted to the family of invariant measures supported on periodic orbits of period at most N. We answer a question of Yuan and Hunt by demonstrating that the error of approximation decays faster than N−k for all k > 0 and show that this is sharp by giving examples for which the approximation error does not decay faster than this.

Speed of d-convergence for Markov approximations of chains with complete connections. A coupling approach☆

We compute the speed of convergence of the canonical Markov approximation of a chain with complete connections with summable decays. We show that in the d-topology the approximation converges at least at a rate proportional to these decays. This is proven by explicitly constructing a coupling between the chain and each range-k approximation.

Geometrical Models for Substitutions

We consider a substitution associated with the Arnoux–Yoccoz interval exchange transformation (IET) related to the tribonacci substitution. We construct the so-called stepped lines associated with the fixed points of the substitution in the abelianization (symbolic) space. We analyze various projections of the stepped line, recovering the Rauzy fractal, a Peano curve related to work in [Arnoux 88], another Peano curve related to the work of [McMullen 09] and [Lowenstein et al. 07], and also the interval exchange transformation itself.

On the eigenvalues of finite rank Bratteli-Vershik dynamical systems

A first embodiment of the present invention is directed to an aqueous disinfectant solution comprising an alkali metal or alkaline earth metal hypochlorite, an amount of base sufficient to raise the pH of the solution to at least 12, and water. A second embodiment of the present invention is directed to a method for sterilizing medical and dental instruments and hard surfaces which comprises contacting the medical or dental instruments or hard surfaces with an aqueous disinfecting solution comprising an alkali metal or alkaline earth metal hypochlorite, an amount of a base sufficient to raise the pH of the solution to at least 12, and water, for a time sufficient to disinfect the medical or dental instruments or hard surface.

Persistence of wandering intervals in self-similar affine interval exchange transformations

In this article we prove that given a self-similar interval exchange transformation T, whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of M. Cobo in [C].

Non Exponential Law of Entrance Times in Asymptotically Rare Events for Intermittent Maps with Infinite Invariant Measure

Abstract. We study piecewise affine maps of the interval with an indifferent fixed point causing the absolutely continuous invariant measure to be infinite. Considering the laws of the first entrance times of a point – picked at random according to Lebesgue measure – into a sequence of events shrinking to the strongly repelling fixed point, we prove that (when suitably normalized) they converge in distribution to the independent product of an exponential law to some power and a one-sided stable law.

A NORMAL FORM FOR BRAIDS

We present a seemingly new normal form for braids, where every braid is expressed using a word in a regular language on some simple alphabet of elementary braids. This normal form stems from analysing the geometric action of braid groups on curves in a punctured disk.

Deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1

Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of mod-ulus 1. The functions γ we consider are the corresponding eigen-functions. In Theorem 1.1 we prove that the limit inferior of the ergodic sums (n, γ(x 0) +. .. + γ(x n−1)) n∈N is bounded for every point x in the phase space. In Theorem 1.2, we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length. Under additional assumptions, we prove that ergodic integrals satisfy the Central Limit Theorem (Theorem 1.3, Theorem 1.9).

STATIONARY PROCESSES WHOSE FILTRATIONS ARE STANDARD

We study the standard property of the natural filtration associated to a 0-1 valued stationary process. In our main result we show that if the process has summable memory decay, then the associated filtration is standard. We prove it by coupling techniques. For a process whose associated filtration is standard, we construct a product type filtration extending it, based upon the usual couplings and the Vershiks criterion for standardness.