Publications

2010

• On the vanishing of some non semisimple orbital integrals
  
  • Chenevier Gaëtan
  • Renard David
  Expositiones Math., 2010, 28, pp.276-289. We prove the vanishing of the (possibly twisted) orbital integrals of certain functions on real Lie groups at non semisimple elliptic elements. This applies to Euler-Poincare functions and makes some results of Chenevier and Clozel unconditionnal.
• Some remarks about semiclassical trace invariants and quantum normal forms
  
  • Guillemin Victor
  • Paul Thierry
  Communications in Mathematical Physics, Springer Verlag, 2010, 294 (1), pp.1-19. In this paper we explore the connection between semi-classical and quantum Birkhoff canonical forms (BCF) for Schrödinger operators. In particular we give a "non-symbolic" operator theoretic derivation of the quantum Birkhoff canonical form and provide an explicit recipe for expressing the quantum BCF in terms of the semi-classical BCF. (10.1007/s00220-009-0920-3)
  DOI : 10.1007/s00220-009-0920-3
• Entropy of semiclassical measures for nonpositively curved surfaces
  
  • Riviere Gabriel
  Annales Henri Poincaré, Springer Verlag, 2010, 11, pp.1085-1116. We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. We follow the same main strategy as in the Anosov case (arXiv:0809.0230). We focus on the main differences and refer the reader to (arXiv:0809.0230) for the details of analogous lemmas. (10.1007/s00023-010-0055-2)
  DOI : 10.1007/s00023-010-0055-2
• Dynamics of meromorphic maps with small topological degree III : geometric currents and ergodic theory
  
  • Diller Jeffrey
  • Dujardin Romain
  • Guedj Vincent
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2010, 43 (2), pp.235-278. We continue our study of the dynamics of mappings with small topological degree on projective complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic "equilibrium" measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalizes results that were known in the invertible case and adds to the small number of situations in which a natural invariant measure for a non-invertible dynamical system is well-understood. (10.24033/asens.2120)
  DOI : 10.24033/asens.2120
• On flows of H^3/2-vector fields on the circle
  
  • Figalli Alessio
  Mathematische Annalen, Springer Verlag, 2010, 347 (1), pp.43-57. (10.1007/s00208-009-0426-5)
  DOI : 10.1007/s00208-009-0426-5
• The structure of popular difference sets
  
  • Wolf Julia
  Israel Journal of Mathematics, Springer, 2010, 179 (1), pp.253-278.
• On square roots of class Cm of nonnegative functions of one variable.
  
  • Bony Jean-Michel
  • Colombini Ferruccio
  • Pernazza Ludovico
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2010, 9 (3), pp.635-644.
• Note on torsion conjecture
  
  • Cadoret Anna
  • Tamagawa Akio
  Séminaires et congrès, Société mathématique de France, 2013, 27, pp.57-68.
• The true complexity of a system of linear equations
  
  • Gowers W. T.
  • Wolf Julia
  Proceedings of the London Mathematical Society, London Mathematical Society, 2010, 100 (1), pp.155-176. It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called "quadratic Fourier analysis", we show that certain linear systems that were previously thought to require quadratic uniformity are in fact governed by linear uniformity. More generally, we conjecture a necessary and sufficient condition on a linear system L which guarantees that any subset A of F_p^n which is uniform of degree k contains the expected number of solutions to L.
• Theorie ergodique des fractions rationnelles sur un corps ultrametrique
  
  • Favre Charles
  • Rivera-Letelier Juan
  Proceedings of the London Mathematical Society, London Mathematical Society, 2010, 100 (1), pp.116-154. We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure m_R which reprensents the asymptotic distribution of preimages of non-exceptional point. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of m_R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction. (10.1112/plms/pdp022)
  DOI : 10.1112/plms/pdp022
• Mass Transportation on Sub-Riemannian Manifolds
  
  • Figalli Alessio
  • Rifford Ludovic
  Geometric And Functional Analysis, Springer Verlag, 2010, 20 (1), pp.124-159. We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the Monge-Ampère equation. (10.1007/s00039-010-0053-z)
  DOI : 10.1007/s00039-010-0053-z
• The skein module of torus knots complements
  
  • Marche Julien
  Quantum Topol., 2010, 1, pp.413-421. We compute the Kauffman skein module of the complement of torus knots in S^3. Precisely, we show that these modules are isomorphic to the algebra of Sl(2,C)-characters tensored with the ring of Laurent polynomials.
• A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions
  
  • Figalli Alessio
  • Gigli Nicola
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2010, 94 (2), pp.107-130. (10.1016/j.matpur.2009.11.005)
  DOI : 10.1016/j.matpur.2009.11.005
• Dynamics of meromorphic maps with small topological degree I : from cohomology to currents
  
