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 [] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the Journées EDP (Port d’Albret-June, 7-11 2010))
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 pour le flot géodésique est bornée inférieurement par la moitié de la borne de Ruelle.
We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of -damped trajectories of the geodesic flow.
The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip...
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