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Vincent Delecroix


Publications

  1. with M. Boshernitzan
    " From a packing problem to quantiative recurrence in [0,1] and the Lagrange spectrum of interval exchanges"
    Discrete Analysis 1749 (2017)
    arXiv:1608.04591
    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|).
  2. 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
    arXiv:1405.3529
    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
  3. with A. Avila
    "Weak-mixing directions in non-arithmetic Veech surfaces"
    J. of AMS 29 (2016) p. 1167-1208
    arXiv:1304.3318
    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)).
  4. 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
    arXiv:1103.1560
    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)
  5. with C. Ulcigrai
    "Diagonal changes in hyperelliptic components. A natural extension to Ferenczi-Zamboni induction"
    Geom. Ded. 176 , n° 1 (2015) p. 117-174
    arXiv:1310.1052
    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))
  6. with V. Berthé
    "Beyond substitutive dynamical systems: S-adic expansions"
    RIMS Kôkyûroku Bessatsu B46 (2014) p. 81-123
    arXiv:1309.3960
    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.
  7. with P. Hubert S. Lelièvre
    "Diffusion for the periodic wind-tree model"
    Ann. Sc. ENS 47 , n° 6 (2014) p. 1085-1110
    arXiv:1107.1810
    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).
  8. "Divergent directions in some periodic wind-tree models"
    J. of Mod. Dyn. 7 , n° 1 (2013) p. 1-29
    arXiv:1107.2418
    We prove that for many choice of rectangular obstacles, there exists divergent directions in the windtree model.
  9. "Cardinality of Rauzy classes"
    Ann. Inst. Fourier 63 , n° 5 (2013) p. 1651-1715
    arXiv:1106.0807
    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.

Prépublications

  • with E. Goujard P. Zograf A. Zorich
    "Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata"
    arXiv:1612.08374
  • with J.-F. Bertazzon
    "Étude d'une équation intégrale avec des méthodes combinatoires"
    arXiv:1403.2235
  • with V. Berthé F. Docle D. Perrin C. Reteunauer G. Rindone
    "Natural coding of linear involutions"
    arXiv:1405.3529
  • with A. Zorich
    "Cries and whispers in wind-tree forests"
    arXiv:1502.06405
  • with A. Avila
    "Some monoids of Pisot matrices"
    arXiv:1506.03692

Articles de conférences

  1. with D. Perrin V. Berthe C. De Felice J. Leroy C. Reutenauer G. Rindone
    "Specular sets"
    Words 2015
    arXiv:1505.00707
  2. with T. Hejda W. Steiner
    "Balancedness of Arnoux-Rauzy and Brun words"
    Words 2013
    arXiv:1308.6694

Présentations

Thèse

J'ai soutenu ma thèse sous la direction d'Arnaldo Nogueira le 16 Novembre 2011 présentation au format pdf. Mon mémoire est disponible en version longue (173 pages) et en version courte (57 pages).

Conférences, groupes de travail, etc

A la rentrée 2017 devrait commencer un groupe de travail sur les cartes combinatoires et la géométrie.

Au printemps 2014, j'ai animé des séances d'introduction au logiciel Sage à l'université de Bordeaux. Pour plus d'informations voir la page wiki.

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.