Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, p. 2465-2523
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.
DOI : https://doi.org/10.5802/aif.2340
Classification:  81Q50,  35Q40,  35P20,  37D40,  58J40,  28D20
Keywords: Quantum chaos, semiclassical measure, ergodic theory, entropy, Anosov flows
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     pages = {2465-2523},
     doi = {10.5802/aif.2340},
     zbl = {1145.81033},
     mrnumber = {2394549},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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/item/AIF_2007__57_7_2465_0/

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