Publications

2010

• Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy
  
  • Burguet David
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2010, 26 (3), pp.873-899. For any integer $r\geq2$ and any real $\epsilon>0$, we construct an explicit example of $\mathcal{C}^r$ interval map $f$ with symbolic extension entropy $h_{sex}(f)\geq\frac{r}{r-1}\log\|f'\|_{\infty}-\epsilon$ and $\|f'\|_{\infty}\geq 2$. T.Downarawicz and A.Maass \cite{Dow} proved that for $\mathcal{C}^r$ interval maps with $r>1$, the symbolic extension entropy was bounded above by $\frac{r}{r-1}\log\|f'\|_{\infty}$. So our example prove this bound is sharp. Similar examples had been already built by T.Downarowicz and S.Newhouse for diffeomorphisms in higher dimension by using generic arguments on homoclinic tangencies. (10.3934/dcds.2010.26.873)
  DOI : 10.3934/dcds.2010.26.873
• Vey theorem in infinite dimensions and its application to KdV
  
  • Kuksin Sergei
  • Perelman Galina
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2010, 27 (1), pp.1-24. We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$, $(F-id)$ is a $\kappa$-smoothing map ($\kappa\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps $F_j^\prime$ such that $F_j-F_j^\prime=O(|u|^2)$ and each $\frac12|F'_j|^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $F^\prime: H\to H$, the germ $(F^\prime-id)$ is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector $(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with $\kappa=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form 'identity plus a 1-smoothing analytic map'. (10.3934/dcds.2010.27.1)
  DOI : 10.3934/dcds.2010.27.1
• Représentations de GL2(Qp) et (φ,Γ)-modules
  
  • Colmez Pierre
  Asterisque, Société Mathématique de France, 2010, 330, pp.281-509.
• The minimum number of monochromatic 4-term progressions in $\mathbb(Z)_p$
  
  • Wolf Julia
  Journal of Combinatorics, International Press, 2010, 1 (1), pp.53-68.
• Bohmian measures and their classical limit.
  
  • Markowich Peter
  • Paul Thierry
  • Sparber Christof
  Journal of Functional Analysis, Elsevier, 2010, 259 (6), pp.1542-1576. We consider a class of phase space measures, which naturally arise in the Bohmian interpretation of quantum mechanics. We study the classical limit of these so-called Bohmian measures, in dependence on the scale of oscillations and concentrations of the sequence of wave functions under consideration. The obtained results are consequently compared to those derived via semi-classical Wigner measures. To this end, we shall also give a connection to the theory of Young measures and prove several new results on Wigner measures themselves. Our analysis gives new insight on oscillation and concentration effects in the semi-classical regime. (10.1016/j.jfa.2010.05.013)
  DOI : 10.1016/j.jfa.2010.05.013
• 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
• KAM for the Non-Linear Schrödinger Equation
  
  • Eliasson Hakan
  • Kuksin Sergei
  Annals of Mathematics, Princeton University, Department of Mathematics, 2010, 172 (1), pp.371-435. We consider the $d$-dimensional nonlinear Schrödinger equation under periodic boundary conditions: $-i\dot u=-\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $ where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is an analytic function with $\hat V$ real, and $F$ is a real analytic function in $\Re u$, $\Im u$ and $x$. (This equation is a popular model for the 'real' NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For $\ep=0$ the equation is linear and has time--quasi-periodic solutions $u$, $$ u(t,x)=\sum_{a\in Å}\hat u(a)e^{i(|a|^2+\hat V(a))t}e^{i\sc{a,x}} \quad (|\hat u(a)|>0), $$ where $Å$ is any finite subset of $\Z^d$. We shall treat $\omega_a=|a|^2+\hat V(a)$, $a\inÅ$, as free parameters in some domain $U\subset\R^{Å}$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: If $|\ep|$ is sufficiently small, then there is a large subset $U'$ of $U$ such that for all $\omega\in U'$ the solution $u$ persists as a time--quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients. (10.4007/annals.2010.172.371)
  DOI : 10.4007/annals.2010.172.371
• Compatible structures on Lie algebroids and Monge-Ampére operators
  
  • Kosmann-Schwarzbach Yvette
  • Rubtsov Vladimir
  Acta Applicandae Mathematicae, Springer Verlag, 2010, 109 (1), pp.101-135. We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between such composite structures. We then show that the non-degenerate Monge-Ampére structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampére operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold. (10.1007/s10440-009-9444-2)
  DOI : 10.1007/s10440-009-9444-2
• 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.
• 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.
• 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
• 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
• 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
• 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.
• Note on torsion conjecture
  
  • Cadoret Anna
  • Tamagawa Akio
  Séminaires et congrès, Société mathématique de France, 2013, 27, pp.57-68.
• 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.