Vincent Delecroix

Bio: Vincent Delecroix is an academic researcher from University of Bordeaux. The author has contributed to research in topics: Dynamical billiards & Moduli space. The author has an hindex of 12, co-authored 40 publications receiving 467 citations. Previous affiliations of Vincent Delecroix include Centre national de la recherche scientifique & Paris Diderot University.

Diffusion for the periodic wind-tree model

Abstract: The periodic wind-tree model is an infinite billiard in the plane with identical rectangular scatterers placed at each integer point. We prove that independently of the size of scatters and generically with respect to the angle, the polynomial diffusion rate in this billiard is 2/3.

Beyond substitutive dynamical systems: S-adic expansions

Abstract: An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. A fundamental example of this relation is between Sturmian sequences and regular continued fractions. We study S-adic words from different perspectives, namely word combinatorics, ergodic theory, and Diophantine approximation, by stressing the parallel with continued fraction expansions.

Interval Exchange Transformations

Abstract: These are lecture notes for 4 introductory talks about interval exchange transformations and translation surfaces given by the author in Salta (Argentina) in November 2016.

Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

Abstract: We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae are derived from lattice point count involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann surfaces. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani through completely different approach. We prove further result: up to an explicit normalization factor depending only on the genus and on the number of cusps, the density of the orbit of any simple closed multicurve computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to the simple closed multicurve. We study the resulting densities in more detail in the special case when there are no cusps. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are exponentially less frequent. We conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. This description is conditional to the conjectural asymptotic formula for the Masur-Veech volume in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of psi-classes in large genera.

Weak mixing directions in non-arithmetic Veech surfaces

Abstract: We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square and hexagon). More generally, we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non-arithmetic Veech surface, and show that the Hausdorff dimension of the set of non-weak mixing directions is not full. We also provide a necessary condition, verified for instance by the Veech surface corresponding to the billiard in the pentagon, for the set of non-weak mixing directions to have positive Hausdorff dimension.

The number of ramified coverings of the sphere by the torus and surfaces of higher genera

Abstract: We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification points. In addition, the conjecture is proved for simple coverings of the sphere by the torus. We obtain corresponding expressions for surfaces of higher genera for a small number of ramification points, and conjecture the general form for this number in terms of a symmetric polynomial that appears to be new. The approach involves the analysis of the action of a transposition to derive a system of linear partial differential equations that give the generating series for the desired numbers.

On inequalities for the gamma function

Abstract: We present various inequalities for the classical gamma function of Euler. Among others, we prove the following result: Let α be a real number. For all x, y ∈ (0, 1] and for all x, y ∈ [1,∞) we have x Γ(x) + yΓ(y) ≤ 1 + (xy)Γ(xy). The constant 1 is sharp.

Beyond substitutive dynamical systems: S-adic expansions

Abstract: An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. A fundamental example of this relation is between Sturmian sequences and regular continued fractions. We study S-adic words from different perspectives, namely word combinatorics, ergodic theory, and Diophantine approximation, by stressing the parallel with continued fraction expansions.

Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations,flows on surfaces and billiards

Abstract: This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'' (from 4 to 16 July, 2011).   In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmuller and moduli space of translation surfaces, the Teichmuller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmuller geodesic flow. We sketch two applications of the ergodic properties of the Teichmuller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmuller flow and the Kontsevich--Zorich cocycle work as renormalization dynamics for interval exchange transformations and translation flows.   In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmuller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).

The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion

Abstract: We study periodic wind-tree models, unbounded planar billiards with periodically located rectangular obstacles. For a class of rational parameters we show the existence of completely periodic directions, and recurrence; for another class of rational parameters, there are directions in which all trajectories escape, and we prove a rate of escape for almost all directions. These results extend to a dense $G_\delta$ of parameters.