Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
[Demi-délocalisation des fonctions propres du laplacien sur une variété d’Anosov]
Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2465-2523.

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.

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.

DOI : 10.5802/aif.2340
Classification : 81Q50, 35Q40, 35P20, 37D40, 58J40, 28D20
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

1 tabacckludge ’Ecole Normale Supérieure Unité de Mathématiques Pures et Appliquées 6, allée d’Italie 69364 LYON Cedex 07 (France)
2 CEA/DSM/PhT Service de Physique Théorique Unité de recherche associé CNRS CEA/Saclay 91191 Gif-sur-Yvette (France)
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Anantharaman, Nalini; Nonnenmacher, Stéphane. Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2465-2523. doi : 10.5802/aif.2340. http://www.numdam.org/articles/10.5802/aif.2340/

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