Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent
Gabriel Rivière
Entropy of eigenfunctions of the Laplacian in dimension 2
Journées équations aux dérivées partielles (2010), Exp. No. 15, 17 p., doi: 10.5802/jedp.72
Article PDF

Résumé - Abstract

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 $\mu $ 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 $37^{\text{èmes}}$ Journées EDP (Port d’Albret-June, 7-11 2010))

Bibliographie

[1] L.M. Abramov On the entropy of a flow, Translations of AMS $\mathbf{49}$, 167-170 (1966)  Zbl 0185.21803
[2] N. Anantharaman Entropy and the localization of eigenfunctions, Ann. of Math. $\mathbf{168}$, 435-475 (2008)  MR 2434883 |  Zbl 1175.35036
[3] N. Anantharaman, H. Koch, S. Nonnenmacher Entropy of eigenfunctions, arXiv:0704.1564, International Congress of Mathematical Physics (2007) arXiv |  Zbl 1175.81118
[4] N. Anantharaman, S. Nonnenmacher Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier $\mathbf{57}$, 2465-2523 (2007) Cedram |  MR 2394549 |  Zbl 1145.81033
[5] D. Bambusi, S. Graffi, T. Paul Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time, Asymp. Analysis $\mathbf{21}$, 149-160 (1999)  MR 1723551 |  Zbl 0934.35142
[6] L. Barreira, Y. Pesin Lectures on Lyapunov exponents and smooth ergodic theory, Proc. of Symposia in Pure Math. $\mathbf{69}$, 3-89 (2001)  MR 1858534 |  Zbl 0996.37001
[7] A. Bouzouina, S. de Bièvre Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm. in Math. Phys. $\mathbf{178}$, 83-105 (1996) Article |  MR 1387942 |  Zbl 0876.58041
[8] A. Bouzouina, D. Robert Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. Jour. $\mathbf{111}$, 223-252 (2002) Article |  MR 1882134 |  Zbl 1069.35061
[9] N. Burq Mesures semi-classiques et mesures de défaut (d’après P.Gérard, L.Tartar et al.) Astérisque $\mathbf{245}$, séminaire Bourbaki, 167-196 (1997) Numdam |  MR 1627111 |  Zbl 0954.35102
[10] Y. Colin de Verdière Ergodicité et fonctions propres du Laplacien, Comm. in Math. Phys. $\mathbf{102}$, 497-502 (1985) Article |  MR 818831 |  Zbl 0592.58050
[11] M. Denker, C. Grillenberger, K. Sigmund Ergodic Theory on Compact Spaces, Springer, Berlin-Heidelberg-New-York (1976)  MR 457675 |  Zbl 0328.28008
[12] M. Dimassi, J. Sjöstrand Spectral Asymptotics in the Semiclassical Limit Cambridge University Press (1999)  MR 1735654 |  Zbl 0926.35002
[13] F. Faure, S. Nonnenmacher, S. de Bièvre Scarred eigenstates for quantum cat maps of minimal periods, Comm. in Math. Phys. $\mathbf{239}$, 449-492 (2003)  MR 2000926 |  Zbl 1033.81024
[14] B. Gutkin Entropic bounds on semiclassical measures for quantized one-dimensional maps, Comm. Math. Physics $\mathbf{294}$, 303-342 (2010)  MR 2579457
[15] B. Hasselblatt, A. B. Katok Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its applications $\mathbf{54}$ Cambridge University Press (1995)  MR 1326374 |  Zbl 0878.58020
[16] D. Kelmer Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus, Ann. of Math. $\mathbf{171}$ 815-879 (2010)  MR 2630057 |  Zbl pre05712745
[17] F. Ledrappier, L.-S. Young The metric entropy of diffeomorphisms I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. $\mathbf{122}$, 509-539 (1985)  MR 819556 |  Zbl 0605.58028
[18] H. Maassen, J.B. Uffink Generalized entropic uncertainty relations, Phys. Rev. Lett. $\mathbf{60}$, 1103-1106 (1988)  MR 932170
[19] G. Rivière Entropy of semiclassical measures in dimension 2, to appear in Duke Math. Jour., hal-00315799 (2008) Article
[20] G. Rivière Entropy of semiclassical measures for nonpositively curved surfaces, hal-00430591 (2009) arXiv
[21] Z. Rudnick, P. Sarnak The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. in Math. Phys. $\mathbf{161}$, 195-213 (1994) Article |  MR 1266075 |  Zbl 0836.58043
[22] D. Ruelle An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat. $\mathbf{9}$, 83-87 (1978)  MR 516310 |  Zbl 0432.58013
[23] R. O. Ruggiero Dynamics and global geometry of manifolds without conjugate points, Ensaios Mate. $\mathbf{12}$, Soc. Bras. Mate. (2007)  MR 2304843 |  Zbl 1133.37009
[24] A. Shnirelman Ergodic properties of eigenfunctions, Usp. Math. Nauk. $\mathbf{29}$, 181-182 (1974)  MR 402834 |  Zbl 0324.58020
[25] P. Walters An introduction to ergodic theory, Springer-Verlag, Berlin, New York (1982)  MR 648108 |  Zbl 0475.28009
[26] L.-S. Young Dimension, entropy and Lyapunov exponents, Ergodic theory and Dynamical systems $\mathbf{2}$, 109-124 (1983)  MR 684248 |  Zbl 0523.58024
[27] S. Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour. $\mathbf{55}$, 919-941 (1987) Article |  MR 916129 |  Zbl 0643.58029
Copyright Cellule MathDoc 2018 | Crédit | Plan du site