  • Diller Jeffrey
  • Dujardin Romain
  • Guedj Vincent
  Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2010, 59 (2), pp.521-561. We consider the dynamics of a meromorphic map on a compact Kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which iterates of the map expand the cohomology class of a Kahler form. Our goal in this article and its sequels is to carry out a program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, using the linear action of the map on cohomology to construct and analyze invariant currents with special geometric structure. We also give some examples and consider in more detail the special cases where the surface is irrational or the self-intersections of the invariant currents vanish. (10.1512/iumj.2010.59.4023)
  DOI : 10.1512/iumj.2010.59.4023
• Harder-Narasimhan categories
  
  • Chen Huayi
  Journal of Pure and Applied Algebra, Elsevier, 2010, 214 (2), pp.187-200. We propose a generalization of Quillen's exact category --- arithmetic exact category and we discuss conditions on such categories under which one can establish the notion of Harder-Narasimhan filtrations and Harder-Narsimhan polygons. Furthermore, we show the functoriality of Harder-Narasimhan filtrations (indexed by $\mathbb R$), which can not be stated in the classical setting of Harder and Narasimhan's formalism. (10.1016/j.jpaa.2009.05.009)
  DOI : 10.1016/j.jpaa.2009.05.009
• Quelques souvenirs sur I. M. Gelfand (1913-2009).
  
  • Guichardet Alain
  Gazette des Mathématiciens, Société Mathématique de France, 2010, 123, pp.95-100.
• Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes
  
  • Bernard Etienne
  • Caglioti Emanuele
  • Golse François
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2010, 42 (5), pp.2082-2113. Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed 1. Assuming that the holes are absorbing --- i.e. that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distribution of free-path lengths for the periodic Lorentz gas. In particular, it is proved that the total mass of the particle system at time t decays exponentially fast as t tends to infinity. This is at variance with the collisionless case discussed in [Caglioti, E., Golse, F., Commun. Math. Phys. 236 (2003), pp. 199--221], where the total mass decays as Const./t as the time variable t tends to infinity.
• Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms
  
  • Sorrentino Alfonso
  • Viterbo Claude
  Geometry and Topology, Mathematical Sciences Publishers, 2010, pp.2383-2403. In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at $0$ of Mather's $\beta$ function, thus providing a negative answer to a question asked by K. Siburg in \cite{Siburg1998}. However, we show that equality holds if one considers the asymptotic distance defined in \cite{Viterbo1992}.
• The ℓ-primary torsion conjecture for abelian surfaces with real multiplication
  
  • Cadoret Anna
  , 2012, B32, pp.195-204.
• Damped-driven KdV and effective equation for long-time behaviour of its solutions
  
  • Kuksin Sergei
  Geometric And Functional Analysis, Springer Verlag, 2010, 20 (6), pp.1431-1463. For the damped-driven KdV equation $$ \dot u-\nu{u_{xx}}+u_{xxx}-6uu_x=\sqrt\nu \eta(t,x), x\in S^1, \int u dx\equiv \int\eta dx\equiv0, $$ with $0<\nu\le1$ and smooth in $x$ white in $t$ random force $\eta$, we study the limiting long-time behaviour of the KdV integrals of motions $(I_1,I_2,...)$, evaluated along a solution $u^\nu(t,x)$, as $\nu\to0$. We prove that %if $u=u^\nu(t,x)$ is a solution of the equation above, for $0\le\tau:= \nu t \lesssim1$ the vector $ I^\nu(\tau)=(I_1(u^\nu(\tau,\cdot)),I_2(u^\nu(\tau,\cdot)),...), $ converges in distribution to a limiting process $I^0(\tau)=(I^0_1,I^0_2,...)$. The $j$-th component $I_j^0$ equals $\12(v_j(\tau)^2+v_{-j}(\tau)^2)$, where $v(\tau)=(v_1(\tau), v_{-1}(\tau),v_2(\tau),...)$ is the vector of Fourier coefficients of a solution of an {\it effective equation} for the dam-ped-driven KdV. This new equation is a quasilinear stochastic heat equation with a non-local nonlinearity, written in the Fourier coefficients. It is well posed. (10.1007/s00039-010-0103-6)
  DOI : 10.1007/s00039-010-0103-6
• On subsets of $\mathbb(F)_q^n$ containing no k-term progressions
  
  • Lin Yuncheng
  • Wolf Julia
  European Journal of Combinatorics, Elsevier, 2010, 31 (5), pp.1398-1403.
• Représentations potentiellement triangulines de dimension 2
  
  • Berger Laurent
  • Chenevier Gaëtan
  Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, 2010, 22 (3), pp.557--574. The two main results of this note are on the one hand that if V is a 2-dimensional potentially trianguline representation of G_Qp then V satisfies at least one of the following properties (1) V is split trianguline (2) V is a direct sum of characters or an induced representation (3) V is a twist of a de Rham representation, and on the other hand that there exists some 2-dimensional representations of G_Qp which are not potentially trianguline.