Allocataire de recherche-Moniteur
Centre de Mathématiques Laurent Schwartz
Ecole polytechnique
91128 Palaiseau Cédex
France
Tel:+33169334930
E-mail: gabriel.riviere(the symbol)math.polytechnique.fr
- Research:
- Articles:
- Entropy of semiclassical measures in dimension 2 (pdf, accepted at Duke Mathematical Journal). Abstract:
In this article, I study the high-energy asymptotic properties of
eigenfunctions of the Laplacian in the case of a compact Riemannian
surface of Anosov type. To do this, I consider families of
distributions associated to them on the cotangent bundle and I derive
entropic properties on their accumulation points in the high-energy
limit (the so-called semiclassical measures). Precisely, I show that the
Kolmogorov-Sinai entropy of a semiclassical measure with respect to the
geodesic flow is bounded from below by half of the Ruelle upper bound. This result answers a question of
Anantharaman and Nonnenmacher (see here) in the case of an unique variable positive Lyapunov exponent.
- Entropy of semiclassical measures for nonpositively curved surfaces (pdf, submitted). Abstract: In this article, I show that the method of the article `Entropy of semiclassical measures in dimension 2' allows to get the same bound on the entropy of semiclassical measures for nonpositively cured surfaces. Compared with the Anosov case, the Liouville measure is not a priori ergodic for the geodesic flow (even if the genus is larger than 1). In particular, this result forbids that the eigenfunctions of the Laplacian only concentrate on a closed unstable geodesic in the large eigenvalue limit but it does not forbid that they concentrate on closed stable geodesics.
- Entropy of quantum limits for symplectic linear maps of the multidimensional torus (pdf, submitted). Abstract: In the case of a linear symplectic matrice A acting on the 2d-torus, semiclassical measures are A-invariant probability measures associated to sequences of high energy quantum states. The main result of this article is an explicit lower bound on the entropy of any semiclassical measure of a given quantizable matrix A in Sp(2d,Z). In particular, the result implies that if A has an eigenvalue outside the unit circle, then a semiclassical measure cannot be carried by a closed orbit of A.
- Notes:
- Entropy of semiclassical measures for quantized cat-maps
(pdf):
In this note, I give a simplified proof of Anantharaman-Nonnenmacher's
result in the case of linear hyperbolic symplectomorphisms of the
2-torus. In this case, the proof only uses results on the propagation
of coherent states (Bonechi, de Bièvre, Faure, Nonnenmacher). This
proof can be adapted in higher dimension and this extension is the
object of the article `Entropy of quantum limits for symplectic linear
maps of the multidimensional torus'.
- Talk at the séminaire X-EDP (pdf): Text (in french) for Actes du séminaire X-EDP. Talk on the entropy of semiclassical measures in dimension 2 (15/12/2009).
- Thesis (pdf): I made a PhD thesis at the CMLS under the supervision of Nalini Anantharaman. I defended my thesis on the 25 of november 2009
in Ecole polytechnique. During my PhD, I studied various properties of
semiclassical measures for chaotic dynamical systems. In the
manuscript, one can find some of notes and articles mentionned above.
There are also semiclassical large deviations results (chapters 3 and
5) included a work in preparation with Nalini Anantharaman on the
solutions of the Schrödinger equation (chapter 5).
- Teaching:
- I give courses to foreign students in Ecole polytechnique (EV2)
which are in their first year. These courses are
made for the students who follow the course `Eléments d'analyse et
d'algèbre' by Pierre Colmez (MAT331).
Topics of the course: finite groups representation, Lebesgue
integration, Fourier transform, holomorphic and meromorphic functions.
- Curriculum Vitæ (in french)