We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.
Nous étudions la limite de haute énergie pour les fonctions propres du laplacien, sur une variété riemannienne compacte dont le flot géodésique est d’Anosov. La localisation d’une mesure semiclassique associée à une suite de fonctions propres peut être mesurée par son entropie de Kolmogorov-Sinai. Nous obtenons pour cette entropie une borne inférieure qui, dans le cas des variétés à courbure négative constante, vaut la moitié de l’entropie maximale. En ce sens, on peut dire que les fonctions propres de haute énergie sont au moins à demi délocalisées.
Keywords: Quantum chaos, semiclassical measure, ergodic theory, entropy, Anosov flows
Mot clés : chaos quantique, mesure semiclassique, théorie ergodique, entropie, flots d’Anosov
Anantharaman, Nalini 1; Nonnenmacher, Stéphane 2
@article{AIF_2007__57_7_2465_0, author = {Anantharaman, Nalini and Nonnenmacher, St\'ephane}, title = {Half-delocalization of eigenfunctions for the {Laplacian} on an {Anosov} manifold}, journal = {Annales de l'Institut Fourier}, pages = {2465--2523}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2340}, mrnumber = {2394549}, zbl = {1145.81033}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2340/} }
TY - JOUR AU - Anantharaman, Nalini AU - Nonnenmacher, Stéphane TI - Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold JO - Annales de l'Institut Fourier PY - 2007 SP - 2465 EP - 2523 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2340/ DO - 10.5802/aif.2340 LA - en ID - AIF_2007__57_7_2465_0 ER -
%0 Journal Article %A Anantharaman, Nalini %A Nonnenmacher, Stéphane %T Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold %J Annales de l'Institut Fourier %D 2007 %P 2465-2523 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2340/ %R 10.5802/aif.2340 %G en %F AIF_2007__57_7_2465_0
Anantharaman, Nalini; Nonnenmacher, Stéphane. Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2465-2523. doi : 10.5802/aif.2340. https://aif.centre-mersenne.org/articles/10.5802/aif.2340/
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