Semi-classical measures and entropy [after Nalini Anantharaman and Stéphane Nonnenmacher]

Séminaire Bourbaki - Volume 2006/2007 - Exposés 967-981, Astérisque, no. 317 (2008), Exposé no. 978, 22 p.

@incollection{AST_2008__317__393_0,
     author = {Colin de Verdi\`ere, Yves},
     title = {Semi-classical measures and entropy [after Nalini Anantharaman and St\'ephane Nonnenmacher]},
     booktitle = {S\'eminaire Bourbaki - Volume 2006/2007  - Expos\'es 967-981},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:978},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {317},
     year = {2008},
     zbl = {1155.37020},
     mrnumber = {2487740},
     language = {en},
     url = {archive.numdam.org/item/AST_2008__317__393_0/}
}

Colin de Verdière, Yves. Semi-classical measures and entropy [after Nalini Anantharaman and Stéphane Nonnenmacher], dans Séminaire Bourbaki - Volume 2006/2007  - Exposés 967-981, Astérisque, no. 317 (2008), Exposé no. 978, 22 p. http://archive.numdam.org/item/AST_2008__317__393_0/

Bibliographie

[1] N. Anantharaman - "Entropy and localization of eigenfunctions", to appear in Annals of Math., 2008. | Numdam | MR 2385191 | Zbl 1175.35036

[2] N. Anantharaman, H. Koch & S. Nonnenmacher - "Entropy of eigenfunctions", preprint arXiv:0704.1564, 2007. | Zbl 1175.81118

[4] V. M. Babič & V. F. Lazutkin - "The eigenfunctions which are concentrated near a closed geodesic", in Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems (Russian), Izdat. Leningrad. Univ., Leningrad, 1967, p. 15-25. | MR 234391 | Zbl 0164.13601

[5] M. Berger, P. Gauduchon & E. Mazet - Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer, 1971. | MR 282313 | Zbl 0223.53034

[6] F. Bonechi & S. De Bièvre - "Exponential mixing and $|l n ℏ|$ time scales in quantized hyperbolic maps on the torus", Comm. Math. Phys. 211 (2000), p. 659-686. | Article | MR 1773813 | Zbl 1053.81032

[7] F. Bonechi & S. De Bièvre, "Controlling strong scarring for quantized ergodic toral automorphisms", Duke Math. J. 117 (2003), p. 571-587. | Article | MR 1979054 | Zbl 1049.81028

[8] A. Bouzouina & S. De Bièvre - "Equipartition of the eigenfunctions of quantized ergodic maps on the torus", Comm. Math. Phys. 178 (1996), p. 83-105. | Article | MR 1387942 | Zbl 0876.58041

[9] A. Bouzouina & D. Robert - "Uniform semiclassical estimates for the propagation of quantum observables", Duke Math. J. 111 (2002), p. 223-252. | Article | MR 1882134 | Zbl 1069.35061

[10] M. Brin & G. Stuck - Introduction to dynamical Systems, Cambridge University Press, 2002. | Article | MR 1963683 | Zbl 1314.37002

[11] Y. Colin De Verdière - "Ergodicité et fonctions propres du laplacien", Comm. Math. Phys. 102 (1985), p. 497-502. | Article | MR 818831 | Zbl 0592.58050

[12] Y. Colin De Verdière & B. Parisse - "Équilibre instable en régime semiclassique. I. Concentration microlocale", Comm. Partial Differential Equations 19 (1994), p. 1535-1563. | Article | MR 1294470

[13] M. Dimassi & J. Sjöstrand - Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999. | MR 1735654 | Zbl 0926.35002

[14] L. Evans & M. Zworski - "Lectures on semiclassical analysis (v. 3)", , 2007.

