Entropy of eigenfunctions of the Laplacian in dimension 2

Rivière, Gabriel

Journées équations aux dérivées partielles (2010), article no. 15, 17 p.

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))

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

[2] N. Anantharaman Entropy and the localization of eigenfunctions, Ann. of Math. $\mathbf{168}$, 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 $\mathbf{57}$, 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 $\mathbf{21}$, 149-160 (1999) | MR | Zbl

[6] L. Barreira, Y. Pesin Lectures on Lyapunov exponents and smooth ergodic theory, Proc. of Symposia in Pure Math. $\mathbf{69}$, 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. $\mathbf{178}$, 83-105 (1996) | MR | Zbl

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

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

[10] Y. Colin de Verdière Ergodicité et fonctions propres du Laplacien, Comm. in Math. Phys. $\mathbf{102}$, 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. $\mathbf{239}$, 449-492 (2003) | MR | Zbl

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

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

[16] D. Kelmer Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus, Ann. of Math. $\mathbf{171}$ 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. $\mathbf{122}$, 509-539 (1985) | MR | Zbl

[18] H. Maassen, J.B. Uffink Generalized entropic uncertainty relations, Phys. Rev. Lett. $\mathbf{60}$, 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. $\mathbf{161}$, 195-213 (1994) | MR | Zbl

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

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

[24] A. Shnirelman Ergodic properties of eigenfunctions, Usp. Math. Nauk. $\mathbf{29}$, 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 $\mathbf{2}$, 109-124 (1983) | MR | Zbl

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