photo

Vincent Delecroix


Publications

  1. with A. Zorich
    "Cries and whispers in wind-tree forests"
    What's Next?: The Mathematical Legacy of William P. Thurston
    arXiv:1502.06405
  2. with E. Goujard P. Zograf A. Zorich
    "Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes"
    Quelques aspects de la théorie des systèmes dynamiques : un hommage à Jean-Christophe Yoccoz (volume I)
    arXiv:1903.10904
    We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials.
  3. with C. Matheus C. G. Moreira
    " Approximation of the Lagrange and Markov spectra "
    Math. Comp. (2020)
    arXiv:1908.03773
    The (classical) Lagrange spectrum is a closed subset of the positive real numbers defined in terms of diophantine approximation. Its structure is quite involved. This article describes a polynomial time algorithm to approximate it in Hausdorff distance. It also extends to approximate the Markov spectrum related to infimum of binary quadratic forms.
  4. with E. Goujard P. Zograf A. Zorich
    "Enumeration of meanders and Masur-Veech volumes"
    Forum of Mathematics, Pi (2020)
    A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with crossings grows exponentially when grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.
  5. with A. Avila
    "Some monoids of Pisot matrices"
    arXiv:1506.03692
    A matrix norm gives an upper bound on the spectral radius of a matrix. Knowledge on the location of the dominant eigenvector also leads to upper bound of the second eigenvalue. We show how this technique can be used to prove that certain semi-group of matrices arising from continued fractions have a Pisot spectrum: namely for all primitive matrices in this semi-group all eigenvalues except the dominant one is smaller than one in absolute value.
  6. with J.-F. Bertazzon
    "Sommes de Birkhoff itérées sur des extensions finies d'odomètres. Construction de solutions auto-similaires à des équations différentielles avec délai"
    Bull. SMF 146 (2018)
    arXiv:1403.2235
    Nous étudions les sommes de Birkhoff itérées de fonctions sur certains systèmes dynamiques substitutifs. Les fonctions que nous regardons ont la propriété d'avoir toutes leurs sommes de Birkhoff itérées bornées (ce sont en particulier des cobords). Nous construisons une fonction continue comme limite de ces sommes de Birkhoff et montrons qu'elle vérifie une équation fonctionnelle. Cet article prolonge l'étude du premier auteur dans arXiv:1201.2502.
  7. 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|).
  8. 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
  9. 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)).
  10. 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)
  11. 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))
  12. 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.
  13. 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).
  14. "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.
  15. "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 M. Bell V. Gadre R. Gutiérrez-Romo S. Schleimer
    "Coding Teichmüller flow using veering triangulations"
    arXiv:1909.00890
  • with J. E. Andersen G. Borot S. Charbonnier A. Giacchetto D. Lewanski C. Wheeler
    "Topological recursion for Masur-Veech volumes"
    arXiv:1905.10352
  • with A. Aggarwal E. Goujard P. Zograf A. Zorich
    "Conjectural large genus asymptotics of Masur-Veech volumes and of area Siegel-Veech constants of strata of quadratic differentials"
    arXiv:1912.11702
  • "Asymptotics of lieanders with fixed composition sizes"
    arXiv:1812.03912

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