with M. Boshernitzan
" From a packing problem to quantiative recurrence in [0,1] and the Lagrange spectrum of interval exchanges"
Discrete Analysis 1749 (2017)
This article provides optimal constants for two quantitative recurrence problems. First of all for recurrence of maps of the interval [0,1] that preserve the Lebesgue measure. On the other hand, we study the bottom of the Lagrange spectrum of interval exchange transformations. Both results are based on a unconventional packing problem in the plane with respect to the "pseudo-norm" N(x,y) = sqrt(|xy|).
with V. Berthé F. Dolce D. Perrin C. Reutenauer G. Rindone
"Return words of linear involutions and fundamental groups"
Erg. Th. and Dyn. Sys. 37 , n° 3 (2017) p. 693-715
We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions as Poincaré maps of measured foliations, a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet A. The set of return words with respect to a subgroup of finite index G of the free group on A is also proved to be a symmetric basis of G
with A. Avila
"Weak-mixing directions in non-arithmetic Veech surfaces"
J. of AMS 29 (2016) p. 1167-1208
In this paper we prove the genericity of weak mixing in non-arithmetic Veech surfaces (arithmetic is synonym for square tiled). We know since the work A. Avila and G. Forni (Ann. of Math. 165 (2007), see also arXiv:math/0406326) that weak-mixing is prevalent in the space of translation surfaces. Nevertheless, there was no known example of surfaces for which the weak-mixing is prevalent in almost every direction. Our result applies in particular to billiard in regular polygons studied by Veech (Inv. Math. 97 (1989)) and billiard in L-shaped tables introduced by C. McMullen (J. Amer. Math. Soc. 16 (2003)).
with C. Mathéus
"Un contre-exemple à la réciproque du critère de Forni pour la positivité des exposants de Lyapunov du cocycle de Kontsevich-Zorich"
Math Res. Lett. 22 , n° 6 (2015) p. 1667-1678
Forni proved that a certain geometric quantity gives a lower bound for the number of positive Lyapunov exponents of the Kontsevich-Zorich cocycle (J. Mod. Dyn. 5, No. 2 (2011), see also arXiv:1009.4655). In this short note, we exhibit an example for which the geometric criterion is not satisfied but for which all Lyapunov exponents are positive. In order to prove positivity, we use a result of C. Matheus, M. Moeller and J.-C. Yoccoz (arXiv:1305.2033)
with C. Ulcigrai
"Diagonal changes in hyperelliptic components. A natural extension to Ferenczi-Zamboni induction"
Geom. Ded. 176 , n° 1 (2015) p. 117-174
In this article, we introduce an induction scheme for translation surfaces in hyperelliptic strata. It can be considered as a geometric counterpart to Ferenczi-Zamboni construction (J. Analyse Math. 112 (2010))
with V. Berthé
"Beyond substitutive dynamical systems: S-adic expansions"
RIMS Kôkyûroku Bessatsu B46 (2014) p. 81-123
Self-similar dynamical systems are example of highly structured dynamical systems. But there are much more systems of low complexity. One way to consider all of them is to slightly weaken the notion of self-similarity: we allow to see different patterns at different scales but all of them belong to a fixed family. The combinatorial counterpart of this construction are the so-called S-adic systems.
with P. Hubert S. Lelièvre
"Diffusion for the periodic wind-tree model"
Ann. Sc. ENS 47 , n° 6 (2014) p. 1085-1110
The wind-tree model is a billiard in the plane where scatterers are rectangles randomly displaced. We study a periodic version and prove that the diffusion rate is 2/3. More precisely the maximum distance reached by a particule before time T is around T^(2/3). It makes a large difference with random walks in the plane for which that quantity equals T^(1/2).
"Divergent directions in some periodic wind-tree models"
J. of Mod. Dyn. 7 , n° 1 (2013) p. 1-29
We prove that for many choice of rectangular obstacles, there exists divergent directions in the windtree model.
"Cardinality of Rauzy classes"
Ann. Inst. Fourier 63 , n° 5 (2013) p. 1651-1715
Rauzy classes are set of permutations that appear in a renormalization scheme of interval exchange transformations introduced by Rauzy and further studied by Veech. In this article we provide a formula for the cardinalities of Rauzy classes and make a conjecture about their asymptotics.
with E. Goujard P. Zograf A. Zorich
"Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata"
with J.-F. Bertazzon
"Étude d'une équation intégrale avec des méthodes combinatoires"
with V. Berthé F. Docle D. Perrin C. Reteunauer G. Rindone
"Natural coding of linear involutions"
with A. Zorich
"Cries and whispers in wind-tree forests"
with A. Avila
"Some monoids of Pisot matrices"
Articles de conférences
- lundi 21 août 2017 (IMA, Minneapolis, USA):
- Conférence: Sage Days: Opening Workshop for a Year of Coding Sprints
- (Page de la présentation avec une vidéo)
- pdf présentation
- feuille de travail sur l'optimisation linéaire ipynb pdf
- feuille de travail sur les fractions continues ipynb pdf
- feuille de travail sur les problèmes avec les expressions symboliques ipynb pdf
Conférences, groupes de travail, etc
A la rentrée 2017 devrait commencer un groupe de travail sur les cartes combinatoires et la géométrie.
J'ai coorganisé avec Pascal Hubert et Erwan Lanneau un groupe de travail sur les travaux de Benoist-Quint (la page contient quelques notes de cours et des références).
Organisation de l'école MathExp 2018 à Saint-Flour au printemps 2018.