Entropie des mesures semi-classiques en dimension $2$

Rivière, Gabriel

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 9, 13 p.

On étudie les propriétés asymptotiques des fonctions propres du laplacien sur des surfaces riemanniennes compactes et lisses de type Anosov (par exemple à courbure strictement négative). Précisément, on répond à une question d’Anantharaman et Nonnenmacher [4] en montrant que l’entropie de Kolmogorov-Sinai d’une mesure semi-classique $\mu$ pour le flot géodésique $g^t$ est bornée inférieurement par la moitié de la borne de Ruelle.

[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] 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

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

[8] 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

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

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

[11] 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

[12] B. Gutkin Entropic bounds on semiclassical measures for quantized one-dimensional maps, arXiv :0802.3400 (2008). | MR

[13] 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

[14] D. Kelmer Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus, to appear in Ann. of Math. | MR

[15] 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

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

[17] G. Rivière Entropy of semiclassical measures in dimension 2, Duke Math. Jour. (à paraître), hal-00315799.

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

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

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

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

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

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

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