[16] F. Faure & S. Nonnenmacher - "On the maximal scarring for quantum cat map eigenstates", Comm. Math. Phys. 245 (2004), p. 201-214. | Article | MR 2036373 | Zbl 1071.81044

[17] F. Faure, S. Nonnenmacher & S. De Bièvre - "Scarred eigenstates for quantum cat maps of minimal periods", Comm. Math. Phys. 239 (2003), p. 449-492. | Article | MR 2000926 | Zbl 1033.81024

[18] P. Gérard - "Microlocal defect measures", Comm. Partial Differential Equations 16 (1991), p. 1761-1794. | Article | MR 1135919 | Zbl 0770.35001

[19] P. Gérard & E. Leichtnam - "Ergodic properties of eigenfunctions for the Dirichlet problem", Duke Math. J. 71 (1993), p. 559-607. | MR 1233448 | Zbl 0788.35103

[20] E. J. Heller - "Wavepacket dynamics and quantum chaology", in Chaos et physique quantique (Les Houches, 1989), North-Holland, 1991, p. 547-664. | MR 1188425

[21] A. Katok & B. Hasselblatt - Introduction to the modem theory of dynamical Systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, 1995. | MR 1326374 | Zbl 0878.58020

[22] D. Kelmer - "Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus", preprint arXiv:math-ph/0510079v5. | Article | MR 2630057 | Zbl 1202.81076

[23] K. Kraus - "Complementary observables and uncertainty relations", Phys. Rev. D (3) 35 (1987), p. 3070-3075. | Article | MR 897714

[24] F. Ledrappier & E. Lindenstrauss - "On the projections of measures invariant under the geodesic flow", Int. Math. Res. Not. 9 (2003), p. 511-526. | Article | MR 1951544 | Zbl 1018.37003

[25] F. Ledrappier & L.-S. Young - "The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula", Ann. of Math. (2) 122 (1985), p. 509-539. | Article | MR 819556 | Zbl 0605.58028

[26] E. Lindenstrauss - "Invariant measures and arithmetic quantum unique ergodicity", Ann. of Math. (2) 163 (2006), p. 165-219. | Article | MR 2195133 | Zbl 1104.22015

[27] H. Maassen & J. B. M. Uffink - "Generalized entropic uncertainty relations", Phys. Rev. Lett. 60 (1988), p. 1103-1106. | Article | MR 932170

[28] R. Mañé - Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 8, Springer, 1987. | MR 889254 | Zbl 0616.28007

[29] S. W. Mcdonald & A. N. Kaufman - "Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories", Phys. Rev. Lett. 42 (1979), p. 1189-1191. | Article

[30] S. W. Mcdonald & A. N. Kaufman, "Wave chaos in the stadium: statistical properties of short-wave solutions of the Helmholtz equation", Phys. Rev. A (3) 37 (1988), p. 3067-3086. | Article | MR 937657

[31] J. V. Ralston - "On the construction of quasimodes associated with stable periodic orbits", Comm. Math. Phys. 51 (1976), p. 219-242. | Article | MR 426057 | Zbl 0333.35066

[32] Z. Rudnick & P. Sarnak - "The behaviour of eigenstates of arithmetic hyperbolic manifolds", Comm. Math. Phys. 161 (1994), p. 195-213. | Article | MR 1266075 | Zbl 0836.58043

[33] R. Schubert - "Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times", Comm. Math. Phys. 256 (2005), p. 239-254. | Article | MR 2134343 | Zbl 1067.81040

[34] A. I. Snirel'Man - "Ergodic properties of eigenfunctions", Uspehi Mat. Nauk 29 (1974), p. 181-182. | MR 402834 | Zbl 0324.58020

[35] L. Tartar - " $H$ -measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations", Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), p. 193-230. | Article | MR 1069518 | Zbl 0774.35008

[36] S. Zelditch - "Uniform distribution of eigenfunctions on compact hyperbolic surfaces", Duke Math. J. 55 (1987), p. 919-941. | MR 916129 | Zbl 0643.58029

[37] S. Zelditch & M. Zworski - "Ergodicity of eigenfunctions for ergodic billiards", Comm. Math. Phys. 175 (1996), p. 673-682. | Article | Zbl 0840.58048