Mersenne banner

Livres, Actes et Séminaires du Centre Mersenne

  • Livres
  • Séminaires
  • Congrès
  • Tout
  • Auteur
  • Titre
  • Bibliographie
  • Plein texte
NOT
Entre et
  • Tout
  • Auteur
  • Titre
  • Date
  • Bibliographie
  • Mots-clés
  • Plein texte
  • Précédent
  • Journées équations aux dérivées partielles
  • Année 2010
  • article no. 15
Entropy of eigenfunctions of the Laplacian in dimension 2
Gabriel Rivière1
1 Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France
Journées équations aux dérivées partielles (2010), article no. 15, 17 p.
  • Résumé

We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure μ for the geodesic flow g t is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the 37 èmes Journées EDP (Port d’Albret-June, 7-11 2010))

  • Détail
  • Export
  • Comment citer
EuDML
DOI : 10.5802/jedp.72
Affiliations des auteurs :
Gabriel Rivière 1

1 Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France
  • BibTeX
  • RIS
  • EndNote
@incollection{JEDP_2010____A15_0,
     author = {Gabriel Rivi\`ere},
     title = {Entropy of eigenfunctions of the {Laplacian} in dimension 2},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {15},
     pages = {1--17},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2010},
     doi = {10.5802/jedp.72},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.72/}
}
TY  - JOUR
AU  - Gabriel Rivière
TI  - Entropy of eigenfunctions of the Laplacian in dimension 2
JO  - Journées équations aux dérivées partielles
PY  - 2010
SP  - 1
EP  - 17
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.72/
DO  - 10.5802/jedp.72
LA  - en
ID  - JEDP_2010____A15_0
ER  - 
%0 Journal Article
%A Gabriel Rivière
%T Entropy of eigenfunctions of the Laplacian in dimension 2
%J Journées équations aux dérivées partielles
%D 2010
%P 1-17
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.72/
%R 10.5802/jedp.72
%G en
%F JEDP_2010____A15_0
Gabriel Rivière. Entropy of eigenfunctions of the Laplacian in dimension 2. Journées équations aux dérivées partielles (2010), article  no. 15, 17 p. doi : 10.5802/jedp.72. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.72/
  • Bibliographie
  • Cité par

[1] L.M. Abramov On the entropy of a flow, Translations of AMS 49, 167-170 (1966) | Zbl

[2] N. Anantharaman Entropy and the localization of eigenfunctions, Ann. of Math. 168, 435-475 (2008) | MR | Zbl

[3] N. Anantharaman, H. Koch, S. Nonnenmacher Entropy of eigenfunctions, arXiv:0704.1564, International Congress of Mathematical Physics (2007) | Zbl

[4] N. Anantharaman, S. Nonnenmacher Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier 57, 2465-2523 (2007) | Numdam | MR | Zbl

[5] D. Bambusi, S. Graffi, T. Paul Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time, Asymp. Analysis 21, 149-160 (1999) | MR | Zbl

[6] L. Barreira, Y. Pesin Lectures on Lyapunov exponents and smooth ergodic theory, Proc. of Symposia in Pure Math. 69, 3-89 (2001) | MR | Zbl

[7] A. Bouzouina, S. de Bièvre Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm. in Math. Phys. 178, 83-105 (1996) | MR | Zbl

[8] A. Bouzouina, D. Robert Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. Jour. 111, 223-252 (2002) | MR | Zbl

[9] N. Burq Mesures semi-classiques et mesures de défaut (d’après P.Gérard, L.Tartar et al.) Astérisque 245, séminaire Bourbaki, 167-196 (1997) | Numdam | MR | Zbl

[10] Y. Colin de Verdière Ergodicité et fonctions propres du Laplacien, Comm. in Math. Phys. 102, 497-502 (1985) | MR | Zbl

[11] M. Denker, C. Grillenberger, K. Sigmund Ergodic Theory on Compact Spaces, Springer, Berlin-Heidelberg-New-York (1976) | MR | Zbl

[12] M. Dimassi, J. Sjöstrand Spectral Asymptotics in the Semiclassical Limit Cambridge University Press (1999) | MR | Zbl

[13] F. Faure, S. Nonnenmacher, S. de Bièvre Scarred eigenstates for quantum cat maps of minimal periods, Comm. in Math. Phys. 239, 449-492 (2003) | MR | Zbl

[14] B. Gutkin Entropic bounds on semiclassical measures for quantized one-dimensional maps, Comm. Math. Physics 294, 303-342 (2010) | MR

[15] B. Hasselblatt, A. B. Katok Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its applications 54 Cambridge University Press (1995) | MR | Zbl

[16] D. Kelmer Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus, Ann. of Math. 171 815-879 (2010) | MR

[17] F. Ledrappier, L.-S. Young The metric entropy of diffeomorphisms I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. 122, 509-539 (1985) | MR | Zbl

[18] H. Maassen, J.B. Uffink Generalized entropic uncertainty relations, Phys. Rev. Lett. 60, 1103-1106 (1988) | MR

[19] G. Rivière Entropy of semiclassical measures in dimension 2, to appear in Duke Math. Jour., hal-00315799 (2008)

[20] G. Rivière Entropy of semiclassical measures for nonpositively curved surfaces, hal-00430591 (2009)

[21] Z. Rudnick, P. Sarnak The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. in Math. Phys. 161, 195-213 (1994) | MR | Zbl

[22] D. Ruelle An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat. 9, 83-87 (1978) | MR | Zbl

[23] R. O. Ruggiero Dynamics and global geometry of manifolds without conjugate points, Ensaios Mate. 12, Soc. Bras. Mate. (2007) | MR | Zbl

[24] A. Shnirelman Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29, 181-182 (1974) | MR | Zbl

[25] P. Walters An introduction to ergodic theory, Springer-Verlag, Berlin, New York (1982) | MR | Zbl

[26] L.-S. Young Dimension, entropy and Lyapunov exponents, Ergodic theory and Dynamical systems 2, 109-124 (1983) | MR | Zbl

[27] S. Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour. 55, 919-941 (1987) | MR | Zbl

Cité par Sources :

Diffusé par : Publié par : Développé par :
  • Nous suivre